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Concept
Variational principle
Summary
In science and especially in mathematical studies, a variational principle is one that enables a problem to be solved using calculus of variations, which concerns finding functions that optimize the values of quantities that depend on those functions. For example, the problem of determining the shape of a hanging chain suspended at both ends—a catenary—can be solved using variational calculus, and in this case, the variational principle is the following: The solution is a function that minimizes the gravitational potential energy of the chain.
Any physical law which can be expressed as a variational principle describes a self-adjoint operator. These expressions are also called Hermitian. Such an expression describes an invariant under a Hermitian transformation.
History of variational principles in physics
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.
The Rayleigh–Ritz method for solving boundary-value problems approximately
Ekeland's variational principle in mathematical optimization
The finite element method
The variation principle relating topological entropy and Kolmogorov-Sinai entropy.
Fermat's principle in geometrical optics
Maupertuis' principle in classical mechanics
The principle of least action in mechanics, electromagnetic theory, and quantum mechanics
The variational method in quantum mechanics
Gauss's principle of least constraint and Hertz's principle of least curvature
Hilbert's action principle in general relativity, leading to the Einstein field equations.
Official source
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