Consider the functional with and fixed

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Variational and asymptotic methods · Variational and asymptotic methods

Action and Symmetry, Ch The action, the first variation and the boundary term

Problem

Consider the functional S[q]=∫0Tq˙2 dtS[q] = \int_0^T \dot q^2\,\mathrm{d}t with q(0)q(0) and q(T)q(T) fixed. Someone computes its first variation as follows: "Write q→q+δqq \to q + \delta q, so q˙→q˙+δq˙\dot q \to \dot q + \delta\dot q and the integrand becomes (q˙+δq˙)2(\dot q + \delta\dot q)^2. Expanding, the change in the integrand is δ(q˙2)=(δq˙)2\delta(\dot q^2) = (\delta\dot q)^2. Hence δS=∫0T(δq˙)2 dt≥0\delta S = \int_0^T (\delta\dot q)^2\,\mathrm{d}t \ge 0 always, with no term linear in δq\delta q; the first variation of SS is identically zero, so every path with the given endpoints is stationary." ((a)) Find the error in this working. ((b)) Give the correct first variation δS\delta S (with its boundary term) and the resulting Euler-Lagrange equation.

Browse the problem library