Microcanonical Entropy and Dynamical Measure of Temperature for Systems with Two First Integrals

We consider a generic classical many particle system described by an autonomous Hamiltonian $H(x^{{}_1},\mathrm{.}\mathrm{.}\mathrm{.},x^{{}_{N+2}})$ which, in addition, has a conserved quantity $V(x^{{}_1},\mathrm{.}\mathrm{.}\mathrm{.},x^{{}_{N+2}})=v$, so that the Poisson bracket $\{H,V\}$ vanishes. We derive in detail the microcanonical expressions for entropy and temperature. We show that both of these quantities depend on multidimensional integrals over submanifolds given by the intersection of the constant energy hypersurfaces with those defined by $V(x^{{}_1},\mathrm{.}\mathrm{.}\mathrm{.},x^{{}_{N+2}})=v$. We show that temperature and higher order derivatives of entropy are microcanonical observable that, under the hypothesis of ergodicity, can be calculated as time averages of suitable functions. We derive the explicit expression of the function that gives the temperature.