History of variational principles in physics

In physics, a variational principle is an alternative method for determining the state or dynamics of a physical system, by identifying it as an extremum (minimum, maximum or saddle point) of a function or functional. Variational methods are exploited in many modern software to simulate matter and light. Since the development of analytical mechanics in the 18th century, the fundamental equations of physics have usually been established in terms of action principles, where the variational principle is applied to the action of a system in order to recover the fundamental equation of motion. This article describes the historical development of such action principles and other variational methods applied in physics. See History of physics for an overview and Outline of the history of physics for related histories.

Variational principles are found among earlier ideas in surveying and optics. The rope stretchers of ancient Egypt stretched corded ropes between two points to measure the path which minimized the distance of separation, and Claudius Ptolemy, in his Geographia (Bk 1, Ch 2), emphasized that one must correct for "deviations from a straight course"; in ancient Greece Euclid states in his Catoptrica that, for the path of light reflecting from a mirror, the angle of incidence equals the angle of reflection; and Hero of Alexandria later showed that this path was the shortest length and least time.

Other extremal principles of classical mechanics were formulated, such as Carl Friedrich Gauss's 1829 principle of least constraint and its corollary, Heinrich Hertz's 1896 principle of least curvature.

Variational principles played decisive roles at critical times in the development of quantum mechanics.

Following Max Planck's proposal that quantum radiators explain the blackbody radiation spectrum and Albert Einstein hypothesis of quantum radiation to explain the photoelectric effect, Niels Bohr proposed quantized energy levels for the orbits in his model of the atom, thereby explaining the Balmer series for absorption of radiation by atoms. However this hypothesis involved no mechanical model. Arnold Sommerfeld then showed that quantization of the action of orbits for Hydrogen predicted the Balmer series, complete with relativistic corrections leading to fine structure in spectral lines. However, this approach could not be extended to atoms with more electrons and, more fundamentally, the quantum hypothesis itself had no explanation from this classical mechanics solution. Dirac was able to show that the wavefunctions probability amplitude at was related to the amplitude at through an complex exponential function of the action.

In 1942, nearly a decade after Dirac's work, Richard Feynman built a new quantum mechanics formulation on the action principle. Feynman interpreted Dirac's formula as a physical recipe for the probability amplitude contributions from every possible path between and These possibilities interfere; constructive interference gives the paths with the most amplitude. In the classical limit with large values of action compared to , the single classical path given by the action principle results.

The breadth of physical phenomena subject to study by action principles lead scientists from all centuries to view these concepts as especially fundamental; the connection of two points by paths lead some to suggest a "purpose" to the selection of one particular path. This teleological viewpoint runs from the earliest physics through Fermat, Maupertuis, and on up to Max Planck, without, however, any scientific backing. The use of colorful language continues in the modern era with phrases like "Nature's command (to) Explore all paths!" or "It isn't that a particle takes the path of least action but that it smells all the paths in the neighborhood ". The Rayleigh method allows approximation of the fundamental frequencies without full knowledge of the material composition and without the requirement of computational power. In Russia, physicists like Ivan Bubnov (in 1913) and Boris Galerkin (in 1915) would rediscover and popularize some of Ritz's methods from 1908. In 1940, Georgii I. Petrov improved these approximations.