Hamiltonian Mechanics

Abstract

We present Hamilton’s formulation of classical mechanics. In this formulation, the n second-order equations of motion of an n-dimensional mechanical system are replaced by an equivalent set of 2n first-order equations, known as Hamilton’s equations. There are problems where it is favorable to work with the 2n first-order equations instead of the corresponding n second-order equations. After introducing basic concepts from symplectic geometry, we consider the phase space of a mechanical system as a symplectic manifold. We then discuss the relation between Lagrangian and Hamiltonian systems. We show that, with appropriate assumptions, the Euler–Lagrange equations of a Lagrangian mechanical system are equivalent to Hamilton’s equations for a Hamiltonian, which can be obtained from the Lagrangian by a Legendre transformation. In the last part, we consider the linearization of mechanical systems as a way of obtaining approximate solutions in cases where the full non-linear equations of motion are too complicated to solve exactly. This is an important tool for analyzing physically realistic theories as these are often inherently non-linear.