A variational principle is a scientific principle used within the calculus of variations, which develops general methods for finding functions which extremize the value of quantities that depend upon those functions. For example, to answer this question: "What is the shape of a chain suspended at both ends?" we can use the variational principle that the shape must minimize the gravitational potential energy.
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Overview
Any physical law which can be expressed as a variational principle describes a self-adjoint operator (according to Cornelius Lanczos). These expressions are also called Hermitian. Such an expression describes an invariant under a Hermitian transformation.
Statements of variational principles are rewarded by the Fermat Prize.
History
Felix Klein's Erlangen program attempted to identify such invariants under a group of transformations. In what is referred to in physics as Noether's theorem, the Poincaré group of transformations (what is now called a gauge group) for general relativity defines symmetries under a group of transformations which depend on a variational principle, or action principle.