26. Introduction to Calculus of Variations#
Contents
Basics. Definition of a functional, definition of a variation, stationariety principle: Lagrange equations, Euler-Beltrami equation
Examples. Shortest path between two points; Fermat principle; Lagrangian formulation of classical mechanics, and special relativity
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Variational principle must have an invariant form, i.e. it must not depent on arbitrary choices of coordinates or parameters that have no geometrical or physical invariant nature
Parametrization of the integral with a parameters whose value depends on the integral itself. E.g.: arc-length parametrization of the length of a curve; arc-length parametrization of time in Fermat principle;…