Derivation of the Ginzburg-Landau equations for a ferromagnetic $p$-wave superconductor

We derive a Ginzburg-Landau free energy for a $p$-wave ferromagnetic superconductor. The starting point is a microscopic Hamiltonian including a spin generalized BCS term and a Heisenberg exchange term. We find that coexistence of magnetization and superconductivity depends on the sign of the energy gradient of the DOS at Fermi level. We also compute the tunneling contribution to the Ginzburg-Landau free energy, and find expressions for the spin currents and Josephson currents across a tunneling junction separating two ferromagnetic $p$-wave superconductors.