Applying the principle of stationary action

The first step in applying the principle of stationary action is to generalize the variation of the action integral to include a variation of the time end-points and to retain the variations at the end-points in order to define the generator F t). We shall express the Lagrangian operator in terms of the complete set of 

To first-order in the infinitesimals this reduces to the change in along the varied path between the unvaried time end-points and the unvaried integrand times the variations in the time at the two time end-points. 

The variation of the Lagrangian ojjerator is formally identical to the variation of the classical action integral as developed in eqns (8.45)-(8.48). We may take over the final result in eqn (8.48) completely in its corresponding operator form, retaining in this case the terms involving Sq at the time endpoints as these variations are no longer required to vanish. The addition of this result to the end-point variations in eqn (8.82) yields, for the general 

The end-point variations can be recast into a more useful form. One first notes that the complete change in a coordinate operator at a time end-point, denoted by the symbol Aq , can arise as a result of a variation in the operator or as a result of a variation in the time 

In accordance with the classical definition of the Hamiltonian, eqn (8.52), and recalling that the derivative of the Lagrangian with respect to a velocity (eqn (8.51)) is the momentum conjugate to the corresponding coordinate, the Hamiltonian operator is defined as 

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