Darwinian evolution: the mutation of a weakly relativistic lagrangian

The work studies Darwin’s order-(v/c)2 approximation to the relativistic interaction of classical charged particles. The first part presents an in- troduction to the methods of symplectic reduction in the Newtonian two- body problem and then applies these to a two-body Darwin interaction. The momentum-dependent interaction of the Darwin system plays an important role in the ability to reduce to a system of one degree of freedom. Circular orbits are sought, and it is shown that two of the three possible orbits are prohibited by velocity conditions. The second part of the work derives a self-consistent Darwin particle theory from a Lagrangian for electromagnetic fields coupled to particles. The resulting particle Lagrangian agrees with previous results. A similar procedure is followed to obtain a Low-Darwin system, coupling the self-consistent Darwin theory to the Vlasov equation.