Equations of Motion - Electrodynamics

Electrodynamics

In electrodynamics, the force on a charged particle of charge q is the Lorentz force:

Combining with Newton's 2nd law gives a first order differential equation of motion, in terms of position of the particle:

or its momentum:

The same equation can be obtained using the Lagrangian (and applying Lagrange's equations above) for a charged particle of mass m and charge q:

where A and ϕ are the electromagnetic scalar and vector potential fields. The Lagrangian indicates an additional detail: the canonical momentum in Lagrangian mechanics is given by:

instead of just mv, implying the motion of a charged particle is fundamentally determined by the mass and charge of the particle. The Lagrangian expression was first used to derive the force equation.

Alternatively the Hamiltonian (and substituting into the equations):

can derive the Lorentz force equation.