Hamilton’s Partial Differential Equation and its Extension to the Isoperimetric Problem

Abstract

Hamilton’s form of the differential equations of motion was derived in Lectures 8 and 9 from the principle that if the initial and final values of the coordinates are given, the variation of the integral ∫(T + U) dt must vanish. One can express this principle more generally so that it holds when not the initial and final values but other conditions which obtain at the limits are given. In this case, namely, it is not the entire variation of the integral ∫(T + U) dt which is to be set equal to zero, but only that part standing under the integral sign; the variation can then be expressed without the integral sign, or what is the same, the variation will be a total differential coefficient. In order to make this clear, we must go back to the derivation given in Lecture 8.