Concisely -- Lagrangian Versus Hamiltonian Operator

The Lagrangian is the gauge-metric of a state or system of energy, whereas -- the Hamiltonian operator is the metric-gauge of a state or system of energy.  Therefore -- the Lagrangian is the actual motion of a state or system of energy, whereas -- the Hamiltonian operator is the actual holonomic substrate (the actual substirngular physical entity) of a state or system of energy.
To Be Continued!  Sam Roach.