Abstract
The (Darwin) Lagrangian, and energy, valid for systems of charged particles when radiation is negligible, are derived in a new way that avoids the usual 
-expansion. This shows more clearly their range of validity. Expressing the energy in terms of canonical momenta gives the corresponding Hamiltonian. When there are many particles it is intractable, but useful approximations are given and general conclusions about magnetism of matter are drawn from these. Macroscopic energy extremizing self-consistent vortex solutions are presented which can be interpreted as corresponding to superconductivity and ferromagnetism. There is a discussion of the quantum mechanics of the Hamiltonian for conduction electrons in a metal and a phase transition is predicted at low temperature.