ABSTRACT
This chapter presents the governing equations for the nonlinear pendulum. The pendulum corresponds to a two dimensional Hamiltonian system in which there is exactly one generalized position and momentum variable, respectively. Hamilton’s equations of motion are the Euler-Lagrange equations associated with possible extremal solutions to the action. The Hamiltonian is related to the Lagrangian through the Legendre transformation. Equivalently, the Lagrangian is related to the Hamiltonian through the Legendre transformation. The least action and Hamiltonian descriptions of the evolution of the dependent variables are equivalent. The least action principle is a powerful tool in finite dimensional mechanics. Hamilton’s equations of motion for the pendulum represent a Lagrangian formulation of the dynamics. It is possible to re-write the dynamical system for the pendulum using a Poisson bracket formulation. A Casimir is a function of the dependent variables that Poisson commutes with every other function of the dependent variables.
