In classical mechanics, this value can be obtained by taking the Legendre transform of the Lagrangian.
The fact that this object commutes with the translation operator in a periodic system is a key step in the proof of Bloch's theorem. The negative derivative of this quantity with respect to p is equal to the time derivative of p, and the derivative of this quantity with respect to p is equal to the time derivative of q, in one system. The eigenvalues of this operator are assumed to change slowly under a perturbation in the adiabatic theorem. The time derivative of the action plus this quantity is equal to zero according to an equation partly named for Jacobi. In classical mechanics, this value can be obtained by taking the Legendre transform of the Lagrangian. For 10 points, name this mathematical function used to define a namesake type of mechanics, which corresponds to the total energy of a system.
The fact that this object commutes with the translation operator in a periodic system is a key step in the proof of Bloch's theorem.
Hamiltonian