On the Extremal Properties of Hamilton’s Action Integral

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Abstract
This Note examines the sufficient conditions for the extremization, in particular the minimization, of the action integral in Hamilton’s principle for a one-degree-of-freedom nonlinear conservative system. It is usually stated in analytical mechanics that the action is actually minimized only over a short-time interval. Here the quantification of these statements is achieved by obtaining an upper bound for this minimizing interval. This is attained by combining results from the sufficiency variational calculus theory, with oscillation/comparison theorems from differential equations.

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