In this chapter we begin by considering canonical transformations. These are transformations that preserve the form of Hamilton's equations. This is followed by a study of Poisson brackets, an important tool for studying canonical transformations. Finally we consider infinitesimal canonical transformations and, as an example, we look at angular momentum in terms of Poisson brackets.

Integrating the equations of motion

In our study of analytical mechanics we have seen that the variational principle leads to two different sets of equations of motion. The first set consists of the Lagrange equations and the second set consists of Hamilton's canonical equations. Lagrange's equations are a set of n coupled second-order differential equations and Hamilton's equations are a set of 2n coupled first-order differential equations.

The ultimate goal of any dynamical theory is to obtain a general solution for the equations of motion. In Lagrangian dynamics this requires integrating the equations of motion twice. This is often quite difficult because the Lagrangian (and hence the equations of motion) depends not only on the coordinates but also on their derivatives (the velocities). There is no known general method for integrating these equations. You might wonder if it is possible to transform to a new set of coordinates in which the equations of motion are simpler and easier to integrate. Indeed, this is possible in some situations.