Hamilton's principal function
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Hamilton's principal function is defined by the Hamilton–Jacobi equations (HJE), another alternative formulation of classical mechanics. This function S is related to the functional 
by fixing the initial time t1 and endpoint 
and allowing the upper limits t2 and the second endpoint 
to vary; these variables are the arguments of the function S. In other words, the action function S is the indefinite integral of the Lagrangian with respect to time.
