Generalized Galilei-Invariant Classical Mechanics

To describe the ``slow'' motions of n interacting mass points, we give the most general 4-d non-instantaneous, non-particle symmetric Galilei-invariant variational principle. It involves two-body invariants constructed from particle 4-positions and 4-velocities of the proper orthochronous inhomogeneous Galilei group. The resulting 4-d equations of motion and multiple-time conserved quantities involve integrals over the world lines of the other n-1 interacting particles. For a particular time-asymmetric retarded (advanced) interaction, we show the vanishing of all integrals over world-lines in the ten standard 4-d multiple-time conserved quantities, thus yielding a Newtonian-like initial value problem. This interaction gives 3-d non-instantaneous, non-particle symmetric, coupled non-linear second-order delay-differential equations of motion that involve only algebraic combinations of non-simultaneous particle positions, velocities, and accelerations. The ten 3-d non-instantaneous, non-particle symmetric conserved quantities involve only algebraic combinations of non-simultaneous particle positions and velocities. A two-body example with a generalized Newtonian gravity is provided. We suggest that this formalism might be useful as an alternative slow-motion mechanics for astrophysical applications.