The Variational Principle is a fundamental concept in physics and mathematics that deals with finding extrema (minimum or maximum values) of functionals, which are mappings from a space of functions to real numbers. It is widely used in various fields including mechanics, quantum mechanics, and calculus of variations.
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A variational principle specifies a function, path or field by requiring a functional to be stationary under an allowed class of variations. The admissible class includes the prescribed boundary conditions and constraints. A stationary value need not be a minimum: the Euler-Lagrange equation expresses the first-order stationarity condition, while a minimum requires additional analysis. Fermat principle applies this idea to optical path length, and the Rayleigh-Ritz method approximates a variational problem in a chosen family of trial functions.