Every Hamiltonian flow X_H preserves the symplectic form ω (hence all powers ω^n, including the phase-space volume ω^n/n!). Proven by Cartan's magic formula: L_{X_H}ω = d(ι_{X_H}ω) + ι_{X_H}dω = d(dH) + ι_{X_H}(0) = 0.
Every Hamiltonian flow X_H preserves the symplectic form ω (hence all powers ω^n, including the phase-space volume ω^n/n!). Proven by Cartan's magic formula: L_{X_H}ω = d(ι_{X_H}ω) + ι_{X_H}dω = d(dH) + ι_{X_H}(0) = 0.