Principles of statistical mechanics: The energy duality

We argue that statistical mechanics of systems with relaxation implies breaking the energy function of systems into two having different transformation rules. With this duality the energy approach incorporates the generalized vortex forces. We show general theorems and their implications and apply the approach to the particle confinement in fields of rotational symmetry. Misleading associations with the theories of quasienergy and generalized thermodynamic potential are exposed.