Conservation Laws for Gauge-Variant Lagrangians in Classical Mechanics

When a physical system has some symmetry properties, it is described by equations of motion invariant under the corresponding transformation group. Its Lagrangian however need not be invariant and may be “gauge-variant,” that is, vary by the addition of a total time derivative. A slightly generalized form of Noether's theorem nevertheless exists in such cases, still leading to conservation laws. The importance of considering such noninvariant Lagrangians and the associated conservation laws is illustrated by several examples: energy conservation, Galilean invariance, dynamical symmetries (harmonic oscillator and Kepler's problem), motion in a uniform electric field.