8.3: Varying Both Ends
( \newcommand{\kernel}{\mathrm{null}\,}\)
The argument given above for the incremental change in action from varying the endpoint is clearly equally valid for varying the beginning point of the integral (there will be a sign change, of course), so
dS(q(2)i,t2,q(1)i,t1)=∑ip(2)idq(2)i−H(2)dt2−∑ip(1)idq(1)i+H(1)dt1
The initial and final coordinates and times specify the action and the time development of the system uniquely.
(Note: We’ll find this equation again in the section on canonical transformations -- the action will be seen there to be the generating function of the time-development canonical transformation, this will become clear when we get to it.)