According to Cornelius Lanczos, any physical law which can be expressed as a variational principle describes an expression which is self-adjoint. 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 Einstein equation also involves a variational principle, according to Stephen Wolfram, (A New Kind of Science, p. 1052.), as a constraint on the Einstein-Hilbert Action.
Further readings
Epstein S T 1974 "The Variation Method in Quantum Chemistry". (New York: Academic)
Nesbet R K 2003 "Variational Principles and Methods In Theoretical Physics and Chemistry". (New York: Cambridge U.P.)
Adhikari S K 1998 "Variational Principles for the Numerical Solution of Scattering Problems". (New York: Wiley)
Gray C G, Karl G and Novikov V A 1996 Ann. Phys. 251 1.
See also
Action principle
History of physics
External links and references
Cornelius Lanczos, The Variational Principles of Mechanics
Stephen Wolfram, A New Kind of Science p. 1052
Gray, C.G., G. Karl, and V. A. Novikov, "Progress in Classical and Quantum Variational Principles (http://arxiv.org/abs/physics/0312071)". 11 Dec 2003. physics/0312071 Classical Physics.
Venables, John, "The Variational Principle and some applications (http://venables.asu.edu/quant/varprin.html)". Dept of Physics and Astronomy, Arizona State University, Tempe, Arizona (Graduate Course: Quantum Physics)
Williamson, Andrew James, "The Variational Principle (http://www.tcm.phy.cam.ac.uk/~ajw29/thesis/node15.html) -- Quantum monte carlo calculations of electronic excitations". Robinson College, Cambridge, Theory of Condensed Matter Group, Cavendish Laboratory. September 1996. (dissertation of Doctor of Philosophy)
Tokunaga, Kiyohisa, "Variational Principle for Electromagnetic Field (http://www.d3.dion.ne.jp/~kiyohisa/tieca/26.htm)". Total Integral for Electromagnetic Canonical Action, Part Two, Relativistic Canonical Theory of Electromagnetics, Chapter VI
Abstract: As is well known, a variational description of a system is very desirable both from mathematical and from physical points of view.
In the Herglotz variationalprinciple the functional, whose extrema are sought, is defined by a differential equation instead of the classical variational integral.
This variationalprinciple is important for a number of reasons.
The variationalprinciple formulated above is the basis of the variational method used in quantum mechanics and quantum chemistry to find approximations to the ground state.
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