Hamiltonian Mechanics - Relativistic Charged Particle in An Electromagnetic Field

Relativistic Charged Particle in An Electromagnetic Field

The Lagrangian for a relativistic charged particle is given by:

Thus the particle's canonical (total) momentum is

that is, the sum of the kinetic momentum and the potential momentum.

Solving for the velocity, we get

So the Hamiltonian is

From this we get the force equation (equivalent to the Euler–Lagrange equation)

from which one can derive

An equivalent expression for the Hamiltonian as function of the relativistic (kinetic) momentum, is

This has the advantage that can be measured experimentally whereas cannot. Notice that the Hamiltonian (total energy) can be viewed as the sum of the relativistic energy (kinetic+rest), plus the potential energy,