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,

Read more about this topic:  Hamiltonian Mechanics

Famous quotes containing the words charged and/or particle:

    God, who gave to him the lyre,
    Of all mortals the desire,
    For all breathing men’s behoof,
    Straitly charged him, “Sit aloof;”
    Annexed a warning, poets say,
    To the bright premium,—
    Ever, when twain together play,
    Shall the harp be dumb.
    Ralph Waldo Emerson (1803–1882)

    Experience is never limited, and it is never complete; it is an immense sensibility, a kind of huge spider-web of the finest silken threads suspended in the chamber of consciousness, and catching every air-borne particle in its tissue.
    Henry James (1843–1916)