Constants of motion for a velocity-dependent potential

Abstract

Geometric continuous symmetries for a closed system of particles are investigated. It is supposed that the interactions are derivable from a velocity-dependent potential. Both the constraints on the form of the potential and the conservation principles resulting from the continuous space- time symmetries are derived. Darwin's Lagrangian is used as an illustration for the case of electric charges in slow motion in Maxwell-Lorentz's formulation of classical electrodynamics and the gauge-dependent linear momentum, angular momentum and energy are obtained.