  We study the quantum mechanical many-body problem of $N$ nonrelativistic
electrons interacting with their self-generated classical electromagnetic field
and $K$ static nuclei. The system of coupled equations governing the dynamics
of the electrons and their self-generated electromagnetic field is referred to
as the many-body Maxwell-Pauli equations. Here we construct time global,
finite-energy, weak solutions to the many-body Maxwell-Pauli equations under
the assumption that the fine structure constant $\alpha$ and the atomic numbers
are not too large. The particular assumptions on the size of $\alpha$ and the
atomic numbers ensure that we have energetic stability of the many-body Pauli
Hamiltonian, i.e., the ground state energy is finite and uniformly bounded
below with lower bound independent of the magnetic field and the positions of
the nuclei. This work serves as an initial step towards understanding the
connection between the energetic stability of matter and the wellposedness of
the corresponding dynamical equations.
