Classical confined phase volume

Set

 context $\langle \mathcal M, \mathcal H,\pi,\pi_0,{\hat\rho},{\hat\rho}_0\rangle$ … classical microcanonical ensemble context $\mathrm{dim}(\mathcal M) = 3N$ context $\hbar$ … Reduced Planck's constant
 definiendum $\varphi(E):=\frac{1}{h^{3N} N!}\int_{\{\langle{\bf q},{\bf p}\rangle\in \Gamma_{\mathcal M}\ |\ H({\bf q},{\bf p})\le E\}} \mathrm d\Gamma$