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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 $K\geq 0$ 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.