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.
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.
Statements of variational principles have been rewarded by the Fermat Prize.
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.
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- Lord Rayleigh's variational principle
- Ekeland's variational principle
- Fermat's principle in geometrical optics
- The principle of least action in mechanics, electromagnetic theory, and quantum mechanics
- Maupertuis' principle in classical mechanics
- The Einstein equation also involves a variational principle, the Einstein–Hilbert action
- Gauss's principle of least constraint
- Hertz's principle of least curvature
- Palatini variation
- The variational method in quantum mechanics
- The finite element method
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