Second law of thermodynamics for macroscopic mechanics coupled to thermodynamic degrees of freedom

Based only on classical Hamiltonian dynamics, we prove the maximum work principle in a system where macroscopic dynamical degrees of freedom are intrinsically coupled to microscopic degrees of freedom. Unlike recent identities between irreversible work and free energy, such as in the Jarzynski relation, the macroscopic dynamics is not governed by an external action but undergoes the back reaction of the microscopic degrees of freedom. Our theorems cover such physical situations as impact between macroscopic bodies, thermodynamic machines, and molecular motors.