Hamiltonian Mechanics - Charged Particle in An Electromagnetic Field

Charged Particle in An Electromagnetic Field

A good illustration of Hamiltonian mechanics is given by the Hamiltonian of a charged particle in an electromagnetic field. In Cartesian coordinates (i.e. ), the Lagrangian of a non-relativistic classical particle in an electromagnetic field is (in SI Units):

where e is the electric charge of the particle (not necessarily the electron charge), is the electric scalar potential, and the are the components of the magnetic vector potential (these may be modified through a gauge transformation). This is called minimal coupling.

The generalized momenta may be derived by:

Rearranging, we may express the velocities in terms of the momenta, as:

If we substitute the definition of the momenta, and the definitions of the velocities in terms of the momenta, into the definition of the Hamiltonian given above, and then simplify and rearrange, we get:

This equation is used frequently in quantum mechanics.