Introduction

This chapter describes an elegant reformulation of the laws of Newtonian mechanics that is due to the French-Italian scientist Joseph Louis Lagrange (1736–1813). This reformulation is particularly useful for finding the equations of motion of complicated dynamical systems.

Generalized coordinates

Let the qi, for i = 1, ℱ, be a set of coordinates that uniquely specifies the instantaneous configuration of some dynamical system. Here, it is assumed that each of the qi can vary independently. The qi might be Cartesian coordinates, angles, or some mixture of both types of coordinate, and are therefore termed generalized coordinates. A dynamical system whose instantaneous configuration is fully specified by ℱ independent generalized coordinates is said to have ℱ degrees of freedom. For instance, the instantaneous position of a particle moving freely in three dimensions is completely specified by its three Cartesian coordinates, x, y, and z. Moreover, these coordinates are clearly independent of one another. Hence, a dynamical system consisting of a single particle moving freely in three dimensions has three degrees of freedom. If there are two freely moving particles then the system has six degrees of freedom, and so on.

Suppose that we have a dynamical system consisting of N particles moving freely in three dimensions. This is an ℱ = 3 N degree-of-freedom system whose instantaneous configuration can be specified by F Cartesian coordinates. Let us denote these coordinates the xj, for j = 1, ℱ.