Generalized Coordinates

CARNEGIE INSTITUTE OF

TECHNOLOGY

LIBRARY

PRESENTED BY J. B. Nathanson

AN INTRODUCTION TO THE USE OF GENERALIZED COORDINATES IN MECHANICS AND PHYSICS

BY

WILLIAM ELWOOD BYERLY PERKINS PROFESSOR OF MATHEMATICS EMERITUS * IK HARVARD UNIVERSITY

GINN AND COMPANY CHICAGO BOSTON - NEW TORK DALLAS COLUMBUS * SAN FRANCISCO ATLANTA -

-

PREFACE This book was undertaken at the suggestion of my lamented colleague Professor Benjamin Osgood Peirce, and with the promise of his collaboration. His untimely death deprived me of his invaluable assistance while the second chapter of the work was still unfinished, and I have been obliged to complete my task without the aid of his remarkably wide and accurate knowledge of Mathematical Physics. The books to which I

am most

indebted in preparing this treatise on Natural Philosophy/' Watson and Burbury's " Generalized Coordinates," Clerk Maxwell's " Electricity and Magnetism," E. J. Routh's "Dynamics of a Rigid Body,' A. Gr. Webster's "Dynamics," andE. B. Wilson's "Advanced are

Thomson and

Tait's u Treatise

7

Calculus."

For their kindness in reading and criticizing my manuscript I am my friends Professor Arthur G-ordon Webster, Professor Percy Bridgman, and Professor Harvey Newton Davis. indebted to

W.

iii

E.

BYERLY

CONTENTS CHAPTER

I

INTRODUCTION ART. ART.

1-37

1.

Coordinates of a Point.

2.

Dynamics of a

Particle.

Number

of degrees of freedom. Free Motion. Differential equa-

tions of motion in rectangular coordinates. Definition of effective forces on a particle. Differential equations of motion in any sys-

tem

of coordinates obtained

from the

fact that in

any assumed

infinitesimal displacement of the particle the work done by the effective forces is equal to the work done by the impressed forces.

ART.

ART. 4. Dynamics of a Particle. ART. 5. Illustrative example in constrained motion. Examples. ART. 6 (a). The tractrix problem, (b). Particle in a rotating horizontal tube. The relation between the rectangular coordinates and the generalized coordinates may contain ART. 7. A System of Particles. the time explicitly. Examples. Effective forces on the system. Kinetic energy of the system. Coordinates of the system. Number of degrees of freedom. The ART. 8. A Sysgeometrical equations. Equations of motion. ART. 9. Rigid tem of Particles. Illustrative Examples. Examples. Bodies. Two-dimensional Motion. Formulas of Art. 7 hold good. ART. 10. Rigid Bodies. ThreeIllustrative Examples. Example. dimensional Motion, (a) Sphere rolling on a rough horizontal plane. billiard ball. Example, (c). The gyroscope, (d). Euler's (6). The ART. 11. Discussion of the importance of skillful equations. ART. 12. Nomenchoice of coordinates. Illustrative examples. clature. Generalized coordinates. Generalized velocities. Generalized momenta. Lagrangian expression for the kinetic energy. ART. 13. Lagrangian equations of motion. Generalized force. 3, Illustrative

examples.

Constrained Motion.

.

Summary of Chapter

I.

CHAPTER

II

THE HAMILTONIAN EQUATIONS. GRANGIAN EXPRESSION.

BOTJTH'S MODIFIED LAIGNOKATION OF COORDINATES 38-61 .

ART. 14. Hamiltonian Expression for the Kinetic Energy defined. Hamiltonian equations of motion deduced from the Lagrangian ART. 15. Illustrative examples of the employment of Equations. ART. 16. Discussion of problems solved Hamiltonian equations. in Article 15. Ignoring coordinates. Cyclic coordinates. Ignorable

problem in hydromechani

Tl

CONTENTS

vii

function in the two configurations and is independent of the paths by which the particles have moved from the first configuraART. 39. tion to the second. Definition of potential energy. Canonical The Lagrangian and the Hamiltonian Functions. forms of the L.agrangian and of the Hamiltonian equations of The modified Lagrangian function 3>. ART. 40. motion. Porms of i, .H", and 3> compared. Total energy ~E of a system ART. 41. When i, JET, or 4? moving under conservative forces. is given, the equations of motion follow at once. When X is given, the kinetic energy and the potential energy can be disor is given, the potential tinguished by inspection. When energy can be distinguished by inspection unless coordinates have been ignored, in which case it may be impossible to separate the terms representing potential energy from the terms ART. 42. contributed by kinetic energy. Illustrative example. Conservation of Energy a corollary of the Hamiltonian canonical ART. 43. Hamilton's Principle deduced from the equations. ART. 44. The Principle of Least Action Lagrangian equations. ART. 45. Brief disdeduced from the Lagrangian equations. ART. 46. cussion of the principles established in Articles 43-44.

H

<

ART. 47. Equations of motion of action. obtained in the projectile problem (a) directly from Hamilton^ principle, (6) directly from the principle of least action.

Another definition

ART.

48. Application of principle of least action to a couple ART. 49. Varying Action. Hamilton's of important problems. characteristic function and principal function.

CHAPTER V 98108

APPLICATION TO PHYSICS ART. 52. ARTS. 50-51. Concealed Bodies. Illustrative example, Problems in Physics. Coordinates are needed to fix the elecmagnetic state as well as the geometrical configuration ART. 53. Problem in electrical induction. ART. 54. Induced Currents. trical or

of the system.

APPENDIX APPENDIX

BODY 109-113

A. SYLLABUS. DYNAMICS OF A RIGID B.

THE CALCULUS OF VARIATIONS

.

.

.

114-118

EKRATA In equation (7), In equation (2),

P. 31

:

P. 40

:

P. 41

:

In

:

In equation (1),

P. 42 P. 50

:

line 10, r

In example

Pp. 77 and 79 P. 118

:

:

In line

2

2 <

1,

Ad

should be

should be



should be

pa

(5')

r<

Ad

<

2

should be apx

should be (5)

In heading, " Third " should be " Fluid 9, d

should be 8

GENERALIZED COORDINATES CHAPTER

I

INTRODUCTION 1.

may #, y,

may

Coordinates of a Point. The position of a moving particle be given at any time by giving its rectangular coordinates % referred to a set of rectangular axes fixed in space. It! be given equally well by giving the values of any threef

specified functions of #, y, and 2, if from the values in questior| the corresponding values of #, y, and z may be obtained uniquely^

These functions may be used as coordinates of the point, and the values of x, y, and z expressed explicitly in terms of them serve as formulas for transformation from the rectangular sys-

tem

new

Familiar examples are polar coordinates in a plane, and cylindrical and spherical coordinates in space, the formulas for transformation of coordinates being to the

system.

respectively

= r cos <, y = r sin " 2 = 2,

x '

=

r sin

(1) " ,

J

, T

It is clear that the

number

I J

= r cos 0, = y r sin 6 cos z = r sin V sin x

1

(2)

,

\~

(3)

<

of possible systems of coordinates if the point is unrestricted in

also clear that

is

unlimited.

its

motion, three coordinates are required to determine it. If it restricted to moving in a plane, since that plane may be taken

is

It

is

as one of the rectangular coordinate planes,

two coordinates

are

required.

The number

independent coordinates required to fix the position of a particle moving under any given conditions ia called the number of degrees of freedom of the particle, and isf of

l

INTRODUCTION

2

equal to the the point.

number

[ART. 2

of independent conditions required to fix

Obviously these coordinates must be numerous enough to the position without ambiguity and not so

numerous

fix

as to render

impossible to change any one at pleasure without changing any of the others and without violating the restrictions of the it

problem. 2.

a Particle.

of

Dynamics

The

Free Motion.

differential

equations for the motion of a particle under any forces we use rectangular coordinates are known to be

when

mx = Jf,

JT,

F,

and

Z, the

components

on the particle

of the actual forces

resolved parallel to the fixed rectangular axes, or rather their equivalents mx, my, mz, are called the effective forces on the particle. They are of course a set of forces mechanically equivalent to the actual forces acting on the particle. The equations of motion of the particle in

terms of any other

system of coordinates are easily obtained. Let j , #2 #8 , be the coordinates in question. ,

The

formulas for transformation of coordinates express and terms of

appropriate

x, y,

and

z in

.

22

y

2s) >

For the component velocity x we have dx

and

fix

.

dx

and z are explicit functions of q^ qz q^ q^ q^ and q^ and homogeneous in terms of q^ q^ and q s

x, y,

linear

,

.

* For time derivatives

we

.

shall write

x for

,

dt

we

shall use the

x for

-dt 2

Newtonian fluxion notation,

so that

CHAP.

FREE MOTION OF A PARTICLE

I]

We may d?^

if,

and

note in passing that

and

z

2

are

dx

Obviously ,

8

d dx

.

follows from this fact that

it

homogeneous quadratic functions


and since

8

=

2i

= c?x *q + .

;

of q v q^

dx ;

2^ tfx

1

.

q*

+

tfx

.

q

,

W

Let us find now an expression for the work q done by the when the coordinate ql is changed by an infinitesimal amount Sq: without changing q2 or j8 If Sa;, 8^, and Ss; are the in and x, y, z, obviously changes thus produced effective forces

.

Sq

W= m

+ y&y + z8z~] coordinates. We

\_xSx

if expressed in rectangular in terms of our coordinates express Sq

W

..

T Now

dx

^r^

d

/. ftiA

.

g^,

need, however, tg

q^ and ^3

d dx

^?i

but by (2) and (3)

Hence

..

x

^-- ._--.

x

and therefore where

and

is

T= the kinetic energy of the particle.

.

INTRODUCTION

4

[ART. a

To

get our differential equation we have only to write the second member of (4) equal to the work done by the actual

when q^ is changed by Sg^ we represent the work in question by

forces If

and

Q^Sq^ our equation is

we

get such an equation for every coordinate. It must be noted that usually equation (5) will contain q& and q s and their time derivatives as well as q^ and therefore canof course

not be solved without the aid of the other equations of the set. In any concrete problem, T must be expressed in terms of q^ and their time derivatives before we can form the expresq^, qs ,

work done by the effective forces. Q-fiq^ Q2 8q2 * work done by the actual forces, must be obtained

sion for the

Q$q^

the

from direct examination of the problem. 3. (a) As an example let us get the equations in polar coordinates for motion in a plane.

Here

x

= r cos 2

and

T

,

y

= r sin <.

=~

.

Sr if

JK is the

W = m\r

r<jF\

Sr

= ItSr

impressed force resolved along the radius vector.

~ cLt

CHAP. if

I]

<> is the

ILLUSTRATIVE EXAMPLES

impressed force resolved perpendicular to the radius

vector.

In more familiar form

m

In cylindrical coordinates where

x = r cos <,

Sr

or

5

y

W = m [r

= r sin <,

2 7-<

] Sr

z

=

=

INTRODUCTION C
In.

where

spherical coordinates

x =* r

cos

= r sin 9 cos $,

y

#,

= mr dr

^

Se TF=

*n [r

m

2

2

<9)

cos

i

-r

2

2

<9<^

sin 6 cos

2

m\df ~- r*dff\ --

r

r

sm

2

m 4. is

Dynamics

of

constrained to

>r sin

r

-

[^

<>,

sin

- r ((9 + sin

m ~ (r

/,

C7

= or

sin

L

=s ^.r 2 sin

-

= ^r

W=

= r sin

"

^ .

z

= mr,

dr

Sr

[ART.

a Particle.

2

.

.

/,

sm(9cos<9(^-

d

o 2

.

o

r sin-

^ <9

-^-

==

^ O,

=

Constrained Motion. If the particle surface, any two inde-

move on some given

pendent specified functions of its rectangular coordinates x, y, z, may be taken as its coordinates q1 and gp provided that by the equation of the given surface in rectangular coordinates and the equations formed by writing q and j2 equal to their values ,

CHAP.

CONSTRAINED MOTION OF A PARTICLE

I]

7

terms of x, y, and z the last-named coordinates may be uniquely obtained as explicit functions of ql and q2 For when this is done, the reasoning of Art. 2 will hold good. If the particle is constrained to move in a given path, any specified function of #, y, and z may be taken as its coordinate in

.

g1 provided that by the two rectangular equations of its path and the equation formed by writing q^ equal to its value in terms of x, y, and z the last-named coordinates may be uniquely obtained as explicit functions of qr For when this is done, the ,

reasoning of Art. 2 will hold good. 5.

(a) For example, let a particle of mass m, constrained to horizontal circle of radius a, be given an

move on a smooth

initial velocity V^ and let it be resisted by the air with a force proportional to the square of its velocity. Here we have one degree of freedom. Let us take as our

coordinate q1 the angle 6 which the particle has described abou*t the center of its path in the time t.

,

= mo,

,

and we have

2

d6

Our

differential equation is

which reduces to

6

dv

or

-7at

-\

m K

ad 2

= 0, *

+ m aO* = 0. .

A<>

Separating the variables,

dd

* --

,, --- H adt on ^2 ,

00

Integrating,

-~\

^

m

= A0.

at= C = ~~

=-

V

INTRODUCTION

8-

ma

1

ma

dt

= m

-j-

k Vat

,

KO,

of the

Vat

mV

dB

and the problem

-f- Jc

[ART. 5

P., .,

log 1

+ ,

-

k Vf\

-_. . ;

(1)

[_

motion

is

completely solved.

(5) If, however, we are interested in It, the pressure of the constraining curve, we must proceed somewhat differently. We

have only to replace the constraint by a force R directed toward the center of the path. There are now two degrees of freedom, and we shall take and the radius vector r as our coordinates

and form two

differential equations of motion.

dr

!fr

dt

m(r

To

these

Whence and

rfr} Sr

we may add 9H

JSSr.

r

= a.

&*

= 0,

It

= ma&.

ka

m

=

as before,

(2)

(3) \ s

(4)

CHAP.

I]

CONSTRAINED MOTION OF A PARTICLE

Let us now suppose that the constraining

9

circle is

rough. Here, since the friction is JUL (the coefficient of friction) multiplied by the normal pressure, R, will be needed, and we must (
replace the constraint

We

by

R

as before.

have now

m

-

= a. It = mat)*,

and

r

Whence

as before.

ma V _p

r

\_m Zyv

Replacing

-m

we have

tfOL

in Art. 5,

6

(),

(1),

by

= _!_ log [l-f (

--r m Jca

,

+ ^1.

\m

L

h

w /

(1)

&\

EXAMPLES 2 ^7 /9

Obtain

1.

,

n

the

familiar

equation ^

1

-

r

dt*

n

= + -sin# a

the

for

simple pendulum. 2. Find the tension of the string in the simple pendulum.

[/dO\ g cos 6 --a(

\dt/

3.

2 ~l -

\

J

Obtain the equations of the spherical pendulum in terms

of the spherical coordinates 6

Ans.

The

SUL0 cos

0<

and 2

<.

+ ^a sin = 0.

sin

2

6<j>

=

C.

constraint may not be so simple as that imposed the moving particle to remain on a given surface by compelling or on a given curve. 6.

(a)

INTRODUCTION

10

[ART. 6

Take, for example, the tractrix problem, when the particle moves on a smooth horizontal plane. Let a particle of mass m, attached to a string of length a, rest on a smooth horizontal plane. The string lies straight on the plane at the start, and then the end not attached to the particle is drawn with uniform velocity along a straight line perpendicular to the initial position of the string and lying in the plane. Let us take as our coordinates #, the distance traveled by that end of the string which is not attached to the particle, and 0, the angle made by the string with its initial position. Let It be the tension of the string and n the velocity with which the end of the string is drawn along. Let JT, Y", be the

rectangular coordinates of the particle, referred to the fixed line and to the initial position of the string as axes.

X

x

a sin0,

Y= a cos 8

X r=

x

1

a cos 00,

a

sin

00".

x

= m [x

-

.

W m cLt

~- Ca2

[x

a cos 001,

= m \j^d

a cos 0#]>

ma sn

a cos 00] Sx a cos 0#0

= It sin

a sin

CHAP.

I]

CONSTRAINED MOTION OF A PARTICLE

x = nt, Adding the condition and reducing, ma [cos 09 sin 66*~\ = It sin

1:

ft

ma*0=Q.

0=0.

= w
It

Integrating,

.

o>

ja

-

mn =i=

2

a

The particle revolves with uniform angular velocity aboui the moving center, and the pull on the string is constant. particle is at rest in a smooth horizontal tube. The (5)

A

then made to revolve in a horizontal plane with unif orn angular velocity to. Find the motion of the particle.

tube

is

Suggestion.

Take

our coordinates, and

the polar coordinates r, <, of the particle as let R be the pressure of the particle or

the tube.

3T

_

,

8T- =

mr2o>. ;

a<^

TO [r

Adding the condition and reducing,

r$

z

Sr

= o>V = 0,

f

2

= 0.

~\

cot>

INTRODUCTION

12

Hence J

we

tions,

each

= A cosh a>t-\-B sinh r= a and r = at the start. r = a cosh cat = a cosh $, = 2 ra#<0 2 sinh cot = 2 raaa> sinh r

Solving,

If

[ART. 6 cot,

2

.

and not in the can be solved more simply.

are interested only in the motions

problems (a) and (J) to use one less coordinate, 9 only

we were

only in (5), rectangular coordinates

in (a)

reac-

If in

and

r

Ir for the particle could given, and therefore could

JT,

,

be obtained whenever the time was or r and t. be expressed explicitly in terms of careful examination of Art. 2 will show that the reasoning is extended easily to such a case, and that the work done by the effective

A

forces

when

q, 21

only

changed &

is

--- -

-

still

is

Sq,. ai

It IS i

[dtdq^ dq\ to be noted, however, that when the rectangular coordinates are functions of t as well as of q^ q^ etc., the energy T is no longer

a homogeneous quadratic in q^ q^

Xz=nt

a'

etc.

a sin

T= n

#,

a cos

=

a sin

00.

r-

==

29 O

.

2

-r(

^

an cos ^-J|,

/

/TT

GU

m

m \a>0 L

an cos

= TnaTt sn

~ ^^ cos 0)

.

aw sin 00

= 0. =

,

as before.

= 0.

CHAP.

I]

CONSTRAINED MOTION OF A PARTICLE

13

BT

-

= morr.

or o>V] Sr

m[r r

= a cosh

= 0. as before.

cot,

EXAMPLES

A

1. particle rests on a smooth horizontal whirling table and attached by a string of length a to a point fixed in the table at a distance b from the center. The particle, the point, and the center are initially in the same straight line. The table is

is

then made to rotate with uniform angular velocity . Find the motion of the particle. Suggestion. Take as the single coordinate 6 the angle made by the string with the radius of the point. Let JT, Y", be the rectangular coordinates of the particle, referred to the line initially joining it with the center and to a perpendicular thereto through the center as axes.

Then and

X= 5 cos + a cos (9 cot

Y= b sin

a>t

-h

a sin (J9

-f-

a>),

+ cot).

+ 0y + 2 abaAcos 0Vco + 0)], and

9+

a

sin<9=:0;

and the relative motion on the table

is

simple

pendulum motion,

the length of the equivalent pendulum being

A particle is attracted

==^2

toward a fixed point in a horizontal with a force table proportional to the distance. It is whirling The table is then made to rotate the center. at at rest initially 2.

INTRODUCTION

14

with uniform angular velocity table

by the

[ART. 6

Find the path traced on the

&>.

particle.

Suggestion. Take as coordinates #, y, rectangular coordinates referred to the moving radius of the fixed point as axis of

abscissas

and to the center

the rectangular

with the

of the table as origin. Let JT, Y, be coordinates referred to fixed axes coinciding

initial positions of the

X= x cos

sin

y

cot

moving

axes.

Y= x sin

cot,

cot -f-

y

cos

a>t.

*

Whence come

If

o>

2

==:

1-

m

,

m [# 2 a>y m \jj -f 2 (ox

the solution

Multiplying (3) by

p (x

ofy\

2, re

2

a>t

+

4

o>

2

(1)

,

(2)

=

|~1 L

2 .

integrating,

V=2

= vers" 4

^

2

and

x = -v

#),

fj,y.

2 coy

o>

2

by

=

cao;

Substituting in (1),

<

:zf]

= aco y + 2 coi = 0. = 0. y+2 x + 4 # = ao)

Integrating (2),

Replacing 2

2

easy and interesting.

is

x

Whence

&>

a
l .

cos 2

ot~\ J9

TfS^-sin 4

x == - [1

cos 5],

2<

CHAP.

I]

MOTION OF A SYSTEM OF PARTICLES

and the curve traced on the table circle of radius

rolling

is

15

the cycloid generated by a

backward along the moving

axis of Y.

7. A System of Particles* If instead of a single particle we have a system of particles, free, or connected or otherwise constrained, mx, mi/, m^ are the effective forces on the particle -P. The effective forces on all the particles are spoken of as the effective forces on the system and are mechanically equivalent to the set of actual forces on the system.

T, the kinetic energy of the system, energies of all the particles.

If

SW is

the

work done by the

is

the

sum

effective forces in

of the kinetic

any supposed

infinitesimal displacement of the system,

If the particles of a

tions

may

moving system are subjected to connecor constraints, these connections or constraints may or not vary with the time. In the latter case a set of any n

such that when they and -, qn the connections and constraints are given, the position of every independent variables y1? j2

-

,

,

particle of the system is uniquely determined, and such that when the positions of all the particles of the system are given, * * follow uniquely, may be taken as coordinates of *z ? 3i 9a'

the system

;

and n

is

called the

number of degrees of freedom

of the system.

such that In the former case a set of variables q^ q^ -, yn the time are given, the position of every particle of the system is uniquely determined, and such that when the positions of all the particles of the system and the time are given, j 1? g2 qn follow uniquely, may be taken as the coordinates of the system and n 'is called the number of degrees of freedom of the system. ,

when they and

,

,

,

;

IKTEODUCTIOIvr

16

[ART. 7

The

equations expressing the connections and constraints in terms of the rectangular coordinates of the particles and of the -, qn , of the system are often called the the of geometrical equations system and may or may not contain the time explicitly. In the latter case the geometrical equations

coordinates j x

make

,

#2 ,

possible to express the coordinates x, y, z, of every point of the system explicitly as functions of the #'s; in the former case, as functions of t and the q's. it

The geometrical equations must not contain explicitly either the time derivatives of the rectangular coordinates of the particles or those of the coordinates q^ q^ qn of the system >

,

unless they can be freed from these derivatives by integration. Examples of geometrical equations not containing the time explicitly are the formulas for transformation of coordinates in and in Art. 6, (a). 3, and the equations for

X

Arts. 1 and

Y

Geometrical equations containing the time are the equations and Y in Art. 6, (V), and in Art. 6, Exs. 1 and 2. for The work, Bq W, done by the effective forces when q^ is changed by Sq^ without changing the other q's is proved to be

X

by reasoning

Here

x dx

and

is

2. For the sake of where the geometrical equations

similar to that used in Art.

variety we take the involve the time.

,

case

=/p, dx

-

&, q#

3x

.

an explicit function of

t,

dx

-,

yj. fix

.

q^ ja

-

-

,

-,

yn

,

.

q^ ja

-

.

qn

.,

,

dx

wr^: d dx

d*z

Px

.

d*x

.

ffx

.

.

CHAP.

MOTION OF A SYSTEM OF PARTICLES

I]

dx

,

and

..

x

= &x ^ d*x a^ ^ A

3

d

d

.

2

x

.

a ^

*

dx

=

d

/ /

-j-

x 3x\ .

1

if

Q1 ^q1

.

x

/ dx\ /

is

St

the

changed by Sqlt

W=-

If the

since x

is

is

^

-

^

-

4^

-4-

/. 9x\

Ix

dx

.

x

J

-

So.

'

work done by the d 8T 8T =

---* 8

result

=

)

and therefore and

2

8

?1

17

X^ N xo by (1) and (2) -t

^

>v

,

actual forces

when

q1 (

.

is

>

)

?l

geometrical equations do not contain the time, the same seen to hold, and in this case it is to be noted that

homogeneous

of the coordinates, that

of the first degree in the time derivatives is, in q ., jn , the kinetic energy l9

T

q^

a homogeneous quadratic in q^ q^ -, qn Generally every one of the n equations of which equation (3) is the type will contain all the h coordinates q^
.

-

,

nected by n simultaneous differential equations no one of which can ordinarily be solved by itself. If the forces exerted by the connections and constraints do no work, they will not appear in our differential equations. Should we care to investigate any of them, we have only to

suppose the constraints in question removed and the number of degrees of freedom correspondingly increased, and then to form replace the constraints by the forces they exert and to the full set of equations on the new hypothesis.

INTRODUCTION

18 8.

A

of Particles. Illustrative

[ART. 8

A

Examples. (cC) rough plank 16 feet long rests pointing downward on a smooth plane inclined at an angle of 30 to the horizon. A dog weighing as much as the plank runs down the plank just fast enough to keep it from slipping. What is his velocity when he reaches its lower end ? Here we have two degrees of freedom. Take x, the distance of the upper end of the plank from a fixed horizontal line in the plane, and y, the distance of the dog from the upper end

System

of the plank, as coordinates, and let exerted by the dog on the plank, and

R be the backward force m the weight of the dog.

m [2 x + y~\ Sap = 2 mg sin 30 Sx. 4- mg sin 30] Sy. m[z + y] Sy = [,ft

By and

hypothesis,

therefore

When y = 16,

y=

Since

y

=

32, nearly.

+ f2

,

CHAP.

I]

MOTION OF A SYSTEM OF PABTICLES

19

A weight 4 m is attached to a string which passes over fixed pulley. The other end of the string is fastened smooth a to a smooth pulley of weight m, over which passes a second __ string attached to weights m and 2 m. The system starts from rest. Find the motion of the weight 4m. Two coordinates, #, the distance of 4 m below the center of the fixed pulley, and y, the distance of 2 m below the center of the movable pulley, will suffice. The velocities are

.

A

,.

x lor 4 m,

x for movable pulley,

x

+y

x

for 2

[8

2m

y for m.

Tz= I [4 md?

?7^

r'n

Ttt,

a;

m [3 y

+ md? + 2 m (ij

Sa;

= [4 mg ==

2

+ m (x + y)

2

]

mg

[2 9??^ w^r] Sy. = 8 a; y 0.

3y

The weight 4 m

mg

x)*

= y.

will descend with uniform acceleration equal

to one twenty-third the acceleration of gravity. dumb-bell problem. Two equal particles, each of mass (
INTRODUCTION

[ART. 8

Find the

loving in any way on a smooth horizontal plane. ibsequent motion.

We have

three degrees of freedom. Let x, y, be the rectangucoordinates of the middle of the bar, and 6 the angle made y the bar with the axis of X. The rectangular coordinates of the two particles are ir

/ (

x

a cos

* 0,

a -

y

A\

(

andj

sin 6\

f

x

,

-h

a ~ cos

a

/i

6, y-*r-^ sin

*\ 0\

leir velocities are

-^[(x

+ 1 sin

00J+ (y

1

cos

id r

T= ^U

2

2

4-

2 1/

+x

2

+ ^.

2

4- y*

-4-

+ (a sin - a cos (9

2 <9

4-

-7T7-

(a cos

==

rt 2

mxx = 0, 2 my^y = 0, 2

*=0,

.

WiB,

(9)

(x

a sin 0)

+ y) J

(i

+ ^)

;

CHAP.

I]

MOTIOK OF A SYSTEM OF PAKTICLES

Hence the middle of the bar velocity, and the bar

uniform

velocity about

its

21

describes a straight line with rotates with uniform angular

moving middle

point.

EXAMPLE

Two

Alpine climbers are roped together. One slips over a Find their motion precipice, dragging the other after him. while falling. Ans. Their center of gravity describes a parabola. The rope rotates with uniform angular velocity about their moving center of gravity.

Two

equal particles are connected by a string which passes through a hole in a smooth horizontal -table. The first particle is set moving^on the table, at right angles with the string, with velocity ^J ag where a is the distance of the particle from the hole. The hanging particle is drawn a short distance (cT)

downward and then

released.

Find approximately the subse-

quent motion of the suspended particle. Let x be the distance of the second particle below its position of equilibrium at the time t, and 6 the angle described about the hole in the time

t

by the

first particle.

a- 2f)

2

2 <9

].

ft rr = 2 mx, .

ex

O rn

r

=

.

30

z + ta-x^d^Sx^mgSx,

(1 )

A [(

(2)

etc

-x)^] 80 = 0.

2x+(a-x)d* = ff

.

(a - xyd=C= a-Jag,

(8) (4)

INTRODUCTION

22

[ART. &

since (2) holds good while the hanging particle is being down as well as after it has been released.

O

2

a; -f

a

a

= 0,

drawn

approximately,

+ 2r^a ^ == 0.

and For small

oscillations of a simple

pendulum

of length

Z,

Therefore the suspended particle will execute small oscillaa. tions, the length of the equivalent simple pendulum being

EXAMPLE

A

a strong golf ball weighing one ounce and attached to " " The table. horizontal a on teed* is up large, smooth, string the from 10 feet in a hole the table, string is passed through

and fastened to a hundred-pound weight which rests on a prop just below the hole. The ball is then driven horizontally, at right angles with the string, with an initial velocity of a hundred feet a second, and the prop on which the weight rests is knocked away. (&) How high must the table be to prevent the weight from ball,

falling to the ground ? golf ball will acquire ?

(5)

What

is

Ans. (a) 8.96

the greatest velocity the ft.

(5) 963.4

ft.

per

sec.

Two-dimensional Motion. If the particles of a system are so connected that they form a rigid lody or a system of rigid bodies, the reasoning and formulas of Art. 7 9.

still

Rigid Bodies.

hold good.

CHAP.

I]

PLANE MOTION OF RIGID BODIES

23

(#) Let any rigid body containing a horizontal axis fixed in the body and fixed in space move under gravity. Suppose that

the body cannot slide along the axis. Then the motion is obviously rotational, and there is but one degree of freedom,

Take

as the single coordinate the angle 6 made by a plane containing the axis and the center of gravity of the body with a

through the axis. Let Ji be the distance of the center of gravity from the axis r and Jc the radius of gyration of the body about a horizontal axis through the center of gravity. Then

vertical plane

T = ~(h* + Jf) fr.

(v.

App. A,

5 and 10>

m Qf + 4 ) 880 = - mffh sin 0S0. s

and we have simple pendulum motion, the length alent simple

of the equiv-

pendulum being

Two

equal straight rods are connected by two equal a fastened to the ends of the rods, the whole a quadrilateral which is then suspended from a horiforming zontal axis through the middle of the upper rod. The system. Find the motion. is set moving in a vertical plane. Take as coordinates <, the inclination of the upper rod to (5)

strings of length

the horizon, and #, the angle made with the vertical by a line joining the point of suspension with the middle of the lower rod. From the nature of the connection the rods are always parallel.

Let k be the radius of gyration of each rod about of gravity.

its

center

INTRODUCTION

24

[ART. 9

(v-

0"+-

sin

App. A,

10)

= 0,

and the rods revolve with uniform angular velocity while the middle point of the lower rod is oscillating as if it were the bob of a simple pendulum of length a. an inclined plane is just rough enough to insure the (c?) If show that a thin hollow rolling of a homogeneous cylinder, of drum will roll and slip, the rate slipping at any instant being one half the linear velocity. Let x be the distance the axis of the cylinder has moved down the incline, 6 the angle through which the cylinder has of rotated, a the radius of -the cylinder, and a the inclination the plane.

Call the force of friction F. T:==

dT ox

&x

If there is

no

mx,

= \mg sin a

= aO, mx = mg sin

slipping,

x

cc

F~\ Sx,

CHAP.

PLANE MOTION OF HIGH) BODIES

I]

- = Fa. a TT

For a

= mg sin a '

Hence

-y^-

&2

solid cylinder,

=

,

F = ^ mg since R = 77?^ cos a, where

JK is the pressure

on the plane ;

F where

y^ is

1

the coefficient of friction.

= a, F = ?ft# sin #,

For a hollow drum,

k

-J

p

<

-

tan

and the drum will slip. For the drum, then,

F = ft-K = = TW^ sin a

/t*m<7

77737

=

cos -^

ce

= -|-

gt sin & TT .

d

= ^ g sin a^ = gt sin a<9

=x

^= o0

1 -=

O

where

^

is

7n^ sin

cr.

mg sin

or,

ct,

4-

r f

sin #,

m^ sin a = |^

ma u = Jia = 2/1

TTZ^

the rate of slipping.

^

ct.

mgt sin

= H* 4w

'

25

INTRODUCTION

26

[ART. 10

EXAMPLES 1.

A

2.

A

sphere rotating about a horizontal axis is placed on a perfectly rough horizontal plane and rolls along in a straight Show that after the start friction exerts no force. line.

plane.

sphere starting from rest moves down a rough inclined Find the motion, (a) What must the coefficient of

prevent slipping ?

friction be to is

Ans. (a) 3.

(5) If there

velocity ?

its

A

wedge

of

p

> | tan a.

mass

(6)

is

S = gt [sin

slipping,

a

what

| ^ cos

].

M having a smooth face and a perfectly

rough face, making with each other an angle a, is placed with its smooth face on a horizontal table, and a sphere of mass m and radius a is placed on the wedge and rolls down. Find the motion. Let x be the distance the wedge moves on the table, and y the distance the sphere rolls

Note that

down

T= ~ [Jf + m\ tf + ^K \ ^ f - 2 xy cos a\. 2i

Ans.

(m +

2i \_

M) x

geneous sphere

is

horizontal plane. x, y,

a,

my cos a =

0,

CL

^y

J

x cos a

Three-dimensional Motion,

10. Rigid Bodies.

Let

the plane.

^gf sin a.

(a)

A

homo-

any way on a perfectly rough Find the subsequent motion. set rolling in

be the coordinates of the center of the sphere

referred to a set of rectangular axes fixed in space; two of and Y, lie in the given plane. Let OA, which, the axes of

X

OB, OC, be rectangular axes fixed in the sphere and passing through its center; let OX, OY, OZ, be rectangular axes through the center of the sphere parallel to the axes fixed in space; and let 6,

,

d,

and

<jE>,

\fr

be the Euler's angles (v. App, A, 8). as our coordinates. The only force

^

F

Take

F

F, the friction, and we shall let x and components parallel to the axes OX, OY, respectively.

consider

is

x, y,

we have y

be

to its

CHAP.

MOTION OF RIGID BODIES IK SPACE

I]

= 6 sin ^ 4- sin 9 cos ty, = 6 cos sili sin ^, a>y

where

cox

<

-*//"

^=

<j>

4 ^.

cos

/yyj

-f- 4>

2

+

z

(6*

get

?ft

= Fx

wy =

2

+ cos 0-^] =

[0 TTiA

2

[0

as there

is

x

y

From

(1)

-F;,

(2)

=

aJ^ sin Q sin

i/r

+ sin 6(r~\ = aFx cos

i/r

no

>

a^ sin ^

i/r,

(4)

;

slipping,

y

(6)

<}>

if

(7)

n/r

(4) and (5),

ml? [sin

(<

-*//"

+ sin

4- cos 0-^)

cos

^ (6 + sin 0^^)J =

m/fc

cos

J^ sin

4-

= x a (j9 cos ^ 4- sin 6 sin ^r) == 0, ao)x = + a ( sin + ^ sin cos ^) = 0. aco

-\-

,

+ cos

~

8)

+ ^ + 2 cos 0^)].

2 <

^J

We

App. A,

(v.

T= f [i + / + k

Hence

and

27

2

[cos -^

-

(^

sin 9 sini/r (0 4- sin

4- cos 0-^)

^

sin

mJc* [cos i/r0

sin i/r0

first

members

of (3), sin

we

^6^

of (8)

(8)

#<^)]

= a^; sin Expanding the ing ^r by the aid

<9,

<9.

(9)

and (9) and eliminat-

get cos

sin

sin

sin

^<

4- sin

cos

- a.Fx i/r^] =

-\-

-f-

+ cos 9 cos sin 6 sin ^t/)^] = aF

cos ->/r^ 4- sin 6 cos

-^r^>

y

.

,

(10)

INTRODUCTION

28

But

the 3

and mk2

first -,

du

members

[ART. 10

of (10) arid (11) are obviously

respectively.

Hence, by

mk*

((3)

mk -~ 2

and (7),

*= ..

,

Substituting in (1) and (2),

mJc2

..

x

we

get

=

max,

=

may

a

mk*

x

whence

From

..

y

i

0,

(7) and (6),

dt

and from

~* - 0.

(3),

ci/TJ

Fx = 0,

Finally,

Hence the center of the sphere moves in a straight line with uniform velocity, and the sphere rotates with uniform angular velocity about an instantaneous axis which does not change its direction, and no friction is brought into play after the rolling begins. (5) The billiard

Suppose the horizontal table in (a) is imperfectly rough, coefficient of friction ^ and suppose the ball.

ball to slip.

Take the same coordinates

as before, and equations (1), (2),

(3), (4), (5), (8), (9), (10),

and (11)

still

hold good.

Let a

CHAP.

THE BILLIABD BALL

I]

29

be the angle the direction of the resultant friction, F = (img, makes with the axis of JT, and let 8 be the velocity of slipping, that is, the velocity with which the lowest point of the ball

moves along the

Of course the

table.

directions of

F and

&

are opposite.

Let

X and

Sx and Sy Y.

We

be the components of S parallel to the axes o have

S cos #=/%. =

aa>

y,

S sin a = Sy = y + aa>x

and

.

Fx s= pmg cos a F = pmg sin a.

y

and

y

dS^x =

tf

=

.

From

.

o,

>S

a

cos

da

cos a

a

dS- =

-

.

sin

a;

..

^

dS = -

a

..

dcox

^+ .

-2

^=

(1),

and from (10), TT Hence

sin

da--

~ ^ C os a.

--^ c&c

^ /Ssm
/c

-- cos^

==

^

M^ cos or,

,^

ON

(12)

and from (2) and (11), >S"

cos

a

tf -- sm ^ dS- = cP + /u# sm r^

<2or

.

.

--

dt

at

K

Multiplying (12) by sin a and (13) by cos

*-.

a,


^o-v (13)

and subtracting, (14)

Multiplying (12) by cos a and (13) by sin a, and adding,

_ <2

dt

s

INTRODUCTION

30

S=S - 2-^ pgt.

Integrating (15),

From

[ART. 10

(16)

Q

=

a

(14),

o?

,

direction of slipping does not change. has the direction the axes are so chosen that the axis of

and the If

X

opposite to the direction of slipping,

a=

Then

0.

These equations are familiar in the theory of projectiles, and the path traced on the table is a parabola so long as slipping lasts.

Should y

happen to be

zero, the

path degenerates into a

straight line.

When

slipping stops,

x

aa>

v

= 0,

and

and we have the case treated

y -f

acox

= 0,

in (#).

EXAMPLE

A

homogeneous sphere is set rolling on a perfectly rough inclined plane. Find the path traced 011 the plane. Ans.

A

parabola.

Suppose a rigid body containing a fixed point and having two of its moments of inertia about its prinObtain the differ* cipal axes through the fixed point equal. ential equations for its motion under gravity. We shall use Euler's angles with a vertical axis of Z. (#) TJie gyroscope.

We

have

^ = $ sin

-^ sin 6 cos <,

<j>

= cos -h ^ sin 9 sin <, = -^ cos d (v, App. A, T=* i I>K +-+

e



2

-f-

3

2

.

2 4ft) 2

2

2

<

8)

10)

CHAP.

THE GYEOSCOPE

I]

dT = _ co O

+

cos e

),

rri

= A sin2 6^r + <7cos <9(^ cos + <),

dT =

A sin

2

cos

C

equations are

{7

0T/r

Ou

Our

-

6"

sin

(-\lr

+ <j>y=

(^ cos

dfc-

where

or

TJrcos0

(1), is

-4^

0,

(1) 0,

(2)

= m^a sin 0.

<) -^

+ = a,

(3) (4)

<j>

^= S^w,

2

0^r

+ Cte cos 6=L. + Ca sin = ^^

^

- -

^4

or substituting

(cT)

-f-

the initial velocity about the axis of unequal moment.

A sin

/?/?/?^-

cos

(^ cos ^ + <)] =

cos

cos ^ 4-

From

31

cos 0-^

sin

^

2

(5) sin 0,

(6)

from (5),

(L v

Cacos0y(L

cos

Ca*)

^rinag

.

-

^+^asmft

_

(7)

Obtain Euler's equations for a rigid body containing a

fixed point.

T=

Here

[^

2

4- ^*>2

+ Ceo

2

(v.

].

(r.App. A,

=tfV /

= Aco ^

l

[6 cos

App. A,



+ ^r sin

sin

<

10) 8)

INTRODUCTION

32

Whence

FC

^-

(A

- ^) aywj

<

[ART. 11

- NSfr

where N" is the moment of the impressed forces about the

The remaining two Eulers equations follow this

by

11.

C axis.

at once

from

considerations of symmetry.

In Arts. 2 and 7

it

was shown that under

slight limi-

tations the coordinates of a

tem

moving particle or of a moving syscould be taken practically at pleasure, and the differential

equations of motion could be obtained by the application of a single formula. It does not follow, however, that when it comes to solving a concrete problem completely, the choice of coordinates is a matter of indifference. Different possible choices

lead to differential equations differing greatly in complication, and as a matter of fact in the illustrative problems of

may

the present chapter the coordinates hav-e been selected with care and judgment. That this care, while convenient, is not essential

may

be worth illustrating by a practical example, and

we

shall

consider the simple familiar case of a projectile in vacuo. Altogether the simplest coordinates are x and y, rectangular coordinates referred to a horizontal axis of axis of

We

Y through have

the point of projection.

T dT

mx =

my =

0,

mg.

X and

a vertical

CHAP.

I]

Solving,

THE CHOICE OF COORDINATES x = VB, X

33

=

Let us now try a perfectly crazy where

set of coordinates, q

and

and Proceeding in our regular way, we have

m sec* ,

4

-

_=__

_

,

tan

(1)

INTRODUCTION

34

[AilT. 11

(2)

Adding (1) and

(2),

+ # ) = 0. = 2 v^ q^ + q

-

(^

Whence

2

2

and

<7.

+-

j

=

Subtracting (2) from (1),

Han'

= - mg sec 4

Multiplying by

(^

#2 ), and integrating,

sec

Let sec

2

%-r =

% 9 tan

^/Vy

:

2,

-2,

t/

a^ja^tan-Hvj-^l.

2

CHAP.

NOMENCLATURE

I]

But from

9l

(3),

~t

q*

=

q

= vxt + tan""

35

vj,

2i

Hence

Of

We

tan

1

~

2

=y=v t~

The parameters q^ q^ position of our moving

the

9 |

answer, and a

first

does.

it

=x=

have

12.

$~

\vyt

course this should agree with our

moment's consideration shows that

fix

1

called generalized coordinates.

r-

that

,

as before.

we have been using

particle or

moving system

to

are

Following the analogy of rectan-

gular coordinates, the time derivative qk of any generalized coordinate qk is called the generalized component of velocity corresponding to qk . It may be a linear velocity, or an angular

many of our problems, or it may be much more complicated than either as in our latest example. The kinetic energy T expressed in terms of the generalized coordinates and the generalized velocities is called the Lagranvelocity as in

gian expression for the kinetic energy. If we are using rectangular coordinates and dealing with a

moving

particle,

dT and

is

the x component of the

momentum

'dT this analogy,

Following

ponent of the momentum coordinate qk

.

It

is

-

is

of the particle.

called the generalized com-

0ft of the system, corresponding to the

frequently represented by

pk and may ,

be a

INTRODUCTION

36

momentum, lems, or

it

[ART. IS

moment of momentum as in many may be much more complicated than or a

our latest example. Equations of the type d

dT

dT

dt dqk

dqk

what are

of our probeither as in

the Lagrangian equations of motion, although strictly speaking the regulation form of the are practically

called

Lagrangian equations is a little more compact and will be given later, in Chapter IV* Qky defined through the property that Qk qk is the work done by the actual forces when qk is changed by Sqk is called the It may be generalized component of force corresponding to qt a force, or the moment of a force as in many of our problems, or it may be much more complicated than either as in our ,

.

latest

13.

example.

Summary

number n

of Chapter I.

of degrees of

If a

freedom

moving system has a finite and n independent

(V* Art. 7)

-, generalized coordinates q^ q^ qn , are chosen, the kinetic can be in of the coordinates and terms energy expressed the generalized velocities q^ q^ qn and when so expressed will be a quadratic in the velocities, a homogeneous quad-

T

-

,

,

ratic if the geometrical equations (v. Art. 7)

do not contain the

time explicitly.

The work done by the effective forces in a hypothetical infinitesimal displacement of the system due to an infinitesimal change dqk in a single coordinate qk is \d_9T_

_9f\

dqk

dqk ^

\_dt

If this is written equal to

Qk Sqk

,

the

(

work done by the

actual

forces in the displacement in question, there will result the

Lagrangian equation

,

^

dt dqk

. **

dqk

CHAP.

SUMMARY

I]

The n equations

of

which

37

this is the type

form a

set of simul-

taneous differential equations of the second order, connecting the n generalized coordinates with the time. When the complete solution of this set of equations has been obtained, the problem of the motion of the system is solved completely* It must be kept in mind that in order to obtain the value of

a single coordinate or of a set of coordinates less in number than n it is generally necessary to form and to solve the complete set of n differential equations. shall see, however, in the next chapter, that in certain important classes of problems some of these equations need not

We

be formed, and that some of the coordinates can be safely ignored without interfering with our obtaining the values of the

remaining coordinates; that, indeed, we

may be

able to handle

satisfactorily some problems concerning moving systems having an infinite number of degrees of freedom.

CHAPTER

II

THE HAMILTONIAN EQUATIONS. KOUTIFS MODIFIED LAGRANGIAN EXPRESSION. IGNORATION OF COORDINATES The Hamiltonian Equations.

If the geometrical equations not do Art. contain the time explicitly system (v. 7) a' is therefore and the kinetic energy homogeneous quadratic in q^ q^ qn , the generalized component velocities, 14.

of the

T

,

Lagrange's equations can be replaced by a set known as the Hamiltonian equations. The Lagrangian expression for the kinetic energy we shall

now

represent by

= dl2, 1

Let

^? 1

T^

-

jp 2

= 3T-%

etc.

be the generalized component

/2

il

Then p^ p^ are homogeneous of the first degree in terms of p^p# in 19 J2 -, q^ q# -, Express q^ ?2 noting that they are homogeneous of the first degree in terms and substitute these values for them in T# which of jp 1? p 2 will thus become an explicit function of the momenta and the coordinates, homogeneous of the second degree in terms of the former. This function is called the Hamiltonian expression for the kinetic energy, and we shall represent it by Tp Of course momenta.

*

-

-

-

,

,

,

,

.

By

Euler's Theorem,

therefore

^=^"

2T.= 2Tp ^p^ +p

Let us try to get

dT -~ a

?l

and

dTdpi 38

2

q2

+

indirectly.

-

(2)

CHAP.

THE HAMILTONIAN EQUATIONS

II]

From

s

(1),

i

T^ =

or

But from

^

_

0j

+

2

(3)

we

get

0'

we have from

(1),

=

2

The Lagrangian

+

+p

Subtracting (6) from (7),

we

*

-

(7)

k

/

. fc

(9)

-

=. ^^T

have also

The equations

..

dqk

dT p k + <70v^ = Q '*

.

get

^ar^ar^ ^ dtdqk

becomes

.

reqriation %

We

.

9It

aj x

(2),

.

(2),

STE

From

J Jh. _^

_f

'+^1" -l-^

Subtracting (3) from (4),

Again,

J _2i

39

q of

dP*

which (9) and (10)

are the type are

(10)

known

as the Samityonian equations of motion. The so-called canonical form of the Hamiltonian equations is somewhat more compact

and

will be given later, in Chapter IV.

THE HAMILTONIAISr EQUATIONS

40

The 2w equations a,

of

which (9) and (10) are the type form

system of 2 n simultaneous

order, connecting the

differential equations of the first

qn and the n comwith the time, and in order to

n coordinates q^ q^

ponent momenta p^ p^

[ART, 15

-,

jp M

,

,

^

any one coordinate we must generally, as in the case the Lagrangian equations, form and make use of the whole

.solve for

of .set

of equations.

In concrete problems there is usually no advantage in using the Hamiltonian forms, but in many theoretical investigations they are of importance. It may be noted that in the process of forming Tp from T^ q : is expressed in terms of the jo's and j's, .and thus equation (10) is anticipated. To familiarize- the student with the actual

working of the

Hamiltonian forms, we shall apply them to a few problems which we have solved already by the Lagrange process. 15. (#) The equations of motion in a plane in terms of polar coordinates (v. Art. 3, (#)).

Here

Whence

=

#

dr

= ^r, m

_, mr*

(1)

OHAP.

II]

ILLUSTRATIVE EXAMPLES

41

(4)

Our Hamiltonian

equations are

(6)

jp*

If

we

eliminate

p r and

_p^,

= H>. we

(8)

get

md -._

our familiar equations. (b) Motion of a bead on a horizontal circular wire (v. Art. 5, (#)).

Here

r.

=

= Tp

2

~%?'

THE HAMILTONIAN EQUATIONS

42

Integrating,

va

h

= C=

5ma

Vkt

(
The

tractrix

2

5

-

+m Vk

(v. Art. 6, (#)).

problem

^ = - [^ +

Here

1

maV

= m-

A

u

[ART.

^-

2 a cos

ftc

^j

=m ^ p = m \_a?d JB^ e

Whence

x

=

a cos

a cos

-

^-5-5 ma sin u

ftr].

^+

cos 6p e ~\,

(1)

cos

We

A = ^ sin ^

get

cos

We


+ (!+ ^s

2

AP J - 0.

tf)

(4)

have the condition

x

With

(5),

= w.

(5)

px = m[n a cos ##'], p = ma \a6 n cos Q

Substituting in (4),

pe or

5

6~\.

we

get

mna sin 60 = ma?0

= 0.

0,

CHAP.

ILLUSTRATIVE EXAMPLES

II]

4=0 = -. a

Whence

px = mn (1 j5x

as in Art.

==

mw sin 66 =

it (v.

Here

-

cos 0).

2

sin 6

a

Art.

of the

two

particles

=

T,

e=

^ [2 + (a 2

*

Pi

j90

p

e

^

x)

+ - 1. m.\Pl L 2 ( x)*\

2

and

7712:,

m (#

=

= 0.

C= ma me?

Whence x

is

and the

8,

^=2

if

= R sin 6.

6.

(&) The problem hole in

48

small, as in Art. 8,

2i"

+

(cT).

= a

table with a

THE HAMILTONIAK EQUATIONS

44:

(V)

The gyroscope

(v. Art. 10, (
& + sin

We

^p

2

6ty )

2

+ C(& cos

C cos 0<>

= 0, & = 0, cos ^ - (1 + cos

get

-f-

cos

(9

<)

2

].

+ <),

(1)

j& 4

d .40 ==

(2) 2

0) ^jp J = mya sin 6.

--

-- ^ [(CV 3LV

or

2

= ^L sin 0^ +

=

[ART. 16

(L

4-

Ohr

-^^^

L^y cos -

<7Ztf (1 \

(Z cos < V3

cos 0)

A/r

sin

2

COS

-f-

0^ +

+ cos

Co?) ^

^

Cte

=

cos

2

0)] ;j .

(3)

= m^a sin
+ wga sin

-

.

,

0.

,

x

(4)

T,

=

as in Art. 10, (V).

The

two problems have a peculiarity that deserves closer examination. Let us consider Art. 15, (V). The kinetic energy in the Lagrangian form T^ and therefore* in the Hamiltonian form Tp fails to contain the coordinates and i/r. 16.

last

<

,

Moreover,

when

=

* Since

either 2

(

v Art. .

dg*

T&

it

fyk is missing also in

Tp

.

of

these

coordinates

is

varied,

14), it follows that if a coordinate is

the

missing in

CHAP.

IGNOKABLE COORDINATES

II]

45

impressed forces do no work. Hence two of our Hamiltonian equations assume the very simple forms

Pi

= 0,

fa

= 0,

which give immediately

= Ca, a constant, p^ = L, a constant. p^

and

These enable us to eliminate p$ and p^ from a third Hamiltonian equation (Art. 15, (V), (3)), which then contains only the third coordinate and its corresponding momentum p e This same result might have been obtained just as well by .

replacing p^ and p^ in Tp by their constant values and then forming the Hamiltonian equations for 6 in the regular way. So that if we are interested in 6 only, and Tp has once been

formed and simplified by the substitution of constants for p^ and ty need play no further part in and p^ the coordinates the solution. Should we care to get the values of these ignored coordinates, they can be found from the equations p$ = Ccc 9

p^

= L^

by the aid

In Art. 15,

(cT),

of the value of 6 previously determined. and p e ma vag, the substitution pe

since

=

=

p e in Tp enables us to solve the problem so concerned without paying further attention to 0. To generalize, it is easily seen that if the Lagrangian form, and therefore the Hamiltonian form, of the kinetic energy fails to contain some of the coordinates of a moving system,* and if the impressed forces are such that when any one of these of this value for

far as

x

is

work is done, the momenta p^ p^ are constant; and that after these coordinates to corresponding the momenta in question in the for these constants substituting

coordinates

is

varied no

Hamiltonian form of the kinetic energy, the coordinates corresponding to them may be ignored in forming and in solving the Hamiltonian equations for the remaining coordinates. * Coordinates that do not appear in the expression for the kinetic energy moving system are often called cyclic coordinates.

of a

MODIFIED LAGKANGIAISr EXPRESSION

46

[ART. 17

Unfortunately the ignored coordinates have to be used in

forming T^ the Lagrangian form of the kinetic energy, and in deducing from it Tp the Hamiltonian form of the energy. Not infrequently this preliminary labor may be abridged considerably by using a modified form of the Lagrangian expression for the kinetic energy of the system, as we shall proceed to show. ,

Routh's Modified Form of the Lagrangian Expression for the Kinetic Energy of a Moving System. In forming the Hamiltonian equations of motion (v. Art. 14) we first changed 17.

the

form of T^ by replacing all the generalized velocities ky their values in terms of the coordinates q^ q^ 2' *

'

q#

'

and the generalized momenta p^ JP 2

8*i = T~>

%

-,

jp 2 ,

dTp = 8 ^,

where

l

*i

, etc.

2

now

try the experiment of replacing in T^ one only of the velocities q^ by its value in terms of the corresponding

Let us

momentum p^ velocities y 2

Call

We

,

yg

the coordinates q^ ja ,

T thus changed in form, Tp Of T,l= TV aild ?1 = F <J>* & ?3' .

have

^ ^ + ^?2 ===

aT^ '

is called

for

^

3jx

T~i?,*-^=

Hence

- ft

=

Transposing, A Agam

and the remaining

-,

,

O

[T,,

i

-

' '

0-

?!' ?2'

.

_ p

course

-^-j^ a

^-^-' 3y,

,

.

N

.

l

- AjJ.

the Lagrangian expression for the kinetic energy modified

the coordinate

q^.

CHAP.

II]

Our

MODIFIED LAGKANGIAN EXPRESSION

47

Lagrangiaii equation


dt 3 j x

becomes

We

Pl

have also

o,1

a

?1

-

= Qf

= --^

(1)

.

C2~)

8pj

It

is

noteworthy that (1) and (2)

tonian

for

equations modified expression

M

q^

qi

from the Hamil-

differ

that

negative of the appears in place of the Hamiltonian

only

in

the

expression Tp Let us go on to the other coordinates. .

,

-~- 2t

whence

a^ -2 = ar,

,

whence

3s,

The

= ^^

%

2

=

P,

\TV

a

dq = -^

p. 18 ?2

-

3?

Lagrangiaii equation for q2

[T p L Pl

p,q~\

=

,?, ' l

2

is

therefore

from the ordinary form of the Lagrangian equation is replaced by the modified expression q^ In forming the modified expression it must be noted that q^ must be replaced by its value in terms of p g2 q^ y1? not only in T4 but in the term ^ 1 ^1 as well. y2 An advantage of the modified form is that when it has once been formed we can get by its aid Hamiltonian equations for one coordinate and Lagrangian equations for the others.

and

differs

M

only in that T^

l<9

,

,

j

,

,

MODIFIED LAGRANGIA3ST EXPBESSIOK

48

[ART. 13

reasoning just given can be extended easily to the case where we wish Hamiltonian equations for more than one coordinate and Lagrangian equations for the rest. The results may be formulated as follows: Let TPt ,p f ..; Pf be

The

the form assumed

by T$ when

their values in terms of

q. ..., jn

we have

if

Then,

.

l9

z,

,

-

pr9

,

qr

,

are replaced

jr+1 , jr + 2 ?

>

qn

->

by <2i>

if

equations of the type

*
and

CIC

if

p p

j l7 y a ,

Jc

>

^^7.

^^Jfc

r.

18. If

we modify

energy for

all

the Lagrangian expression for the kinetic

the coordinates,

and we get Hamiltonian equations

for all the coordinates,

and

as

of the

form

we have nowhere assumed

in

our

a homogeneous quadratic in the generalreasoning that T^ ized velocities, we can use these equations safely when the geometrical equations contain the time explicitly (V. Art. 7). If the geometrical equations do not involve the time, in which is

case T^

is

a homogeneous quadratic in q^ j2 2? =

,

,

ILLUSTRATIVE EXAMPLE

CHAP. II]

by Euler's Theorem and Jf ..., = and (2) assume the familiar forms ;

ffl ,

fftt

Tf

49

Tf =

2

Tp

;

and (1)

A + f|=&,

(3)

*

dT?

,A1

*"*'

important to note that the modified Lagrangian expression is not usually the kinetic energy of the system, qr although, as we shall see later, in some special problems it reduces to the kinetic energy. As we have just seen, when the time does not enter the geometrical equations, the completely modified Lagrangian expression (that is, the Lagrangian expression modiIt

M

is

Qv q 2

.

,

.,

fied for all the coordinates) 19.

As

is

the negative of the energy.

an illustration of the employment

of the

Hamiltonian

equations when the geometrical equations contain the time, let us take the tractrix problem of Art. 6, (#')

T=

Here and

is

6 3

2

[n L

2

+ a*0

not homogeneous in PQ

=-

-^

2

2 an cos

$.

= wi Yd Q

an cos

01,

^(9

A

and -,

r

= ~PQ am

.

j

rnn* \\ 2

n a

cos 2

sin B

n

.,*

+

L

-

arm\

.-,--i--

and we have

p&

^

= mn

2

sin 6 cos 6

2 ran sin 6 cos

x

fl)

a.

+

_-

sin 0p

;

-- sin dp = &

;

(2)

and (1) and (2) are our required Hamiltonian equations. Let as solve them.

MODIFIED LAGBANGIAN EXPRESSION

50

From

p Q = mcPQ

(1),

mna

cos

[ART. 20

#,

p = ma*d + mna sin 99.

whence

e

Substituting in (2),

ma

2

-f-

mn

mna sin 66

2

mna sin #$ -j- mn2 sin # cos 6 = 0,

cos

sin

= 0,

or

which agrees with the result

of Art. 6, (V).

EXAMPLES 1.

2.

Work Work

20.

form,

(#)

we

Art.

6,

(6*5,

Exs. 1 and

by the Hamiltonian method. Art.

2,

As an example

by the Hamiltonian method.

6,

of the

employment of the modified problem of Art. 6 and modify

shall take the tractrix

for the coordinate x.

We

have

(y. Art.

T-

6)

= ^ [ + a?8* - 2 a 2

^j

_px

= w[

cos

+

f = ^ [a

2

sin

2

2L

JE

^

2

-

=^

dM = ma GU

2

m?|

2

-m

2

sin ^^"

sin

a,

-

2

- a cos

cos

a cos

sin 6 cos 60

^,

+ a sin 66pv *

.

CHAP.

ILLUSTRATIVE EXAMPLES

II]

We

have for x the Hamiltonian equations

px x

and

51

= R sin 0,

(1)

= & 4- a cos 00, m

(2") ^ y

for # the Lagrangian equation

ma? [sin 2 00 +

cos

sin

2 <9<9

a cos 0px

]

= 0.

(3)

Of course

(1), (2), and (3) must be solved as simultaneous and we can simplify by the aid of the condition equations,

x Solving,

As

we

get

= nt. = 0,

(4)

a second example we shall take the problem of the table with a hole in it (v. Art. 8,

two particles and the and modify for 0.

We

have

T>

= ~ [2 x + (a - x) 2

=

whence rri

__

^*

" J

~

cy

2 <9

].

^ 2

f2

|

(1) jr 9

p 1

'2|

Our Hamiltonian

equations are (1) and

A-0.

(2)

MODIFIED

52

Our Lagrangian equation

m\2x+ m L Ey

P

y (3) v

.,1=gMg8

(#--#)

J

p 9 =C=ma ^Tag

whence (3) becomes 2

-

--^~- = #,

H

(

CL ~~~ CC

(4)

s

)

8, (d*).

As

a third example we shall take the Ex. 3, Art. 9, and modify for x. of problem (
[ART. 20

is

2

(2),

as in Art.

EXPRESSION

LAGRAJtfGIAJST

wedge and sphere

t

We

have

my cos a.

.

a2

2

Our Hamiltonian

y

2

jg

+ mp^y cos

o:

equations are (1) and

p.= Our Lagrangian equation

Q.

(2)

is

a\

cos ^ _.

..

y

g

J^

Bj

^,= ^=0;

(2),

whence (3) becomes

["+ a

L

as in Art. 9,

Ex.

3.

-

(4)

^^1 M+m y =^ J

sin

,

(5)

CHAP.

ILLUSTRATIVE EXAMPLES

II]

(cZ)

gram

We

53

As

a fourth example we shall take the flexible-paralleloproblem (V. Art. 9, (5)) and modify for .

have

T.

=~

V& + a #]. a

[2

_

**~

Our Hamiltonian

'

equations are (1) and

P*

Our Lagrangian

22

+

2

equation

= 0-

(2)

is

sin

or

^"

ff,

+ a sin = 0. (9

(3)

(4)

EXAMPLES 1.

Take

the dumb-bell problem of Art.

Ans.

T*> A

= m \z? 4- v

77^a;== 0.

<my

= 0.

2

-

8, (c),

and modify for

0.

IGNORATION OF COORDINATES

54 2.

Take

the dumb-bell problem of Art.

A.

r

Tff>Pv

2

\px 4 L

3.

and modify for

+P *0*\ m v+a J

\ma e= z

8, (
2

I

"1

0.

Take the gyroscope problem, Art.

10, (e),

and modify for

^

, 21.

We

[ART. 21

proceed to comment on the problems of the pre-

ceding section. in the (a) No one of them involves the time explicitly kinetic the and therefore energy T^ in geometrical equations, all

of

them

is

a

homogeneous quadratic

in

the generalized

velocities.

(J)

The momenta,

therefore, are

homogeneous

of the first

degree in the velocities, and consequently the eliminated velocities are homogeneous of the first degree in the corresponding the remaining velocities, and the energy 3^ v ^,... are homogeneous quadratics and the modified function q^ qy ... in the introduced momenta and the velocities not eliminated.

momenta and

M

CHAP. (
IGNOKATION OF COORDINATES

II]

In

all

55

the problems the coordinates for which

we have

modified the Lagrangian expression for the kinetic energy are In all of them except the first no work is done when cyclic. of the coordinates in question is varied. Consequently one any one of the Hamiltonian equations for that coordinate is of the form pk = 0, and the momentum p k = ck where ck is a constant. Therefore it is easy to express the energy Tp ^ p ^... and the ,

M

coordimodified expression q ^ q^... in terms of the remaining constants the and the nates, corresponding velocities, c^ 2 and when so expressed they are quadratics in the velocities but not necessarily homogeneous quadratics. When the modified function has been so expressed, it may be used in forming the
,

,

Lagrangian equations for the remaining coordinates precisely as the Lagrangian expression for the kinetic energy is used, and the coordinates that have been eliminated may be ignored in the rest of the

work

of solving the

problem unless we are

interested in their values (v. Art. 16). (j) To generalize : If some of the coordinates of a

moving sys-

are cyclic, and if the impressed forces are such that when any of these coordinates is varied no work is done, the momenta one corresponding to these coordinates are constant throughout the

tem

motion. The substitution of these constants for the momenta in the Lagrangian expression for the kinetic energy modified for the coordinates in question will reduce it to an explicit function of the remaining coordinates, the corresponding velocities, and the constants substituted, which will be a quadratic in the velocities

but not necessarily a homogeneous quadratic. When the modified function has been so expressed, it may be used in forming the Lagrangian equations for the remaining coordinates precisely as the Lagrangian expression for the kinetic energy is used, and the coordinates that have been eliminated may be ignored in the rest of the work of solving

the problem (v. Art. 16). In the important case where the system starts from rest, the constant momenta corresponding to the ignorable coordinates

IGNOHATION OF COORDINATES

56

[ART. 22

being zero at the start are zero throughout the motion, and the modified expression is identical with the Lagrangian expression for the kinetic energy, which therefore is a function of the

remaining coordinates and the corresponding velocities and is a homogeneous quadratic in the velocities (V. Art. 24, (#)) (e) The fact that the Lagrangian expression for the kinetic energy modified for ignorable coordinates is expressible in terms of the remaining coordinates and the corresponding velocities and is a quadratic in terms of those velocities is often of great

we

importance, as

shall see later.

Let us take the gyroscope problem of Art. 10, (


<

<

T/T

are ignorable.

Pt

= 0,

We

have

and

jfy

= 0,

so that

=

p$

fiT

^ = rl2 = C(^r cos

and


1?

+ <)=


t

= A sin 0^+C cos (^ cos 2

whence

and

_ C. v^ ^ ^ (7

<>i

coe

(?)

.4sin 2


cos g

p^

=


2

CHAP.

TOTAL IGNOBATION OF COORDINATES

II]

Forming the Lagrangian equation

~ A sin

for

in the usual way,

57

we

cos e

8

6

or

which

is

identical with (T), Art. 10, (V).

EXAMPLE

Work 23.

Both

the problem of Art. 20,

(5),

ignoring the coordinate

&.

of Art. 20, (d) 9 has an interesting peculiarity. coordinates are cyclic, and there is no term in the kinetic

The problem

energy that

is

linear in <, the velocity corresponding to the

the constant momentum ignorable coordinate . Consequently itself a constant. therefore is which of a constant is $, multiple p in the energy term the of and of also This is true mZ?^ p$<j> ma?0 2 by confrom then differ must T and which involves (^>.

M+

P4>

stants,

and

Lagrangian

as only the derivatives of M^ are used in forming the equation for 0, we have merely to disregard the term

mTffi* in the energy In cases like this

T

4

and use what

we

tribution to the energy as the coordinate itself,

is left

of T$ instead of M^.

are able practically to ignore the conmade by the ignored coordinate as well

and of course we can conclude that the motion we may thus disregard has no effect on the motion we are studying, but that the two can go on together without

interference.

EXAMPLE Examine Exs. 1 and

2,

Art. 20, from the point of view of

the present article. 24. (a)

The wedge and

to a very important class.

x

is

ignorable.

sphere problem of Art. 20, (V), belongs

Both x and y

are cyclic coordinates,

and

IGISTORATIOX OP COORDINATES

58

The momentum p x

is

constant,

and as

it

is

[ART. 2 initially

zero

it is zero throughout the (since the system starts from rest) kinetic the is aa energy fm and the motion, pjk. Consequently as they are both and are identical, modified J4

T

expression

homogeneous quadratics in y and p x and do not contain y, they 2 reduce to the form Li/ where L is a constant. Therefore our Lagrangiaii equation for y is Ly = rng sin a, and the sphere rolls down the wedge with constant acceleration. ,

we care only for the motion of the sphere on the wedge, we may then ignore x completely and yet know enough of the form of MX to get valuable information as to the required motion. Of course we know that the energy of the whole are able by system can be expressed in terms of y, and if we solve we can so to means it, completely for y withexpress any If

out using the ignored coordinate x at any stage of the process. of this complete ignoration of (5) As a striking: example coordinates, and of dealing with a moving system having an infinite number of degrees of freedom, let us take the motion

homogeneous sphere under gravity in an infinite incomand liquid being initially at rest. pressible liquid, both sphere of a

considerations of symmetry, the position of the sphere can be fixed by giving a single coordinate x, the distance of the center of the sphere below a fixed level, and x is clearly a

From

cyclic coordinate.

The positions of the particles of the liquid can be given in terms of x and a sufficiently large number (practically infinite} of coordinates q^ ja - -, in a great variety of ways. Assume ? that a set has been chosen such that all the j s are cyclic.* Then, since gravity does no work unless the position of the -

,

sphere of

is

varied, the #'s are all ignorable.

them p k =

starts ck = 0.

from

M

q ,g

0,

and the momentum p k

rest the initial value of ,.. .,

the energy

of the

= ck pk

,

That and

is

is,

for every

since the

zero,

one

system

and therefore

system modified for

all

* That this is possible will be shown later, in connection with the of Impulsive Forces (v. Chap. Ill, Art. 36).

the

treatment

CHAP.

SUMMARY

II]

59

then identical with the energy ^> ,p 2 ,... and must be expressible in terms of the remaining coordinate x and the corresponding velocity x (v. Art. 21, (tT)) ; and as x is cyclic the energy of the 2 system will then be of the form Lx where L is a constant. Forming the Lagrangian equation for x, we have is

a

,

Lx = mff, and we learn

the

that

descend with

sphere will

constant

acceleration.

Of course this brief solution is incomplete, as it gives no information as to the motion of the particles of the liquid, and since we do not know the value of Z, we do not learn the magnitude of the acceleration.

The energy

Still

of the

the solution

moving

is

interesting

liquid, calculated

and valuable. by the aid of

2 hydromechanics, proves to be Jw'# where m is one half the mass of the liquid displaced by the sphere (Lamb, Hydromechanics, Art. 91, (3)), and therefore the energy of the system is (m + m')x\ and this agrees with our result. f

,

25.

Summary

of

Chapter

The

II.

ing system which has n degrees

of

kinetic energy of a movfreedom can be expressed and the n general-, qn

n coordinates q^ q2 ized momenta p^ p^ pn9 and when so expressed it is a homogeneous quadratic in the momenta if the geometrical is called equations do not involve the time (v. Art. 14), and in terms of the

-

,

-

,

,

the Hamiltonian expression for the kinetic energy. If the Hamiltonian expression for the kinetic energy, and Lagrangian expression for the kinetic energy,

dTp = fy*

8T*-

%

,

and

9

dTp =

Tp

T

is

the

qk .

%>*

the effective forces in a hypothetical infinitesimal displacement of the system, due to an infinitesi-

The work done by

mal change Sqk this

in a single coordinate qk , is

be written equal to Qk qk

<>

the

I

jP*

+

-rr-

2 1

work done by the

S?*-

^

actual

THE HAMILTONIAST EQUATIONS

60

[ART. 25

forces In the displacement in question, there will result the differential equation of the first order,

The 2n equations

of which, this

and

given above, are the type are known as the Hamiltonian equations of motion for the system. P 2 9 3 ----- pAn ?x 9> ?r If, in the expression Ti are replaced by their values in terms of p lt p 2 r p gr+3L " ?' the result -^ ,* is the Lagrangian ? 2i> 9a modified for the coordinates expression for the kinetic energy For the coordinates for which the expression has , yr q

p^

>

,

>

,

,

,

>

-

,

-

-

.

been modified, that equations of the

is,

for

k
have Hamiltonian

type

For the remaining coordinates, that Lagrangian equations of the type

Whether we work from

is,

for

k

> r, we

the Lagrangian expression

T4

have

for the

T

kinetic energy, or the Hamiltonian expression p9 or the modiin general led to a set fied , we are expression 9v

Lagrangian

M

.

. .

whose number depends freedom in the moving all. system, and such that to solve one we must form and solve Art. are coordinates the of some 16), cyclic (v. If, however, and the impressed forces are such that when any one of them of simultaneous differential equations merely upon the number of degrees of

CHAP.

SUMMARY

II]

work

61

done, the corresponding momenta are constant throughout the motion, and these coordinates are ignorable in the sense that if constants are substituted for the varied no

is

corresponding tions for the

is

momenta

in

remaining

Tp

or

M

q

.,

coordinates in

the Hamiltonian equathe former case (or

the Lagrangian equations. in the latter case) can be formed and, if capable of solution, can be solved without forming the equations corresponding to the ignored coordinates. If the system starts from rest and there are ignorable coorthe Lagrangian expression modified for the dinates, ., q

M

ignorable coordinates, is identical with the kinetic energy of the system; and whether the system starts from rest or not,

M

is a quadratic, but not necessarily a homogeneous quadin the velocities corresponding to the coordinates which ratic, qi ,...

are not ignorable.

CHAPTER

III

IMPULSIVE FOKCES Moments.

If a hypothetical infinitesimal displaceto a system, the product of any force by the given distance its point of application is moved in the direction of

26. Virtual

ment

is

the force

is

called the virtual

of all the virtual

moments

is

moment

of the force,

called the virtual

and the sum

moment

of the

set of forces. If the forces are finite forces, the virtual

moment

is

the vir-

work that is, the work which would be done by the forces in the assumed displacement. If the forces are impulsive forces, the virtual moment is not virtual work but has an interpretation as virtual action^ which we shall give later when we take tual

up what

;

is

called the action of a

moving system.

27. For the motion of a particle under impulsive forces have the familiar equations

are called the effective impulsive forces on the particle mechanically equivalent to the actual forces. If the point is

is

the virtual

we

and are

given an infinitesimal displacement,

moment of the effective forces and moment of the actual forces.

equal to the virtual

62

of course is

CHAP.

COMPONENT OF IMPULSE

Ill]

-

63

If the generalized coordinates of a moving system acted on and a displacement caused by impulsive forces are q^ qz to the is system, by varying q l by S^ given ,

,

4- (

where

As

Sqi A represents the virtual

hi Art.

=

7,

-

Therefore

and

moment

-

of the effective forces.

-

dx fix mx ^ = mx -^.

.

m

^ -

<

Hence

I

^

I

- ^* I

I

^

(1)

(where -P^S^ is the virtual moment of the impressed impulsive forces and Pq is called the component of impulse corresponding to q^) is our Lagrangian equation, and of course we have one such equation for every coordinate qk Equation (1) can be written in the equivalent form .

28. Illustrative

Examples,

(a)

A lamina of

mass

m

rests

on

a smooth horizontal table and is acted on by an impulsive force of magnitude JP in the plane of the lamina. Find the initial

motion.

be the of gravity of the lamina, let the axis of by a perpendicular to the line

Let (X y) be the center

angle made with of action of the force, and from the center of gravity.

X

let

a be the distance of the force

IMPULSIVE FORCES

64

Then

T,

= ? [i + y + *"#].

[ART. 28

2

2

(v.

App. A,

10)

30

If the

axes are chosen so that #

=0, y =

0,

and #

== 0,

we have

mx =

Hence the

0,

initial velocities are

i-O,

m

velocities of a point on the axis of the center of gravity are

The from

The

point in question will have no

initial

X at

the distance b

velocity

if 5

= a

It follows that the lamina begins to rotate about an instantaneous center in the perpendicular from the center of gravity a2 -f-&2 from to the line of action of the force at a distance a

that line and situated on the same side of the line of the force center of gravity. This point is called the center of

as the

percussion.

ILLUSTRATIVE EXAMPLE

CHAP. Ill] (5)

A

wedge

perfectly rough

A

65

of angle a and mass M^ smooth above, rests on a horizontal plane.

below and

sphere of radius a and mass m is rotating with angular about a horizontal axis parallel to the edge of the velocity and is placed gently on the wedge. Find the initial wedge

H

motion (V. Art.

9, Ex. 3). as coordinates x, the distance of the edge of the wedge from a fixed axis parallel to it in the horizontal plane; y^ the distance of the point of contact of the sphere down the wedge ;

Take

and

#,

We

the angle through which the sphere has rotated.

have __

T.

+ m 3? + m , - 2~ cos a + ft-0 *A>N = M 3_ xy
.

t

2

.

.

,

7

2

both before and after the sphere is set down. After the sphere and ad = 0. Before it is set down, x = y = is set down, y 6 = li. Since the sphere cannot slip, it exerts an impulsive force P up the wedge, and an equal and opposite force P is exerted on

it

by the wedge

at the instant the

two bodies come

in contact.

We

have

px =

-

= (J^T+ m) x my cos a,

2

<m

fy

x COS

(

dy

Our

equations are

(Jf+m) i

my cos^ = 0,

m(y x cos a) = P, ?nF (0 - L) = oP, and we have

From

ad^y.

also

(2), (3),

(1)

(2} (3) (4)

and (4),

,^

(5>

IMPULSIVE FOBCES

66

From

[ART. 28

(1) and (5),

and the weight 4 ra be jerked down with a Take the coordinates x velocity v. Find the initial motion. be the magnitude of the jerk. and y as in Art. 8, (6), and let (
In Art.

8, (5), let

P

Our

y) = P,

m(8

equations are

= 0. 23 #z = 3 P, 23 wy = P. 3 P ^ = 2;=

m (3 y

and

Whence

But

ir)

23m'

y=\

and

v.

a (d) Four equal rods freely jointed together in the form of is struck at blow table. a horizontal on at rest are square one corner in the direction of one of the sides. Compare the initial velocities of the middle points of the four rods. be the mass and 2 a the length of a rod. Take a pair Let of rectangular axes in the table. Let (x, y) be the center of the

A

m

figure at

any time and

and

<j>

the angles

made by two

adjacent

rods with the axis of X. x, y, 6, <, are our generalized coordinates. The rectangular coordinates of the four middle points are

obviously

_ a cog ^ (x + a cos 0, .

(a

y _ a gin y

+

^

a sin 0),

CHAP.

ILLUSTRATIVE EXAMPLE

Ill]

_ Tr

We

nave

(x

+ a cos <,

(a?

a cos

0,

67

y

+ a sin <),

(3)

y

-r

a sin 0).

(4)

=4 = 4 m#,

Let the values struck,

Our

and

let

P

be

of x, y, 0, <,

0, 0, 0,

before the blow ^, 2i

is

be the magnitude of the blow.

mx = P,

4

equations are

= 0,

4 my 2

2m(> 4-^)^=0, 2 2m( + ^)^ = aP.

and

Whence

y

= 0,

P 4m

-:

Let v^ v^ v z v^ be the required ,

>

velocities of the four

Then

points.

;

3

2

+F P = X = 1P ^ 4m -^ P

2P 8m IP

= =--=: 4m

-~

-

4

and ^

:

:

v

:

=5

:

2

:

-1

:

2.

5P ~~"

Q"

8

'

m*

middle

IMPULSIVE FORCES

68

29. General Theorems.

[ART. 29

Work done by an

Impulse. If a particle Sx, (#, y, z) initially at rest is displaced to the position (x y -f- Sy, z &), the displacement can be conceived of as brought

+

+

about in the interval of time St by imposing upon the particle a velocity whose components parallel to the axes are u^ v^ w^ where 8x = ufit, Sy = vJSt, and Sz = w^t. If the particle is initially in motion with a velocity whose components are u, v, w, the displacement in question could be brought about by imposing upon it an additional velocity whose w. u, v v, w l components are obviously u^ Let a moving system be acted on by a set of impulsive forces. Let m be the mass of any particle of the system; Pa Py9 Pz the components of the impulsive force acting on the u, v, w, the components of the velocity of the particle particle and before, u^ v^ w^ after, the impulsive forces have acted. If any infinitesimal displacement is given to the system by which the coordinates of the particle are changed by Sx, Sy, and &, we have the virtual moment of the effective forces l

,

i

;

equal to the virtual that

moment

of the actual forces (v. Art. 27);

is,

Sm [_(u

:

u) x 4- (v {

v) Sy

+ (w^

w~) $z]

= 2 IPX $X + Py Sy + P Sz]. n

If the velocity that would have to be imposed upon the partim^ were it at rest, to bring about its assumed displacement

cle

in the time St has the

Sz

= w &t, 2

2ra [(wx

components u^ vz w^ Sx uj&t, and the equation above may be written u) u2 4-

(v^

,

- v) v + (w^

Si/

= v^St^

w) wj

2

^luf. + vfi + wf.}.

(1)

Interesting special cases of (1) are v^

v) v

-t-

(w l

w) w~\

= 2[P. + tP, + wPJ, ^

(2)

w) wj .],

(3)

CHAP.

THOMSON'S THEOREM

Ill]

69

the displacement used in (2) being what the system would have had in the time Bt had the initial motion continued, and that in (3) what it has in the actual motion brought about by the impulsive forces. If we take half the sum of (2) and

we

get

a system 's gain in

or,

kinetic

energy caused by the

action

of

sum of the terms obtained ~by 'multiplying the sum of the initial and final velocities of every force by half

impulsive forces its

is

the

point of application,

both being resolved in the

direction of

the force.

This sum

is

usually called the work done

by the impulsive

forces,

30.

u=

v

Thomson's Theorem. If our system starts from rest, and formulas (1) and (3), Art. 29, reduce re-

= w = 0,

spectively to

2m [w^ + v^ + wj0J = 2 [> PX + vfy + w^,], J,m[u* + v* + <] = 2 [u^ + v^ + w^']. 2

and

But

-

the

first

member

of (1)

is

(1)

(2)

identically

and subtracting (2) from (1) we get 2

]S ~$ {[^2

+ ^2 + W|] ~ [U* + Vl + W*^\

u^ vs w^ are the components of velocity of the particle m in any conceivable motion of the system which could give the If

,

IMPULSIVE FORCES

70

[ART. 31

forces the same velocities points of application of the impulsive that they have in the actual motion, then the second member of

(3)

is

zero and

we

get Thomson's Theorem

:

If a system at rest is set in motion by impulsive forces, its kinetic the velocities of the energy is less than in any other motion where the forces in question are the same as in the points of application of actual motion, by an amount equal to the energy the system would have in the motion which, compounded with the actual motion, would

produce the hypothetical motion. 31. Bertrand's Theorem.

Qx

If

,

Qy and Qz ,

are

the com-

would have to act on the ponents of the impulsive force which in Art. 29 to change its considered of the m system particle w to from u, v, u# v^ w# formula (3), component velocities Art- 29, gives us 9

(1)

JSubtracting (1) from (1) in Art. 29,

2m [O^ + v w 2

The

first

y _/ f

member

x

a

4-

wp)^)

of (2)

is

-

we

get

+

(v. Art.

30) identically

O

If the second

member

of (2) is zero, as will be the case

if

the

from the P-forces only by the impulsive actions and reactions due to the introduction of additional constraints which have no virtual moment in the hypothetical motion into the original system, we have Bertrand's Theorem*. ^-forces differ

is acted on by impulsive forces, the kinetic the subsequent motion is greater than it would be if the energy of were subjected to any additional constraints and acted on system

If a system in motion

THOMSON'S THEOREM

CHAP. Ill]

71

by the same impulsive forces, by an amount equal to the energy it would have in the motion which, compounded with the first motion,

would give 32.

the second.*

Theorem many problems

the aid of Thomson's

By

in-

volving impulsive forces can be treated as simple questions in maxima and minima. of giving the (ct) If, for example, in Art. 28, (a), instead we give the velocity v of the foot of the perpendicular force from the center of gravity upon the line of the force, so that v, then to find the motion we have only to make the y -f. ad

P

energy T^

=

2

[re

+

2

/

4-

&2 #"2 ] a minimum.

dx

x

Whence

=

0,

a

and these

results agree

entirely with the results

obtained in

Art. 28, (a). * Gauss's Principle of Least Constraint If a constrained system is acted on fr constraint " what is forces, Gauss takes as the measure of the with the motion combined motion the of kinetic the which, energy practically that the system would take if all the constraints were removed, would give the :

by impulsive

actual motion. It follows easily from Bertrand's Theorem that this "constraint" is less than in any hypothetical motion brought about by introducing additional con391-393). straining forces (v. Eouth, Elementary Rigid Dynamics,

IMPULSIVE FOBCES

72

(J) In Art. 28, (V), since x

To make Tg

a minimum,

and the problem (
Then

T.

t;,

the energy

we have

solved.

is

In Art. 28,

=

[ART. 33

(cT), let

^=x+

= ~ [4 (v -

2

ac^)

be given.

a<j>

+ 4 y + 2 (a + & )(0 + 2

2

2

2

,

80

-4a

- a<= 0.

+ 3 a2

and

v~

=x

In using Thomson's Theorem we may employ any valid form in the expression for the energy communicated by the impulsive forces. For instance, in the case of any rigid body, 33.

T - 2 [# + f + z* +Aa>? +Ba>l +

Co?]

is permissible and is much simpler than the corresponding form in terms of Euler's coordinates.

CHAP.

THOMSON'S THEOREM

Ill]

73

Take, for example, the following problem : An elliptic disk is at rest. Suddenly one extremity of the major axis and one extremity of the minor axis are made to move with velocities

U

and

F

perpendicular to the plane of

the disk.

motion of the disk. Let us take the major axis as the axis of axis as the axis of

We

X

Find the

and the minor

Y.

T = ~ [x* + f + z* + Aa>? + Ba>% -f /VYi

have, then,

2t

<X].

the conditions of the problem, since the components of the velocity of the point (&, 0, 0) are 0, 0, Z7, and those of the

By

point (0,

6,

0) are

0,

0,

F",

we have =0,

y + aeo = 0, = Z7; z Jo> = 0, x s

aot>

and

8

=

+ Hence

since

2

T=

^

*

= -T-

i

+

^ + | C^-

(

andT

4

F.

-~

jS

==--. 4

F>

We

have also

^ = ^-7 (5 F

*)'].

IMPULSIVE FORCES

74

[ART. 34

EXAMPLE to

One extremity of a side of move perpendicular to the

/,

the

a square lamina is suddenly made plane of the lamina with velocity

while the other extremity is made to move in the plane of lamina and perpendicular to the side with velocity V.

Show

that the center will

move with

V

velocity

-

perpendicular

/-

and with velocity -^-^2 in the plane, toward the corner on which the velocity V was impressed.

to the plane,

system does not start from rest, it is often easy to frame and to solve a problem in which the system is initially at rest and is acted on by the same impulsive forces as in the actual problem, and where consequently the resulting motion, compounded with the actual initial motion, will give the actual 34. If our

final motion.

For example, consider the following problem: A sphere rotating about any axis is gently placed on a perfectly rough horizontal plane. Find the initial motion. Here, in the actual case, the lowest point of the sphere is immediately reduced to rest.

Take rectangular axes

of

X and

F, parallel to the plane

and

Let lx vQ^, Cl z be the comco velocities and before, x co y cos after, the sphere ponent angular Let x, y, be the velocities of the center is placed on the plane. through the center of the sphere.

,

of the sphere. Then, in the actual case, x

,

,

=

,

and y

+ aa>x =

are our given conditions. Initially the velocities of the lowest point of afl y and al x If the sphere were at rest, the the sphere are in the actual case destroys these velocities force which impulsive would give to the lowest point the negatives of these velocities aa>

y

.

;

aO y and

Let us then solve the following auxiliary problem: A sphere is at rest. Suddenly the lowest point is made to move with velocities afly and a!\, parallel to a pair of horizontal axes. Find the initial motion of the sphere. that

is,

#!,,.

CHAP. If

ILLUSTRATIVE EXAMPLE

Ill]

u and

component

respectively, the x of the velocity of the center, v

are,

75

component and the y and co^ 2 o> g are the ct>

,

,

angular velocities,

The

velocities of the lowest point are alx . as afl y and

u

aco

2,

v

+ aco^

but

they were given

u

Therefore

ao>

= aH,

v

f {* [(, + To make

^y

minimum, we have

this a

S7

7

-

= m [a (, + nr) + * J = 0, 2

= m [a

2

(

2

+ %) + tfvj = 0,

2

BT _ Hence

Compounding actual problem,

these with, the initial angular Telocities in the

Ox

,

fl v ,

lz ,

we

get

Jc*

These equations, together with completely solve the original problem.

<

=

and y +

aa> x

= 0,

IMPULSIVE FORCES

76 35.

A

That

is,

[ART. 35

Problem in Fluid Motion. Let us now consider an interof this chapter which was esting application of the principles made "by Lord Kelvin to a problem in fluid motion. It is shown in treatises on hydromechanics that if an incomeither infinite in pressible, frictionless, homogeneous liquid, fixed or moving, surfaces closed finite or bounded by any extent in it, is moving immersed bodies flexible or and with any rigid under the action of conservative forces (v. Chap. IV) and has ever been at rest, the motion will be what is called irrotational. if

a;,

y,

s,

are the rectangular coordinates of

any fixed

there will be a funcpoint in the space occupied by the liquid, the are if that such u, v, w, tion <(#, y, z) components of the velocity of the liquid at the point d
u = ~~,

v

dx

The function

<j>

is

x, y, z, d
= ~, d

oz

oy

called the velocity-potential function.

Since throughout the motion the liquid is always supposed an incompressible continuum, u, v^ w, must satisfy the equa-

to be

tion of continuity for an incompressible liquid,

du -- dw -- ~ Bv o

ox

and therefore



satisfies

o

oy

rTT~

~

n ^

'

oz

Laplace's equation,

and will be uniquely determined except for an arbitrary constant the velocity normal to the surface, is term if the value of ,

on

given at every point of the boundary of the liquid, however irregular that boundary. Therefore the actual motion at every point of the liquid at any instant is uniquely determined, if the motion is irrotational, when the normal velocities at all points of the

boundary are given. now to prove that the kinetic energy of the actual wish We motion is less than that of any other motion, not necessarily

CHAP.

PROBLEM IN

Ill]

TIIOilMP

MOTIOK

77

with the equation of continuity and with the actual normal velocities at the boundary, irrotational, consistent

If u, let

u

v,

+ #,

w^ are the velocities at (x, y, z) in the actual motion, v -f j8, 20 7, be the velocities in the hypothetical

+

motion, and

let v n

be the actual normal velocity at any point O I

Then we have -2 =

of the boundary.

^

lu

+ ww> 4- ?M0 = tu

where

m, n, are the direction cosines of the normal, and u, v, w^ are the components of the velocity at the point in question. In the hypothetical motion, Z,

I

at the

(u

+ #) + m (v + /?) + n (w + 7) = vn

same point; therefore la -f m/3

+ ny ==

(1)

at every point of the boundary.

As must

the hypothetical velocities as well as the actual velocities obey the law of continuity,

and therefore

T is

where p integral

,

j

j

&B

at every point in the If

-- --

t

+ dy

bounded

+ dz =

^ ' (2)

space.

the energy of the actual motion,

the density of the liquid, and where the volume

is is

If T' is

taken throughout the space filled by the liquid. the energy of the hypothetical motion,

dxdydz

+ p ii \jxu + @v + yw\ dx dy d& I

IMPULSIVE FORCES

78

By

the aid of Green's I

Theorem we can prove

M[

fiv 4-

TW] dx dy dz

T dxdydz =

We have

[ART. 35

that

= 0.

u cos ads

Newtonian Potential Function, p. 92 (143)), where the volume integral is taken throughout any bounded space and the surface integral over the boundary of the space, cos# normal to the boundary. being the x direction cosine of the (v. Peirce,

=

Now and In

to

CCCau dx dy dz = like

=

3

.da

,,

,

te(f>-*te>

CladS

CfC$ ~ dx dy dz.

manner,

CCC/3v dx dy dz

and

8d>

=

Cm/3d>S

CCCyw dx dy dz = Cny^dS CCC[au + @v

Hence

C\lcc

-

CCC J^ dx dV d*> I

j

I

$ -jL dx dy dz.

+ 7^] dx dy dz

+ m/3 + nry~\ $ dS

-ff$\^ + ^ + ^

dxdydz.

surface integral vanishes by (1), and the volume f Therefore the energy T is greater integral vanishes by (2). than the actual energy T.

But the

It follows

that the irrotational motion of any frictionless homogeneous liquid under the action of con-

incompressible servative forces

motion a from rest which would have been suddenly generated by set of the liquid of in the boundary impulsive forces applied at points and such that they would suddenly give all the points of the boundary the normal velocities that these points actually have at the instant in question (v. Thomson's Theorem in Art. 30). is

at every instant

identical with the

CHAP.

PBOBLEM IN

Ill]

now we have

36. If

9ffifflRrB

MOTION

79

a liquid contained in a material vessel bodies, a set of generalized coordinates

and containing immersed

can ke c h sen equal to the number of degrees of *' ? ?i' ?2> freedom of the material system formed by the vessel and the immersed bodies, and the normal velocity of every point of the surface of the vessel and of the surfaces of the immersed bodies can be expressed in terms of the coordinates q^ j2 qn and '

*

>

-

,

,

,

the corresponding generalized velocities q^ q# -, qn can now choose other independent coordinates q^ q'^ -, practically infinite in number, which, together with our coordiwill give the positions of all the particles nates q^ q2 -, qn .

We

-

-

-

-

,

,

of the liquid.

Suppose the system (vessel, immersed bodies, and liquid) at Apply any set of impulsive forces, not greater in number than n, at points in the surface of vessel and of immersed bodies and consider the equations of motion. For any of our coordinates qk we have the equation rest.

is the generalized momentum corresponding to qk since (as in varying q'k no one of the coordinates q^ qz , , #, is changed, and therefore no one of the impulsive forces has its

where p'k

,

point of application moved) the virtual moment of the component impulse corresponding to qk is zero. As the actual motion at every instant could have been generated suddenly from rest by such impulsive forces as we have just considered, the momentum p'k is zero throughout the actual motion ; and the impressed forces being by hypothesis conservative, and the liquid always forming a continuum, no work is done when q'k is varied. Consequently, in the Hamiltonian

equation

pk +

dT f

= Q& pk

^ and Q'k

vQk

0,

and therefore

$T 7

= 0.

'Q*:

Hence every coordinate q'k is cyclic, and it is also completely ignorable. The energy modified for the coordinates qk is identical with the energy, which, being free

from the coordinates qk

IMPULSIVE FOBCES

80

and the momenta p& ?2

?i

and

'

*

>

is

*s

expressible in terms of the

[ART. 37

n coordinates

-, jn , ?n an(* tlae corresponding velocities q^ j2 , a homogeneous quadratic in terms of these velocities -

*

(v. Art. 24, (a)).

Summary

37.

of

Chapter

which a moving system

we

care only for the state of motion brought about in the forces question, since on the usual assumption that

sive forces,

by

is

In dealing with problems in supposed to be acted on by impulIII.

there

is

no change

we

in configuration during the action of the are not concerned with the values of the

impulsive forces, coordinates but merely with the values of their time derivatives.

The

moment

virtual

(v. Art.

26) of the effective impulsive

forces in a hypothetical infinitesimal displacement of the system, due to an infinitesimal change Sqk in a single coordinate qk , is

or

P

written equal to k $qk the virtual moment of the actual impulsive forces in the displacement in question, we have one of the equivalent equations If either of these

is

)

_

The n equations of which either of these is the type are n simultaneous linear equations in the n final velocities (j^, * an( as kke configuration and the initial state of "' C?a)i "

^-

motion are supposed to be given, no integration is required, and the problem becomes one in elementary algebra.

A

skillful

use of Thomson's or of Bertrand's Theorem re-

motion under impulsive forces to simple problems in maxima and mimima. duces

many problems

in

CHAPTER IV CONSERVATIVE FORCES 38. If X,

Y",

Z,

are the components of the

forces

acting

on a moving particle (coordinates #, y, 2), the work, W, done by the forces while the particle moves from a given position P^, (# y 2 Q ), to a second position P^ (x^ y^ z^), is equal to ,

,

-PI

\Xdx + Ydy + ^2]

;

)

and

since every one of the quantities JT, F, general case a function of the three variables

and

Z

is

x, y^ z,

in the

we need

know

the path followed by the moving particle, in order to find W. Let /(#, y, 2) = 0, < (x, y, z) = 0, be the equations of the path. can eliminate z between these equations and then express y explicitly in terms of #; we can then eliminate y

to

We

between the same two equations and express z in terms of xi and we can substitute these values for y and z in X, which will then be a function of the single variable

found by a simple quadrature. with

Y

and

Z we

By

I

and

Xdx can

/

proceeding in the same

/* y\

can find

x,

f*

Ydy and

I

U=

exact differential^

(x, y, z)

that

is,

Zd&>

that there

such that

=Z

=Y ~~

9

dy 81

'

"""

dz

way

zi

and the sum

of these three integrals will be the work required. It may happen, however, that Xdx -f- Ydy -h Zdz is

called an

be

is

what

is

a function

CONSERVATIVE FORCES

82

[AKT. 38

Since the complete differential of this function

is

3U or Xcfa; -f I"t%

-f-

we have

Zdz^

1% +

f\Xdx 4-

=

fe]



O,

y, z)

C \Xdx+Ydy + Zdz~]

and

J? and

=

cf>

(xl9 y lf

in obtaining this result

O- O <

,

y

=

O=

,

IT9

Z7,

we have made no use

-

i7

;

of the

path

followed by the moving particle. When the forces are such that the function 17 = (#, y, 5;) exists, they are said to be conservative, and U is called the <

force function. can infer, then, that the work done by conservative forces on a particle moving by any path from a given initial position to a given final position is independent of the path and

We

is

equal to the value of the force function in the final position its value in the initial position. If instead of a moving particle we have a system of particles,

minus

the reasoning given above applies. Let (XM y*, 2*) be any particle of the S3r stem, and JT Tt ZkJ "be the forces applied at the particle. Then the whole work, W, done on the system as it moves from one configuration to fc ,

another

equal to

is

If there -,

then

2

and

is

. ,

\X^ dxk +

U= 4>(x

.

l9

x^,

27= <^(x^ #2

a function

is

Zf z2

-

,

.,

.,

zt9

.

.

.)

y^ ya

,

the force function ;

.

,

xk

,

such that

Yk dyk + Zk dz^\ .

-

,

an exact

differential and indefinite ., z^ z# -) integral the forces are a conservative set; and is

-

is its

POTENTIAL ENERGY the

work done by the

forces as the system

83

moves from a given

initial to a

given final configuration is equal to the the force function in the final configuration minus its

value of

value in the initial configuration, no matter by what paths the particles

may have moved from their initial to then: final It is well known and can be shown without

positions. difficulty that

such forces as gravity, the attraction of gravitation, any mutual attraction or repulsion between particles of a system which for every pair of particles acts in the line joining the particles and is a function of their distance apart, are conservative while such ;

forces as friction, or the resistance of the air or of a liquid to the motion of a set of particles, are not conservative.

The negative of the force function of a system moving under conservative forces is called the potential energy of the system, and we shall represent it by V.

we

under conservative forces and are using generalized coordinates, and the geometrical equations do not contain the time, we can replace the rectangular coordiIf

are dealing with motion

nates of the separate particles of the system in the force function or in the potential energy by their values in terms of the generalized coordinates q^ q# qn and we can thus get Z7, and con,

->

sequently V, expressed in terms of the generalized coordinates. If U is thus expressed, QjBq^ (the work done by the impressed forces

when the system

and therefore

is

is

displaced by changing qk

approximately ==

-

Sq^ or

^_

*~3y*~ 39.

by Sqk

Sqk

,

*)

is

S9Jb U

and hence

%*

The Lagrangian and the Hamiltonian Functions.

If the

forces are conservative, our Lagrangian equation ,-,, C) *

,

may be

...

written

d dTt* dt d

= dQd

CONSERVATIVE FOKCES

84

[ART. 39

V is

the potential energy expressed in terms of the coordi*. not containing the velocities q^ q^ nates q^ , and q' *

where

,

L=T,-V,

If

L

(3)

an explicit function of the coordinates and the velocities and is called the Lagrangian function. is

.

1 Obviously

dL

%

= dT^ -^

,

and

cL

--= 3T

^fc

*fe

%

^?*

Hence our Lagrangian equation (1) can be written very neatly

If our forces are conservative,

and we are using the Hamil-

tonian equations and express the kinetic energy Tp in terms of the coordinates and the corresponding momenta, and if we let

H= Tp + F,

H .

is

called the Samiltonian function and

.., ql,

andjt?^^,

-,

pn

(5) is

a function of ql9 ya ,

.

Our Hamiltonian equations

can

now

be written dt

J

^ _ ^' dqh

and these are known

2

as the Samiltonian canonical equations.

If our forces are conservative,

and we are using instead of

the kinetic energy the Lagrangian expression for the energy modified for some of the coordinates q^ q^ *, qr (v. Art. 17), if

we

let

4>

= ^,..., s,-r,

C8)

THE LAGEANGIAZsT FUNCTION

CHAP. IV]

we have

and

for

any coordinate

qr+k the

Lagrangian equation

any coordinate qr _ t the pair

for

85

of Hamiltonian equations

40. The Lagrangian function L is the difference between the kinetic energy T^ (expressed in terms of the coordinates q^ q^ and the velocities q^ q^ , qn , yn , and homogeneous of the ,

terms of the velocities) and the potential -, jw ). -energy V (expressed in terms of the coordinates q^ y2 The Hamiltonian function is the sum of the kinetic

second degree

in

,

H

energy

Tp

(expressed in terms of the coordinates q^ q^

and the corresponding momenta p^ p^ eous

of

?*

,

JB

-,

,

pn and homogen,

the second degree in terms

the potential energy i

-

V

of the momenta) and (expressed in terms of the coordinates

'' ?)

The sum

of the kinetic energy and the potential energy, however expressed, is sometimes called the total energy of the system, and we shall represent it by j&, so that

&=T+V. The

function

(1)

of the preceding section

<3>

between the kinetic energy (expressed nates q^ qz velocities yr+1

-

,

,

,

<jr

jn ,

+
(similarly expressed),

the '?

in

is

terms of the coordi-

momenta p^ p^

?n )>

the difference

,

j?r ,

and the

miaus the terms

and the potential energy (expressed

terms of the coordinates q^ js modified Lagrangian function.

,

,

gn )*

We

in

shall call it the

CONSERVATIVE FORCES

86

[AKT. 41

except those contributed by the potential energy are homogeneous of the second degree in the momenta introduced and the velocities not It

is

to be observed that all the terms of

<2>

eliminated by the modification.

In dealing with the motion of a system under conservative forces, we may form the differential equations of motion in any one of three ways, and the equations in question are 41.

function practically given by giving a single or Hamiltonian the or function, H, function,

,

the Lagrangian the modified

3>,

Lagrangian function.* Every one of these functions consists of two very different on the parts : one, the potential energy F, which depends merely forces, which in turn depend solely on the configuration of the system; the other, the kinetic energy T or the modified

Lagrangian expression

M

q

...,

either of

which involves the veloci-

ties or the momenta of the system as well as its configuration. If we are using as many independent coordinates as there are

degrees of freedom, a mere inspection of the given function enable us to distinguish between the two functions of which it is formed, the potential energy or its negative being will

the terms not involving velocities or momenta. If, however, we are ignoring some of the coordinates (v. Arts. 16, 21, and 24) and are using (the Hamiltonian function) or <3>

composed of

all

H

(the Lagrangian function modified for the ignored coordinates), the portion contributed to (the modified by Tp or to <E> by

H

M

expression for the kinetic energy) is no longer necessarily a homogeneous quadratic in the velocities and momenta (v. Art. 21) and may contain terms involving merely the coordinates and therefore indistinguishable from terms belonging to the potential energy ; and consequently the part of the motion not ignored would be identical with that which would be pro*

duced by a or

<

set of forces quite different

from the actual

forces.

* Indeed, for equations of the Lagrangian type, any constant multiple of will serve as well as JL or <.

L

ILLUSTRATIVE EXAMPLE

CHAP. IV]

87

"We may note that the last paragraph does not apply if the system starts from rest, so that the ignored momenta are zero throughout the motion (V. Art. 24, (#)) Let us consider the problem of Art. 8, (<#), where the potential energy V is easily seen to be mgx. If we are using the Lagrangian method, as in Art. 8, (af), we have Z = ~[2:r2 + (a--a;)0 2 -h2<^]. (1)

we use

If

the Hamiltonian method, as in Art. 15,

we

(/T),

'

have

r

we use

If

Art. 20, (5),

2

2

method modified

the Lagrangian

for

#,

as in

we have

A mere inspection of any one of these three functions enables us to pick out the potential energy as V = mgx. If, however, we are ignoring #, as in Art. 22, Example, we have and

here, so far as the function

may

be



-

shows, the potential energy 2


mgx,

as,

in fact,

it is,

or

it

may be - 7W/ (a 2i

~

^

mgx.

5C }

Indeed, the hanging particle moves precisely as if its mass were 2 and it were acted on by a force having the force function

m

that If,

is,

a force vertically J

as in

downward equal

many important

to

mg

--

-

-

m(a are we Chap. V), problems (v.

r^

xy

unable

and measure the impressed forces directly and are attempting to deduce them from observations on the behavior of to discern

CONSERVATIVE FORCES

88

[ART. 42

a complicated system, which for aught we know may contain undetected moving masses, the fact that we cannot discriminate with certainty between the terms contributed to the modified function by the kinetic energy of the system and

Lagrangian

those contributed by the potential energy may lead to entirely different and equally plausible explanations of the observed phenomena (v. Art. 51).

we

are dealing with a system -, gn9 moving under conservative forces, the coordinates q l9 q^ , qn are functions of t, the time, as are also the velocities q^ 2 , 42. Conservation of Energy.

If

-

.

Therefore V, the potential energy, and T9 the kinetic energy, are functions of the time, as is their sum, the Hamiltonian function H. T us nnd Let

As the

dH

-

r

at

H depends explicitly on the coordinates qv }

momenta p^ p^

-

,

-

-,

qn9

and

p n9
But by our Hamiltonian equations

-dt

-

3,

,

A and

d k

p

(v. Art. 39,

-~ = dt

cqk

dt

(6) and (7)),

?

dpk

and (1) becomes

&K = V F dt

Therefore Ji is a constant.

-

1

=

^[dqk ^pk dpk 8qk \ T + V = H = h,

(3)

where

Hence in any system moving under conservative sum of the kinetic energy and the potential energy

forces the constant

is

during the motion. This is called the Principle of the Conservation of Energy. Since by (3) any loss in potential energy during the motion is just balanced by an increase in the kinetic energy, and the loss in potential energy is equal to the work done by the actual

HAMILTON'S PRINCIPLE

CHAP. IV]

89

forces during the motion, our principle is a narrower statement of the familiar principle : If a system is moving under any forces, conservative or not^ ike gain in kinetic energy is always equal to tlie

work done by

the actual forces.

Let a system move under conserv-

43. Hamilton's Principle.

ative forces

from

its

figuration at the time

configuration at the time t^.

We d_

dL ;

t

Q

to its con-

have

=

dt oq k

3L dqk

where L, the Lagrangian function, is equal to T V. Suppose that the system had been made to move from the to the second configuration so that the particles traced slightly different paths with slightly different velocities, but so

first

that at any time t every coordinate qk differed from its value in the actual motion by an infinitesimal amount, and so that

every velocity qk differed from its value in the actual motion by an infinitesimal amount, or (using the notation of the calculus of variations)* so that Sqk and 8qk were infinitesimal; and suppose it had reached the second configuration at the time t^ Then, if at the time t the difference between the value of L in the hypothetical motion and its value in the actual motion is SL 9

Now,

at the time

and 2?*

\ = d /dL zq*~

d

81,

,,

q*

j^/f^ * For a brief introduction to the calculus of variations, see Appendix B.

CONSERVATIVE EOBCES

90

by

Therefore,

Si =

(2),

=S

and

f ART.

Af\Ldt Af^Ldt

= [ V |^ a& L

Since the terminal configurations are the same in the actual motion and in the hypothetical motion and the time of transit is

the same,

S& = S

when

=t

t

and when

=S

AC^Ldt Af \T

-

and

t

t

l9

V] dt =

0.

(3)

the necessary condition that L should be either a a minimum and is sometimes stated as follows: a system is moving under conservative forces, the time inte-

But (3) is maximum or

When

gral of the difference between the kinetic energy and the potential ** " in the actual motion. This stationary energy of the, system is is

known 44.

as Hamilton's principled

The Principle

ceding section that the time of transit

Least Action. If the limitation in the preand the hypothetical motions from the first to the second configuration is of

^

in the actual

* Hamilton's principle plays so important a part in mechanics and physics that it seems worth while to obtain a formula for it which is not restricted to conservative systems. shall use rectangular coordinates, and we shall make the hypothesis as to the actual motion and the hypothetical motion which has been employed above. For every particle of the system we have the familiar equations

We

mcc

=

my =

-5T,

mz

T",

r=r](** +

Since

$5$

+ zdz] = _

But

5x

mx, dt

Hence and

= Z.

=m dt

mxdx

(xdx)

5T+ S [X&c +

TSy

= dt

+

f "'{sr + S [XSz + ^l/

ZSz] 4-

= dt

Zto] } dt

fi

.M3 at

at

at

(mxSx)

Sm [xdx 4- $$y + zSz],

= S [TTIXSO; +

*/<

If the system is conservative, get the formula (3) in the text.

),

S [JSTffoj 4- ^2/ +

252]

=

&2/

+ ^^j!o1 ~ 0.

5Z7=

c

5T 7

",

and we

PRINCIPLE OF LEAST ACTION

CHAP. IV] the same

"

91

be removed and the variations be taken not at the

same time but

at arbitrarily corresponding times, t will no longer be the independent variable but will be regarded as depending

We now have

upon some independent parameter r.

%=j

t

dL

...

Sfe

dL d

- ft

I &,

..

.

(v.

dL d

(v. Art.

App. B,

6,

43)

(1))

...

aad dt .

^

Sj '

since

Hence

If now we impose the condition that during the hypothetical motion, as during the actual motion, the equation of the coa servation of energy holds good, that is, that

then

T .SL = S^T - SF =

and (1) becomes

COKSEBVATIVE FOBCES

92

f **2 Tdt =

5

and, finally,

0.

[ART. 45

(2)

A)

The equation and the action.

^= r I

A

i

2

Tdt

defines the action, A, of the forces,

fact stated in (2) is usually called the principle of least a matter of fact, (2) shows merely that the action is

As

e*

stationary." 45. In establishing Hamilton's principle we have supposed the course followed by the system to be varied, subject merely to the limitation that the time of transit from initial to final

configuration should be unaltered ; consequently, as the total energy is not conserved, the varied course is not a natural course. That is, to compel the system to follow such a course

we should have

to

introduce

additional forces that

would

do work. In establishing the principle of least action, however, we have supposed the course followed by the system to be varied, subject to the limitation that the total energy should be unaltered; consequently the varied course is a natural course. That the system to follow such a course we need is, to compel introduce merely suitable constraints that would do no work. We have deduced both principles from the equations of motion. Conversely, the equations of motion can be deduced from either of them. Each of them is therefore a necessary and sufficient condition for the equations of motion.

DEFINITIONS OF ACTION

CHAP. IV]

93

The action, A, of a conservative set of forces acting on a moving system has been defined as the time integral of twice 46.

the kinetic energy.

.

A = i 2Tdt l

(1)

J**

X*i

2m (x*

Therefore

2 -f- ?/

4- if) dt

A=

=

r**.

C'Zmvds,

A7\

2w (j-

\

2

dt

=

/**i

/

^mv zdt. (2)

might just as well have been defined as the of which is the line integral of the momenone any tum of a particle taken along the actual path of the particle. There is another interesting expression for the action, which does not involve the time even through a velocity. so that the action

sum

of terms

Since

As

=r We

have stated without proof that the differential equations of motion for any system under conservative forces could be deduced from Hamilton's principle or from the principle of 47.

stationary action* Instead of giving the proof in general, we will give it in a concrete case, that of a projectile under gravity.

CONSERVATIVE FORCES

94

We

X

[ART. 47

Y

horizontal and shall use fixed rectangular coordinates, vertically downward, taking the origin at the starting point

of the projectile.

In

this case, obviously,

TfL

y

By

(a)

2

),

Hamilton's principle, s

r Jo

-

and

V

mgx.

we have

1

r

or

Jo /*! I

[#&

+ y^y 4- g&%\ dt = 0.

*/0

Integrating by parts,

but since fe and Sy are wholly arbitrary, this is impossible unless the coefficients of Sx and Sy in the integrand are both zero. 3?K

TT Hence

(i)

By

the principle of stationary action,

l

whence

--<>>

C' Jr Q

(if

dr + + ?) S ^ &r

P^

Jr

Mr

S (#

we have

+ f) dr = 0.

(1)

LEAST ACTION

CHAP. IT]

95

But Hence

=

S

ff $x,

2i

and

S (i?

*

since

=*

=

+ f) =

S

&*--*;

dt*d? + y* = d % _s__^ _g t

:B

App. B,

6,

* _ /^ -2^ # ^ + y _s, (^ + y ) _^ ,

a;

(v-

<#

,

/

and (1) becomes

'" f

J

*^ rO

r

ir.

X

\

L

f-s xda? L

+ ^oy I

Jr Q

g* .

c?

dt

Tr

we

Integrating by parts, 7

r -1-

"

f

1

,

P

T^ he

J

Jro

Jr.

get

L

dr

.

dr

dr

J

But S& and Sy are zero at the beginning and at the end of the actual and the hypothetical paths, so that 1

T [(^~ and

as Sx

and Sy

are wholly arbitrary, this necessitates that

as in Art. 47, (a).

CONSERVATIVE FORCES

96 48.

Although we have proved merely that

[ART. 48 in a

system under

conservative forces the action satisfies the necessary condition of a minimum, namely, SA = 0, it may be shown by an elaborate investigation that in most cases it actually is a minimum, "

and that the name principle,

least action," usually associated

with the

is justified.

A very pretty corollary of the principle

comes from its application to a system moving under no forces or under constraining forces that do no work. In either of these cases the potential is zero, and consequently the kinetic energy is constant, energy

V

that

T=h.

is,

A=

2Tdt = 2h

I

dt^ZhQ

t^

J*o

J*o

proportional to the time of transit. Hence the actual motion is along the course which occupies the least

and the action

is

possible time. For instance,

energy A v

2

if

a particle

is

moving under no

constant, the velocity of the particle

is

forces, the is

constant,

and as it moves from start to finish in the least possible time, must move from start to finish by the shortest possible path,

it

that

is,

by a

straight line.

If instead of

moving

move on a given must

freely the particle

surface, the

trace a geodetic

on the

is

constrained to

same argument proves that

it

surface.

Varying Action.* The action, A, between two configurations of a system under conservative forces is theoretically expressible in terms of the initial and final coordinates and the total energy (V. Art. 46, (3)), and, when so expressed, Hamilton 49.

called

it

the characteristic function.

In like manner,

C ,

if

h

Ldt*=

r\TV*)dt = r\hA>

A>

* For a detailed account of Hamilton's method, see Routh, Advanced Kigid Dynamics, chap, x, or Webster, Dynamics, 41.

CHAP. IV]

S may

VARYING ACTION

9T

initial and final coordinates and the time of transit. When

be expressed in terms of the

of the system, the total energy, so expressed, Hamilton called

it the principal function. By the variation considering produced in either of these functions by varying the final configuration of the system, Hamilton showed that from either of these functions the integrals of

the differential equations of motion for the moving system could be obtained, and he discovered a partial differential equation of the first order that S would satisfy and one that A

would

and so reduced the problem of solving the equamotion for any conservative system to the solution of

satisfy,

tions of

a partial differential equation of the first order. It must be confessed, however, that in most cases the advantage thus gained is theoretical rather than practical, as the solving of the equation for S or for A is apt to be at least as difficult as the direct solving of the equations of motion.

CHAPTER V APPLICATION TO PHYSICS 50. Concealed Bodies.

In

many problems

in mechanics

the

configuration of the system is completely known, so that a set of coordinates can be chosen that will fix the configuration

completely at any time; and the forces are given, so that if the system is conservative the potential energy can be found. In that case the Lagrangian function L or the Hamiltonian can be formed, and then the problem of the motion function of the system can be solved completely by forming and solving the equations of motion.

H

In most problems in physics, however, and in some problems in mechanics the state of things is altogether different. It may be impossible to know the configuration or the forces in their entirety, so that the choice of a complete set of coordinates or

the accurate forming of the potential function is beyond our powers, while it is possible to observe, to measure, and partly to control the phenomena exhibited by the moving system which

we are studying. If from results which have been observed or have been deduced from experiment we are able to set up indirectly the Lagrangian function, or the Hamiltonian function, or the modified Lagrangian function, we can then form our differential equations and use them with confidence and profit. motion considered in the problem two equal particles and the table with a hole in it (v. Art. 8, (^)), and suppose the investigator placed beneath the table and provided with the tools of his trade, but unable to see what is going on above the surface of the table. He sees the hanging particle and is able to determine its mass, its 51. Take, for instance, the

of the

98

THE DANGLESTG PARTICLE

CHAP. V] velocity,

and

to measure

its

acceleration

;

99

to fix its position of equilibrium

;

motion under various conditions to apply additional forces, finite or impulsive, and to note their effects. His system has apparently one degree of freedom. It possesses an apparent mass m and is certainly acted on by gravity with a downward force mg. He determines its position of equilibrium, and taking as his coordinate its distance x below that position, he painfully and laboriously finds that x, the accelits

;

eration of the hanging mass,

He

is

If

is

equal to

now ready

T is

to call into play his knowledge of mechanics. the kinetic energy and L is the Lagrangian function for

the system which he sees,

L=T- F=*-

and Q Since .

mx = an, * Therefore

The motion, then, can be accounted for as due downward force of gravity combined with a second force having the potential energy

-~ ^ 2t

I

u

-

(ct

-^

+x

Xj

upward having the

intensity

-^ A

vertical

that

,

is,

a

I

r~

force vertically

to the

-

\_(a

s

~i

^ + 1J x)

;

and of course this force must be the pull of the string and may be due to the action of some concealed set of springs.

APPLICATION TO PHYSICS

100

[ART. 51

the other hand, the moving system may contain some concealed body or bodies in motion, and that this is the fact is force JP is applied strongly suggested if a downward impulsive

On

to the hanging particle when at rest in its position of equilibrium. For such a force is found to impart an instantaneous velocity x

=

T>

just half

what we should get

if

the hanging

were the only body in the system, and just what we should have if there were a second body of mass m above the a stretched stringtable, connected with the hanging particle by

particle

of fixed length.

Obviously this concealed body is ignoralle, since we have obtain the differalready found a function from which we can have just called we function the for ential equation x> namely, the Lagrangian function L*

It is

and contains the single coordinate x* But if we have ignored a concealed moving body forming part of our system, this expression is not the Lagrangian function L> but is equal or proportional to , the Lagrangian function modified for the coordinate or coordinates of the concealed body and in that case some or all of the terms which ;

we have regarded as representing the potential energy of the system may be due te the kinetic energy of the concealed body (v. Art 41). Of course the complete system may have two or more degrees of freedom. Let us see what we can do with two. Take x and a second coordinate S and remember that as 6 is ignorit must be a cyclic coordinate and must not enter into

able

the potential energy. Let us now form the Lagrangian function and modify it for 0. have T~Ai? Bx6 CO^ where A^ B, and C are functions

We

of

x.

+

+

THE

CHAP. V]

DAISTGLIKG PABTICLE

dT Q== p e A

'

101

A

Bx

~^Q-

B To modify

we must

for

subtract 6p Q .

We

get

Suppose that no external force acts on the concealed particle, so that the potential energy of the system is mgx. Then, if the Lagrangian function modified for 6 is <&,

on our ignoration hypothesis 6 does not enter the K, a constant, and potential energy, the momentum p Q Since

But we know

that <& ~

equal or proportional to

is

m

X-2

ff"

^

the function which on our hypothesis of no concealed bodies we called L.

We see that if B = 0, A = m, and if -4(7 = 2

2

if

,

whence

Then we have

-

C=

2

T = ^^ +

(a

x)

,

APPLICATION TO PHYSICS

102

[ART. 51

where IK. is the momentum corresponding to the coordinate 6 and may be any constant. If

is

we

take for JT 2 the value

obviously the kinetic energy of a mass

x and

and having a function

m

moving

as polar coordinates.

in a plane

The Lagrangian

equal to

is

and the x Lagrangian equation 2

mx

is

m(a

2

x)6

-)r

mg*

If the angular velocity of the concealed mass when the hanging particle is at rest in its position of equilibrium is Q since ,

in that case

and x =

x=

0,

equation (1) gives us & Q

= -o-

-

The

i Cff

observed motion of the hanging particle is then accounted for completely by the hypothesis that it is attached by a string to an equal particle revolving on the table and describing a circle

a about the hole in the table with angular

of radius

velocity ->J~ ^(X>

-

when

the hanging particle

We

tion of equilibrium. o

*j

I

o~7 A (^ct

\2

xj

"+"

x

'

is

at rest in its posi-

see that on this hypothesis the

term

which on the hypothesis that the system

I

contained only the hanging particle was an unforeseen part of is due to the kinetic energy of the con-

the potential energy, cealed moving body*

K

It may seem that giving a different value might lead to a different hypothesis as to the motion of the concealed body that would account for the motion of the hanging particle.

PHYSICAL COORDINATES

CHAP. V]

Such, however,

not the

is

case.

We

103

have

Let

T = - \tf + i? + (a - xf /VV)

Then nrvt

f

2

~*

2

^)

r

J

ig?

as above, the kinetic energy of a

body

.

of mass m, with polar coordinates a x and ; and the concealed motion is precisely as before. In using the form (2) we have merely used as our second generalized coordinate #, a perfectly suitable parameter but one less simple than the polar angle. <

Problems in Physics. In physical problems there may be present electrical and magnetic phenomena and concealed molecular motions, as well as the visible motions of the material parts of the system. In such cases, to fix the configuration of the system even so far as it is capable of being directly observed we must employ not only geometrical coordinates required to fix the positions of its material constituents, but also 52.

parameters that will

we

are

motions, tion

we

rarely

sure

we must

fix its electrical or

of

magnetic state ; and as

the absence of concealed molecular

often allow for the probability that the funcby the aid of observation and

are trying to form

experiment and on which we are to base our Lagrangian equaof motion may be the Lagrangian function modified

tions

for the

ignored coordinates corresponding molecular motions. 53. Suppose, for instance, that

to

the concealed

* ,

we have two

similar, parallel, straight, conducting , wires, through which electric currents due It is found exto applied electromotive forces are flowing.

perimentally that the wires attract each other if the currents have the same direction, and repel each other if they have opposite directions, and that reversing the currents without

APPLICATION TO PHYSICS

104

[ART. 53

forces does altering the strength, of the applied electromotive It is known not affect the observed attraction or repulsion.

that an electromotive force drives a current against the resistance of the conductor, and that the intensity of the current is it is

known

proportional to the electromotive force. Moreover, that an electromotive force does not directly cause any motion of the conductor. It is found that, as far as electric currents are concerned, the phenomena depend merely on the intensity

and

direction of the currents.

our configuration we shall take & as the distance between the wires and take parameters to fix the intensities of These parameters might be regarded as the two currents. coordinates or as generalized velocities, but many experiments

To

fix

We

shall call them y r and y^ of electricity that have units of yl since a given epoch. first wire the of section crossed a right

suggest that they are velocities.

and

define

As

as the

all effects

number

depend upon yt and y2 and not on

^

and y#

and y^ are cyclic coordinates. Let us suppose that there are no concealed motions. Then the kinetic energy T is a homogeneous quadratic in y^ and Let {/ T = Ai? + Ltf + Jtfyj/a + Nyl + Bxy^ + C&y* (1)

y^

,

.

where the

coefficients are functions of x* Since reversing the directions of the currents, that is, reversing the signs of i/l and ya , does not change the other phenomena, and C are zero. it must not affect T; therefore

B

If the wires are not allowed to move, x

to

=

and

T

reduces

+ Ny^

Lyl -f My^ij^ of the system. Since

which is called the eleetroldnetic energy from considerations of symmetry this can-

NL.

not be altered by interchanging the currents, Let us now suppose that the first wire is fastened in position and that the only external forces are the electromotive forces El and E^ producing the currents in the two wires ; the resistances Ry^ and Hy^ of the wires, equal, respectively, to JS^ and when the currents yx and y2 are steady; and an ordinary 2

J

PARALLEL LINEAR CONDUCTORS

CHAP. V]

105

mechanical force F, tending to separate the wires. have

We

now

.

and for our x Lagrangiaii equation

dA

.

2

dA

.

dL

2

.

d

2

M

.

dL

.

.

Let us study this equation. First suppose the electromotive = = 0, and the impressed force F all zero, so that

^ ^

forces

and j^=

0.

Equation (2) reduces to

1

=

#

or

2 If,

A

dA^ <E% ax

then, a transverse velocity were impressed on the second

wire, the wire

would have an

acceleration unless

both wires, on our hypothesis, being attract nor repel each other, therefore

T=Ax* when

^ = ^ = 0, A

is

dA-

0.

ax

But

they can neither a constant ; and as

inert,

A

is

a positive constant.

Therefore

(2) reduces to

Let us

now

suppose that yz ~

Z

becomes

dL

...

Ax

This

is

is

is

I

dL

attracted

and

F~

0.

Equation (3)

= A0, , a

or repelled

by the

first,

flowing through the second wire.

contrary to observation.

a constant; and as

.

ax y*

..^

and the second wire even when no current

0,

TLyl

Therefore

when x and

-

ax

= 0,

and

y^ are zero,

L

L is

is

a

APPLICATION TO PHYSICS

106

positive constant.

y and

If

[AKT. 54

Equation (3) now becomes

y^ are of the

same

sign,

x and -j- have the same

observation the wires attract if the currents flow in the same direction, therefore x is negative and

sign.

But according

dM -

negative.

.

is

ax

If

x=

sity y^

0,

+ y^ <& +

comes

is less

a single wire carrying a current of intenThe electrokinetic energy Lyl My$^~\- Ly% be&)* or Lyf 2Zy^ Ly^ so that the value of is a decreasing function, .M is 2 L. Hence, as

we have

M when x = x

to

than 2

is infinite, it

We

+

+

+

always.

must be

As

M

seen to be zero

J&f is easily

positive for all values of

have, then,

T - Ai? 4- i# 4- My& + Lyl where

when

x.

A

and

ing function of

(4)

M

is a positive decreasare positive constants and is called the coefficient x always less than 2 L.

L

M

the coeffiof self-induction of either wire per unit of length, and cient of mutual induction of the pair of wires per unit of length.

Our Lagrangian

equations are

(5) (6)

[2 Zy, -M#J

= 2?, - JJy,.

(7)

(#) Suppose that no current is flowing in either of the wires considered in the last section, and that the first wire is suddenly connected with a battery furnishing an electromotive force J^ and that thereby a current 54. Induced Currents.

INDUCED CURRENTS

CHAP. V]

107

impulsively established. Then, by Thomson's Theorem Art. (v. 30), such impulsive velocities must be set up in the system as to make the energy have the least possible value

yx

is

consistent with the velocity caused by the applied impulse. If the current y^ set up hi the first wire has the intensity

T

and making

we have

a minimum,

Mi

-f-

i>

2

y2

=

whence x = 0, and the second wire Mi and y = TT-T and a current will

;

have no

will

initial velocity,

be set up impulsively in the 5 2iL second wire, of intensity proportional to the intensity of the current in the first wire. Since, as we have seen, and L are both 2

M

negative, and

induced current will be positive, y z in direction to current the impressed opposite y^ This impulsively induced current is soon destroyed by the resistance of the wire. (5) Suppose that a steady current y f caused by the electrois

motive force JS^ wire

is

this so-called

flowing in the

and that

first

wire while the second

and consequently y^

is impulsively the battery. This amounts destroyed by suddenly disconnecting to impulsively applying to the first wire the additional electro-

is inert,

J^,

motive force E^ If the system were initially inert, this, as we have just seen in (a), would immediately set up the induced current y^ _ _.

=~

= Mi in the 2L

jyT't

y

this

2i

L

,

as the

with the

second wire, and

we

should have

immediate result of our impulsive

initial

action.

^=

Combine

motion in our actual problem, y^ = z,

and we get for our actual result y

= 0,

Mi y2 = ^

i>

y^

= 0,

(v. Art. 34).

So that if our first wire is suddenly disconnected from the battery, an induced current whose direction is the same as that of the original current is set up in the second wire. It is, however, soon destroyed

by the

resistance of the wire.

APPLICATION TO PHYSICS

108

Suppose that we have a current y^

[ART. 54

our fixed wire, caused by a battery of electromotive force JS^ and no current in our second wire, and that the second wire is made to move away from the first. Equations (6) and (7) of the preceding section. (c)

in

give us

(4 L*

- If*) Ctt

= 2 (^ - JZy,) - If(^ - JZy,) x^>

-r

-*- N

-

-(ZLy^-My^x When we and we have

are starting to

^s^

As we have

^s^

'

seen, i,

move

Jf",

4X,

2

the second wire,

-2^-^ = 0,

'

2

JHf

,

are positive

and

is

nega-

Ct^C

tive.

Hence the current

^

will decrease in intensity,

and a

current y^ having the same direction as y^ will be induced in the moving wire. of induced currents which we have just from our Lagrangian equations are entirely confirmed by observation and experiment.

The phenomena

inferred

APPENDIX A SYLLABUS.

DYNAMICS OF A RIGID BODY

In a moving system of particles the and internal that act on any particle is called the effective force on that particle. Its rectangular components are mx mi/, and mz. The science of rigid dynamics is based on D'Alembertfs prinIn any moving system the actual forces impressed and ciple internal, and the effective forces reversed in direction, form a set of forces in equilibrium, and if the system is a single rigid body, the internal forces are a set separately in equilibrium and may D'Alembert's Principle.

1.

resultant of all the forces external

}

:

be disregarded. It follows from this principle that in any moving system the actual forces and the effective forces are mechanically equivalent.

Hence, (a)

2mx

= 2X,

= 2 [yX xY], 2m [xSx -h 3% + &Q = S [-ST&e +

() 2m [yx (c)

xy]

These equations (a) is

may be

FSy

+ Z&i].

put into words as follows

:

The sum

the same

of those components which have a given direction for the effective forces and for the actual forces.

The sum of the moments about any fixed line is the same for the effective forces and for the actual forces. of the system, actual or hypothetical, the (o) In any displacement work done by the effective forces is equal to the work done by the (by

actual forces.

Equations (a) and

(5)

are called differential equations of motion

for the system. 2.

the

px =

X

l&mVy.

= 2mx and is the linear momentum of the system in = 2m \jjvx xvy = 2w[?/a; xy~\ and is the

direction

;

7iz

^\

moment of momentum about the

axis of Z.

109

APPEKDIX A

110 Equations (a) and

(Z>)

of

1

may

be written, respectively,

and 1, (6), may now be stated as follows In a moving system the rate of change of the linear momentum (a) in any given direction is equal to the sum of those components of the actual forces which have the direction in question. (6) In a moving system the rate of change of the moment of mo?nentum about any line fixed in space is equal to the sum of the

and

1, (&),

moments 3.

:

of the actual forces about that line. ==

^ px = xinx

T-T

Hence or,

x

Center of Gravity.

~~

the linear

momentum

= M dX

in the

at

X

;

direction

the whole system were concentrated at

its

is

the moment of 'momentum about the axis of Z

or,

what

it

would be

if

center of gravity.

is

what

it

would be

if

the whole mass were concentrated at the center of gravity plus what it would be if the center of gravity were at rest at the origin and the actual motion were

of gravity really 4.

The

same as

what the relative motion about the moving center

is.

'motion of the center of gravity of a

if all

moving system is the the mass were concentrated there and all the actual

unchanged in direction and magnitude, were applied there. The motion about the center of gravity is the same as if the center gravity were fixed in space and the actual forces were unchanged magnitude, direction, and point of application.

forces,

of in

5.

If the system

is

hs

a rigid body containing a fixed axis,

= Mk^u = M(h

z

2 -f & )


where Mk = M(7i* + & ) and is the moment of inertia about the axis, 2

and where


Equation of the

is

the angular velocity of the body.

(&)

moments

of

1 becomes

Mk'2 -p OjU

= N,

where

JV"

is

the

of the impressed forces about the fixed axis.

sum

APPENDIX A 6. <* rel="nofollow">

y

111

If the system is a rigid body containing a fixed point are its angular velocities about three axes fixed in

and

c^,

space and

0)3,

,

passing through the fixed point,

where C

is the moment of inertia about the axis of are the products of inertia about the axes of and that is, C *%m (x 2 -f- T/2 ), 'Sflny&y

X

=

Equation

dh y

where

(&)

D=

of

^ dw-

N

is

the

y

Y

y

and

Z>

and

E

respectively

;

E=

1 becomes

_, d<j)v

sum

Z

_

of the

c?o>,,

,

.

moments

of the impressed forces about

the axis of Z. 7. Euler's Equations. If the system is a rigid body containing a fixed point and 2?
the

body and moving with

it),

equation () of

C?co 3

(

8. Euler's Angles. Euler's angles ing system of rectangular axes A, Xy Yy Zy having the same is the colatiorigin 0. tude and ^ the longitude of the moving axis of C

the fixed system (regarded as a spherical system with the fixed axis of in

Z

as the polar axis), is

the

the

angle

and

made by

moving CA -plane with

the plane through the fixed and the moving axis of

Z

axis of C.

)

0, ,

0)^2

=

^, <, are

1 becomes

.

coordinates of a mov-

C, referred to a fixed system

APPENDIX A

112

We

^ = 9 sin

have

<j>

if/

sin 6 cos

<j>,

= 6 cos $ + & sin 6 sin <, 2 = ^ cos + == # s^ + sm # cos sin sin cos ^ o^ = = COS ^ 4o>


and

<

;

11

o>jc

-I-

<03

9.

2

^-?7ii;

<

^r.

2 y* 4- ^ ] in rectangular coordinates,

[f

-f-

2 ^<^ ] in polar coordinates,

[^2

-|_

^2(^2

+ sin ^ )] 2

^

2

2

M ^^co

2

-+

if

^

+s) 2

in rectangular coordinates

a rigid body contains a fixed axis

+ (i?Tl TrR^iy \cLt / J \_\cLt/ two-dimensional

+ ?* :

2t

^ [A v* + Bu* + Ceo/ rigid and contains a 2

[A o^

a

11

if

the body

is

;

;

free and the motion

;

is

-J

in spherical coordinates.

,

(ic

body

2

^2 the kinetic energy of a 'moving system, becomes

-

is

i/r,

-{-

[ic

2

10.

<j>

t/r,

the kinetic energy of a particle, becomes

?

2 2i

<

*A

2

-f- .?
+ Cto

2 3 ]

2 -D^^ fixed point if

2 -Ea>^

2

J^coa.^ ]

if

the

;

the body

is

rigid

and the axes are

the principal axes for the fixed point. 11. Impulsive Forces. In a system acted on by impulsive forces, the resultant of all the impulsive forces external and internal that act on any particle is called the effective impulsive force on that particle.

(vn&^

Its ).

rectangular components are m(cc a ) ^(^ D'Alembert's principle holds for impulsive forces. 5

2/0),

It

APPENDIX A follows that in forces

and the

113

any system acted on by impulsive forces, the actual effective forces are mechanically equivalent. Hence,

sc

(c)

These equations may be put into words as follows (a) The sum of the components which have a given direction the same for the effective impulsive forces and for the actual :

is

impulsive forces. () The sum of the moments about a given line is the same for the effective impulsive forces and for the actual impulsive forces. (c) In any displacement of the system, actual or hypothetical, the sum of the virtual moments of the effective impulsive forces is equal to the sum of the virtual moments of the actual impulsive forces.

Equations (a) and (>) are the equations for the initial motion under impulsive forces, (a) and (&) may be restated as follows In a system acted on by impulsive forces, the total change in the :

momentum in any given direction is equal to the sum of those components of the actual impulsive forces which have the direction

linear

in question.

In a system acted on by impulsive

forces, the total

change in the

moment of momentum about any fixed line is equal to the sum the moments of the actual impulsive forces about that line. Section 4 holds unaltered for impulsive forces.

of

APPENDIX B THE CALCULUS OF VARIATIONS 1. The calculus of variations owed its origin to the attempt to solve a very interesting class of problems in maxima and minima in which it is required to find the form of a function such that the

definite integral of an expression involving that function derivatives shall be a maximum or a minimum.

Take a simple case

:

If

y =f(x),

let

mine the form of the function /, so that or a minimum. Let f(x) and F(x) be two

& maximum

it

\

^x

and

its

be required to deter-

\x,y,-~\dx shall be

o

possible forms of the function. Consider their graphs y =f(x)

and y

= F(x).

If

,(*)* F(x)-f(x) t1,(x) can be regarded as the increment given to y by changing the form, of the function from f(x) to F(x), the value of the independent variable x being held

fast.

This increment of y, 77 (x), is called the variation of y and is written by it is a function of x, and usually a wholly arbitrary function of x. ;

The corresponding increment in y\ where y = -~ can be shown to be x dir y (#), and is the variation of y\ and is written By or B ~- Obviously, aif

1

?

f

,

If an infinitesimal increment

S?/

is

given to y y

it is

proved in the

(y) y differs by an infinitesimal of dy from the increment produced in <(?/). This

-differential calculus that

higher order than

y

ax



114

APPENDIX B approximate increment

is

115

called the variation of

<j>

(y), so that

(2)

8* (y,

Similarly,

or since x

is

y')

=

Sy

+

8y',

(3)

not varied,

As

we can calculate variations by the familiar formulas and processes used in calculating differentials. Let a be an independent parameter. Then y f(x) + ccy (x), is of one a of curves 4ay, any ~f(x) family including y =/(#) (corresponding to or = 0) and y =/(#) 4- 17 (#) (corresponding to a: = 1). If a; and x l are fixed values, and if 2.

or y

^or

A

I (a) is a function of the parameter a only. necessary condition that /(#), a function of a single variable a} should be a maximum is /'(#) when a 0. or a minimum when #

=

-?'()

when a That

A

=

=

=

'

jf

* (, y

+ ^Sy,

y

r

+

= 0. ^'(0)=

is,

/

~4>dx.

necessary condition, then, that

maximum

or a

/

minimum when y =f(x)

(v.

<(#,

y,

y*}dx should be a f

is

I

1, (4))

S<^>(^,

?/,

y )dx

= 0.

APPENDIX B

116 I

taken as the definition of the variation of

<j>dx is

our necessary condition

How

cf>dx,

and

usually written

is

toward the determination of the form

this condition helps

of f(x) can be seen

/

from an example.

3. Let it be required to find the form of the shortest curve t/=f(x) joining two given points (# y ) and (xl9 y^). Here, since ,

ds

= Vl /*i

/=/ Vl Jx and J

is

to be

n

made a minimum.

r

S/

-r

= dx

/1

+y

12

y'4-sy J dx J

"'-

" 'i-

Vl + 2/2

3^CC

-r^* Integrating by

parts, this last reduces to

^ xo

since, as the

when #

=

sc

.

1

ends of the path are given,

Then S/

= ri

/

^_

Jxn

but since our 8y (that is,

77

8y

=

when x

=

ar

and

if

(x)), is

y_

x

=

;

a function which

is

wholly arbitrary,

APPENDIX B d the other factor, -

--

is to

ClX

,

.

.

*

i

must be equal

*

f~t

--'2

i

,

to zero if the integral

vanish.

=

J

This gives us

y

and the required curve 4.

117

is

C.

= cx

a straight

-f-

d

;

line.

In our more general problem it may be shown in like manner that

f

8

Jxn leads to a differential equation between

y and x and so determines y

as a function of x,

Of course 8/ =

not a sufficient condition for the existence of minimum and does not enable us to discriminate between maxima and minima, but like the necessary condition

~=

f)f

it

or a

maximum

for a

variable, 5.

is

maximum

either a

often

is

or

enough

minimum

to lead us to the solution of the problem.

Let us now generalize a

an independent variable Suppose we have a

value in a function of a single

r,

function

little.

and

let

Let x y y

x' =

?

dr

y

z,

y

1

-

-,

be functions of

= -^ dr

7 <

(r, #, 37, #,

,

re

#'

>

=

*

.

dr

-

,

?/',

).

',

the forms of the functions but holding r fast, let ^ e given the increments (r), 77 (r), (r),

By changing x

y

y> &9

>

x then becomes x -f- f (r), z/ becomes 2/ + ^ (r), # becomes # + ? (^"), c becomes x + becomes y' + rj'(r), ^ becomes # + ^ (?'), '(r), y of x and #' and are written &c are the variations (r), | (^), and 8x'. Obviously, d *

*

>

r

f

f

f

f

.

f

$x

= -7-

r

E.

c?r

The increment produced r

in

<

when

infinitesimal increments &r, and to their

to the dependent variables -, , 8y', , are given 8y, derivatives with respect to r is known to differ from -

$x

by terms of higher order than the variations involved. This approxiand is written B. It is mate increment is called the variation of <

APPENDIX B

118

found in any7 case precisely as

the complete differential of

d

found.

is

r

f

i -

B(r, x, y,

?

x

f

-

-

y'}

,

)

dr

=

is

a necessary condition

u

that

<(V

I

?

minimum /**

x, y,

)

x

r

can be established

)

by the reasoning used

^t

maximum

or a

in the case of

/^* **i

I

<^>

(a:,

y, T/)

c?cc.

The

integral

I

dr

S

is

called the variation

tV rn

v/ a* ,

/^ r i

/^ rl

of

^r should be a

-

y',

,

I

Jr

<^>

dr, so that

Jr

dr

/* ri

=

I

dr.

S<j>

-^ rO

Q

Q

6.

Si

It should be noted that our important formulas

d

- = Sax dr

SIr

and

r

*

dr

-

dr

6x

= c

ri

I

S<^> cZ?*,

hold only when r is the independent variable which is held fast when the forms of the functions are varied; that is, when Sr is supposed to be zero. .

If

x and y are functions of r and we need

directly thus:

dlf

If o* >-.

we need

get

it in-

~~

dx

to

S~y we

t

S

/

*^

<^>^ ?

x

f

we must change our

// <^

dx

8 Cdx

=

I

J

=$

variable of integration

^ c??\



c^r

f^dr.

Date Due

Carnegie Institute of Technology Library PITTSBURGH, PA.

>

CD

1

38 473

-<