Hamiltonian Description

In this chapter we introduce a new way of deriving the equations of motion for a physical system, one that results in 2f first-order differential equations for a system with f degrees of freedom—the Hamilton equations—rather than f second-order equations from the Euler—Lagrange equations resulting from Hamilton’s principle. The advantage of this formulation is that it further generalizes the description of the physical system by using the generalized momenta p i and coordinates q i , with i = 1...f, instead of the coordinates and their derivatives, the velocities. In this way we describe the system with 2f independent variables, and we can represent the evolution of the system with a trajectory in the 2f-dimensional phase space defined by the q i and p i Each set of initial conditions will identify a trajectory, and a set of systems with different initial conditions will be identified by a family of trajectories. This representation leads to a geometric description of the dynamics of the system that permits one to identify easily some important characteristics, such as stability with respect to small perturbations of the initial conditions. This geometrical description is also well suited for the description of a system subject to nonlinear forces, as we will see in some of the examples.