Properties of the quantum mechanical Lagrangian

The quantum mechanical Lagrangian density for a system of many particles interacting via a many-particle potential energy operator V is 

That this is the correct Lagrangian density is demonstrated by showing that the variation of the resulting action integral with respect to T and T yields 

There is no loss in generality by considering a Lagrangian density for a single particle in this demonstration. The variation of using the methods 

We may retain a formal analogy with the variation of the classical Lagrangian (eqns (8.48) and (8.49)) and with the variation of the quantum Lagrangian integral operator (eqn (8.83)) by defining the functional derivative of with respect to T (and correspondingly P ) to be 

The variation of with respect to 4 is already of the same form for both the classical and quantum cases. 

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