By T. T. Taylor and D. ter Haar (Auth.)

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This is seen to be true because the latter are also expressions of the qj in terms of q p and t. 004) serves as a check on the derivation of the Hamiltonian for a particular system. Consider next the partial derivative of 76 with respect to a particular one of the generalized coordinates, say qy. 008) Notice that d/dqj is being taken in the Hamiltonian sense and that the result of this operation on the Lagrangian is represented by everything in the above equation to the right of the first minus sign.

2 a F This is quadratic in a and is minimum at a = 0 ; the definite action is clearly stationary for the true p a t h a n d is, in fact, least for this path. T o prove Hamilton's principle generally, one may write for the Lagrangian on the varied p a t h : <&k

Of great interest in this section is the definite action between an initial configuration C at time t and a final configuration C at time t , both of which lie on the same dynamic path. Q. 08 Fig. 13. The true (dynamic) path and a varied path in configuration space. in Fig. 13. Any other path such as P which connects the same two configurations (and same times) is called a "varied" p a t h ; varied paths which are quite close to the true path are most interesting in this context. ) T h e definite action along a varied path, such as P , is defined b y : a a AS = a j Mdt.

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