Hamilton Equations

In this chapter we introduce the Hamiltonian formalism of mechanics. After reviewing the Legendre transform, we deduce the canonical Hamilton equations of motion first from the Lagrangian ones and then from the action variational principle. We define the phase space and the Poisson bracket. We discuss in detail the connection between conservation laws and symmetries in the canonical framework; in this context we introduce the notion of Killing tensor and state some negative results. We end the chapter by proving the Liouville theorem on the canonical volume of phase space and the Poincaré theorem of eternal return.