  In the general case of a many-body Hamiltonian system, described by an
autonomous Hamiltonian $H$, and with $K\geq 0$ independent conserved
quantities, we derive the microcanonical thermodynamics. By a simple approach,
based on the differential geometry, we derive the microcanonical entropy and
the derivatives of the entropy with respect to the conserved quantities. In
such a way, we show that all the thermodynamical quantities, as the
temperature, the chemical potential or the specific heat, are measured as a
microcanonical average of the appropriate microscopic dynamical functions that
we have explicitly derived. Our method applies also in the case of
non-separable Hamiltonians, where the usual definition of kinetic temperature,
derived by the virial theorem, does not apply.
