Hamiltonian energy balance ($\frac{d}{dt}H(t,X,V)={\partial }_{t}H(t,X,V)$)

Along Hamilton's equations, spatial derivatives of the Hamiltonian function cancel in its total time derivative. Thus $dH/dt={\partial }_{t}H$. An autonomous Hamiltonian function is conserved; explicit time dependence generally prevents conservation even though phase space is still preserved.