In analytical mechanics and quantum field theory, minimal coupling refers to a coupling between fields which involves only the charge distribution and not higher multipole moments of the charge distribution.
Higher moments of particles are consequences of minimal coupling and non-zero spin.
In Cartesian coordinates, the Lagrangian of a non-relativistic classical particle in an electromagnetic field is (in SI Units): where q is the electric charge of the particle, φ is the electric scalar potential, and the Ai, i = 1, 2, 3, are the components of the magnetic vector potential that may all explicitly depend on
This Lagrangian, combined with Euler–Lagrange equation, produces the Lorentz force law and is called minimal coupling.
Note that the values of scalar potential and vector potential would change during a gauge transformation,[2] and the Lagrangian itself will pick up extra terms as well, but the extra terms in the Lagrangian add up to a total time derivative of a scalar function, and therefore still produce the same Euler–Lagrange equation.
The Hamiltonian, as the Legendre transformation of the Lagrangian, is therefore This equation is used frequently in quantum mechanics.
Solving for the velocity, we get So the Hamiltonian is This results in the force equation (equivalent to the Euler–Lagrange equation) from which one can derive The above derivation makes use of the vector calculus identity: An equivalent expression for the Hamiltonian as function of the relativistic (kinetic) momentum, P = γmẋ(t) = p - qA, is This has the advantage that kinetic momentum P can be measured experimentally whereas canonical momentum p cannot.