Coordinates

Notes on Kinematics in Different Coordinate Systems Ulrike Zwiers Spring 2003

1

Cartesian Coordinates

As illustrated in Figure 1, the position of a point in space is defined in terms of Cartesian coordinates as   x  (1) r= y  . z Alternatively, the position vector r may be expressed with respect to the basis unit vectors ex , ey , ez defining the axes of the Cartesian coordinate system. As these basis unit vectors are time-invariant, i. e., they do not change with time, the position, velocity and acceleration of a particle in space are r(t) = xex + yey + zez , v(t) = xe ˙ x + ye ˙ y + ze ˙ z, a(t) = x¨ex + y¨ey + z¨ez ,

(2) (3) (4)

and the kinetic energy is 1 1 T = mv 2 = m(x˙ 2 + y˙ 2 + z˙ 2 ) . 2 2

2

(5)

Cylindrical Coordinates

Using a cylindrical coordinate system, the position of a particle in space is defined in terms of the cylindrical coordinates ρ, φ, z as   ρ cos φ (6) r =  ρ sin φ  , z

1

2

2 CYLINDRICAL COORDINATES

ez ez eφ r ρ φ

eρ z

ex y

ey x

Figure 1: Cartesian and cylindrical coordinates where ρ denotes the distance from the z axis to the point of interest, φ describes the angle from the positive xz plane to the vector defining the point with respect to the origin of the reference frame, and z coincides with the Cartesian z coordinate as illustrated in Figure 1. The basis unit vectors associated with a cylindrical coordinate system are eρ = cos φ ex + sin φ ey , eφ = − sin φ ex + cos φ ey , ez = ez ,

(7) (8) (9)

defining a right-handed, orthogonal reference frame. However, the basis unit vectors depend on time since their direction changes as the point moves. The time derivatives of the basis vectors are e˙ ρ = φ˙ eφ , e˙ φ = −φ˙ eρ , e˙ z = 0 ,

(10) (11) (12)

Thus, the position, velocity and acceleration of a particle in space expressed in terms of cylindrical coordinates are r(t) = ρ eρ + z ez , v(t) = ρ˙ eρ + ρφ˙ eφ + z˙ ez , ˙ eφ + z¨ ez , a(t) = (¨ ρ − ρφ˙ 2 ) eρ + (ρφ¨ + 2ρ˙ φ)

(13) (14) (15)

and the kinetic energy is 1 1 T = mv 2 = m(ρ˙ 2 + ρ2 φ˙ 2 + z˙ 2 ) . 2 2

(16)

3

3 SPHERICAL COORDINATES

ez eρ eφ r θ ρ ex

φ

eθ ey

Figure 2: Spherical coordinates

3

Spherical Coordinates

Using a spherical coordinate system, the position of a particle in space is defined in terms of the cylindrical coordinates r, θ, φ as   r sin θ cos φ (17) r =  r sin θ sin φ  , r cos θ where r denotes the distance from the origin of the reference frame to the point of interest, θ is the angle between the z axis and the position vector, and φ describes the angle from the positive xz plane to the plane in which the position vector is lying, as illustrated in Figure 2. The basis unit vectors associated with a spherical coordinate system are er = sin θ cos φ ex + sin θ sin φ ey + cos θ ez , eθ = cos θ cos φ ex + cos θ sin φ ey − sin θ ez , eφ = − sin φ ex + cos φ ey ,

(18) (19) (20)

defining a right-handed, orthogonal reference frame. However, the basis unit vectors depend on time since their direction changes as the point moves. The time derivatives of the basis vectors are e˙ r = θ˙ eθ + sin θφ˙ eφ , e˙ θ = −θ˙ er + cos θφ˙ eφ , e˙ φ = − sin θφ˙ er − cos θφ˙ eθ ,

(21) (22) (23)

Thus, the position, velocity and acceleration of a particle in space expressed in terms of cylindrical coordinates are r(t) = r er , v(t) = r˙ er + rθ˙ eθ + r sin θφ˙ eφ , a(t) = (¨ r − rθ˙2 − r sin2 θφ˙ 2 ) er + (rθ¨ + 2r˙ θ˙ − r sin θ cos θφ˙ 2 ) eθ + ˙ eφ , +(r sin θφ¨ + 2 sin θr˙ φ˙ + 2r cos θθ˙φ)

(24) (25) (26)

3 SPHERICAL COORDINATES

4

and the kinetic energy is 1 1 T = mv 2 = m(r˙ 2 + r2 θ˙2 + r˙ 2 sin2 θφ˙ 2 ) . 2 2

(27)