Hamiltonian Mechanics

The Hamiltonian Formulation of Mechanics is equivalent to Newton's Laws and to the Lagrangian Formulation. Like the Lagrangian Formulation, one can use generalized coordinates with the Hamiltonian, however, the Hamiltonian is written in terms of coordinates and their conjugate momenta rather than the coordinates and their time derivatives as with the Lagrangian.

The coordinate and conjugate momentum plays an important role in the quantization of both mechanical and other problems so we will find that the Hamiltonian is very commonly used in Quantum Mechanics, where the commutation relations be coordinates and their conjugate momenta form the basis for quantization. In addition, for most problems, the Hamiltonian is the (conserved) total energy of the system.

The Hamiltonian formalism is another tool with which physicists can solve complex problems. An example of this is Liouville's theorem discussed in this section. It is necessary to understand many advanced textbooks.