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.

Read more about this topic:  Hamiltonian Mechanics

Famous quotes containing the words charged, particle and/or field:

    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)

    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)

    The birds their quire apply; airs, vernal airs,
    Breathing the smell of field and grove, attune
    The trembling leaves, while universal Pan,
    Knit with the Graces and the Hours in dance,
    Led on th’ eternal Spring.
    John Milton (1608–1674)