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 such functions which optimize the values of quantities that depend upon 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.[verification needed] These expressions are also called Hermitian. Such an expression describes an invariant under a Hermitian transformation.
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.
- Palatini variation
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