9 - Analytical dynamics

Summary

The approach to classical dynamics proposed so far has been a more or less direct application of Newton's laws of motion. In such an approach the motion of a body or particle can be predicted on the assumption of a given set of external forces which act on it, simply by integrating the equations of motion. However, for complex systems of particles or rigid bodies, it is not always easy to determine appropriate equations for each component, let alone perform the required integration. In practice, using this approach, it is found that each individual type of problem requires its own particular insights and techniques.

In this chapter the Lagrangian approach to classical dynamics is developed. This approach is based upon two scalar properties of a system, its kinetic energy and work. It leads to a powerful and general method for the solution of dynamical problems which is found to be particularly useful in the analysis of mechanical systems which contain a number of rigid bodies that are connected in some way, but which may move relative to each other. In the traditional approach each component would have to be treated separately in terms of the forces acting on it. However, the Lagrangian approach enables such a system to be considered as a whole.

The aim here is to develop a general approach which may be applied to any dynamical system. It is found that the equations of motion can be presented in a standard and convenient form.