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, particle and/or field:

    When a man hath taken a new wife, he shall not go out to war, neither shall he be charged with any business: but he shall be free at home one year, and shall cheer up his wife which he hath taken.
    Bible: Hebrew Deuteronomy 24:5.

    Standing on the bare ground,—my head bathed by the blithe air, and uplifted into infinite space,—all mean egotism vanishes. I become a transparent eye-ball; I am nothing; I see all; the currents of the Universal Being circulate through me; I am part and particle of God.
    Ralph Waldo Emerson (1803–1882)

    We need a type of theatre which not only releases the feelings, insights and impulses possible within the particular historical field of human relations in which the action takes place, but employs and encourages those thoughts and feelings which help transform the field itself.
    Bertolt Brecht (1898–1956)