Notes.docx

Lagrangian Formulation of Mechanics Descriptions of Motion in Configuration Space

4 Introduction: We have gained considerable experience in setting up Newton’s equations of motion in a variety of problems. If the system is not subject to external constraints, the equations of motion are usually easy to set up in Cartesian coordinates. If either the system is subject to external constraints or Cartesian coordinates are not used, then the equations of motion may be difficult to solve or even to formulate. Lagrange found a way to circumvent this problem by the use of generalized coordinates qi (to be defined soon). In terms of these generalized coordinates, we could write the equations of motion in a form that is equally suitable for all coordinates. Furthermore, the introduction of generalized coordinates can take advantage of constraints on a dynamic system.

What are generalized coordinates? Any convenient set of parameters or quantities that can be used to specify the configuration or state of the system can be used as generalized coordinates. The generalized coordinates can be any quantities that can be observed to change with the motion of the system, and they need not be geometrical quantities (lengths or angles). In suitable circumstances, they can be electric currents (see Example 5.8). We shall write the generalized coordinates as qi, i = 1,2,3,…, n. It should be emphasized that we have generalized coordinates but not a generalized coordinate system. Degree Of Freedom ‘’The number of degrees of freedom is the smallest number of coordinates required to specify completely the configuration or state of the system.’’ For a free particle, the number of degree of freedom is 3. For a system of N particles that is free from constraints, a total of 3N coordinates qi (i = 1, 2,…, 3N) is required to describe its configuration completely, that is, the system has 3N degrees of freedom. But if the N particle system is subject to k constraints, the number of independent coordinates for the system is reduced to S =3N - k, and the system is said to have S degrees of freedom

How to summarize the idea of Generalized coordinates? (1) They are general in the sense that they need not be lengths or angles in particular. (2) They are in number just equal to the number of degrees of freedom of the system. (3) They are independent of one another.

Constraints When the motion of a dynamic system is not permitted to extend freely in three dimensions, the system is said to be subject to constraints.

Configuration Space The configuration of a system can be specified by the values of the n independent generalized coordinates q1, q2,…, qn. It is convenient to think of these n numbers as the coordinates of a single point in an n-dimensional space where the q’s form the n coordinate axes. This n-dimensional space is known as configuration space

Kinetic Energy in Generalized Coordinates For a system of N point particles in three spatial dimensions the kinetic energy expressed in Cartesian coordinates is always given by

with xi (i = 1, . . . , 3N) being the ith Cartesian component. For the masses mi an obvious notation has been employed. If there are R constraints on the system, the Cartesian coordinates may be replaced by f = 3N - R suitably chosen generalized coordinates qk

This coordinate transformation does not in general depend on the velocities q˙. Accordingly the total time derivatives of the Cartesian coordinates read

and depend on both the generalized coordinates and velocities and the time. Inserting this expression into (B.1) one obtains

This is the most general expression for the kinetic energy T in terms of the generalized coordinates. As soon as other than Cartesian coordinates are ployed, T may no longer depend only on the velocities but also on the coordinates itself. Because of its definition by Eq. (B.4) the mass matrix is symmetric, mkl = mlk Equation (B.4) is simplified to a large extent if the constraints do not explicitly depend on time, i.e., the coordinate transformation given by Eq. (B.2) itself does not explicitly depend on time,

Then the partial derivatives ∂xi/∂t vanish and the kinetic energy is given by

Virtual Displacement A virtual displacement of a system refers to a change in the configuration of the system as the result of any arbitrary infinitesimal change of the coordinate 𝜹ri consistent with the forces and constraints imposed on the system at a given instant of time t. A virtual displacement is an arbitrary, infinitesimal and instantaneous displacement of the coordinates that is consistent with the constraints.

Virtual displacement 𝜹xj will not correspond to any actual displacement dxj accruing during in dt when forces and constraints are changing.

Virtual Work

Virtual Work Principle ’’The principle of virtual work states that in equilibrium the virtual work of the forces applied to a system is zero.’’