1. Here and subsequently, the term “arbitrary function” is to be understood in a physical rather than a mathematical sense. It connotes a function which is arbitraryexceptfor such restrictions of continuity, differentiability, positive definiteness, etc., as may be necessary to exclude situations which are clearly not sensible in view of the physical significance of the function.
2. V. Volterra,Theory of Functionals(Blackie and Son, Limited, London, 1931), pp. 22–24. When the functionalRis just an integral, this reduces to the familiar variational derivative of the calculus of variations, (see Courant-Hilbert,Mathematische Physik I(Julius Springer, Berlin, 1931), second edition, p. 159).
3. P. A. M. Dirac,Quantum Mechanics(Clarendon Press, Oxford, England, 1947), third edition, p. 58.
4. It is understood that the limiting process implied in the use of the delta function is carried out only after taking the limit ɛ→0.
5. Starting with the straightforward expression for rb as a double integral over time, we reverse the order of integration. One of the integrations can then be carried out, resulting directly in Eq. (5). Alternatively, Eq. (5) may be verified by noting that it satisfies both the initial condition, r(0) = r0, and the relation drb∕dTb = vb.