Magnetism of matter and phase-space energy of charged particle systems

The (Daruin) Lagrangian, and energy, valid for systems of charged particles when radiation is negligible, are derived in a new way that avoids the usu al v/c-expansion. This shows more clearly their range of validity. Expressi ng the energy in terms of canonical momenta gives the corresponding Hamilto nian. When there are many particles it is intractable, but useful approxima tions are given and general conclusions about magnetism of matter are drawn from these. Macroscopic energy extremizing self-consistent vortex solution s are presented which can be interpreted as corresponding to superconductiv ity and ferromagnetism. There is a discussion of the quantum mechanics of t he Hamiltonian for conduction electrons in a metal and a phase transition i s predicted at low temperature.