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.
== Overview ==
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 ==
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.
== Action principle ==
== Examples ==
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
