  We present some novel thermodynamic ideas based on the Maupertuis principle.
By considering Hamiltonians written in terms of appropriate action-angle
variables we show that thermal states can be characterized by the action
variables and by their evolution in time when the system is nonintegrable. We
propose dynamical definitions for the equilibrium temperature and entropy as
well as an expression for the nonequilibrium entropy valid for isolated systems
with many degrees of freedom. This entropy is shown to increase in the
relaxation to equilibrium of macroscopic systems with short-range interactions,
which constitutes a dynamical justification of the Second Law of
Thermodynamics. Several examples are worked out to show that this formalism
yields the right microcanonical (equilibrium) quantities. The relevance of this
approach to nonequilibrium situations is illustrated with an application to a
network of coupled oscillators (Kuramoto model). We provide an expression for
the entropy production in this system finding that its positive value is
directly related to dissipation at the steady state in attaining order through
synchronization.
