In analytical mechanics, minimal coupling refers to a coupling between fields which involves only the charge distribution and not higher multipole moments of the charge distribution. This minimal coupling is in contrast to, for example, Pauli coupling, which includes the magnetic moment of an electron directly in the Lagrangian.
- Taken almost verbatim from Doughty's Lagrangian Interaction, pg. 456
See the Hamiltonian mechanics article for a full derivation and examples.
In studies of cosmological inflation, minimal coupling of a scalar field usually refers to minimal coupling to gravity. This means that the action for the inflaton field is not coupled to the scalar curvature. Its only coupling to gravity is the coupling to the Lorentz invariant measure constructed from the metric (in Planck units):
where , and utilizing the Gauge covariant derivative.