Minimal coupling

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 onThis 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.
analytical mechanicsquantum field theoryfieldschargemultipole momentsmagnetic momentelectronLagrangianelectrodynamicsCartesian coordinatesSI Unitselectric chargeelectric scalar potentialmagnetic vector potentialEuler–Lagrange equationLorentz forcegauge transformationcanonical momentagauge invariantkinetic momentaHamiltonianLegendre transformationquantum mechanicswave functionrelativistic Lagrangianrest masstotal energypotential energycosmological inflationinflaton fieldscalar curvatureLorentz invariantmeasuremetricPlanck unitsgauge covariant derivativeBibcode