Geometric microcanonical thermodynamics for systems with first integrals

In the general case of a many-body Hamiltonian system described by an autonomous Hamiltonian H and with K?0 independent conserved quantities, we derive the microcanonical thermodynamics. Using simple approach, based on 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, such as the temperature, the chemical potential, and 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 nonseparable Hamiltonians, where the usual definition of kinetic temperature, derived by the virial theorem, does not apply.