What is the definition of energy for quantum systems with time dependent Hamiltonians? Is it the eigenvalue of the Hamiltonian? (The eigenvalue is, in general, time dependent). However, the eigenstates of the Hamiltonian (even if it is time dependent) are stationary states, and hence no quantities must change with time. What is the reason for this inconsistency? This leads us to this general question in classical mechanics: What is the general definition of energy when non-conservative forces are present? We'd defined energy as a quantity that remains unchanged with the time translational invariance of the Lagrangian, but that does not hold when time dependent potentials are present.