History of variational principles in physics
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.
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.
This principle was developed by Johann Bernoulli in a letter to Pierre Varignon in 1715, but never separately published.
) is zero: A similar condition but valid for dynamics (systems in motion) introduces, for each force, the change in momentum
[1]: 580 In 1696 Johann Bernoulli posed a puzzle to European mathematicians: derive a curve for motion of a frictionless bead falling between a higher and a lower point in the least possible time.
He named the curve the "brachistochrone", (from brachystos, "shortest", and chronos, "time")[5]: 31 Isaac Newton, Gottfried Wilhelm Leibniz and others contributed solutions, and in 1718 Johann Bernoulli published an analysis based on the solution created by his brother James Bernoulli.
All of these works, especially the approach taken by the Bernoullis, involved reasoning about small deviations in the path taken by the falling bead.
[5]: 68 In 1744[6] and 1746,[7] Pierre Louis Maupertuis generalized Fermat's concept to mechanics,[8]: 97 in the form of a principle of least action.
Maupertuis argued metaphysically, he felt that "Nature is thrifty in all its actions", and applied the principle broadly: The laws of movement and of rest deduced from this principle being precisely the same as those observed in nature, we can admire the application of it to all phenomena.
The movement of animals, the vegetative growth of plants ... are only its consequences; and the spectacle of the universe becomes so much the grander, so much more beautiful, the worthier of its Author, when one knows that a small number of laws, most wisely established, suffice for all movements.This notion of Maupertuis, although somewhat deterministic today, does capture much of the essence of variational mechanics.
Euler continued to write on the topic; in his Reflexions sur quelques loix generales de la nature (1748), he called the quantity "effort".
Lagrange introduced the idea of variation of entire curves or paths between the endpoints than of individual coordinates.
[14] Among Jacobi's results was the extension of Hamilton's method to time-dependent potentials (or "force functions" as they were known at that time).
Variational principles played decisive roles at critical times in the development of quantum mechanics.
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.
In 1933 Paul Dirac published a paper seeking an alternative formulation based on Lagrangian mechanics.
[25]: 1025 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
[25]: 1027 In 1950, Julian Schwinger revisited Dirac's Lagrangian paper to develop the action principle in a different direction.
Its formulation started in the 1970s and has successfully explained almost all experimental results related to microscopic physics.
[27] This teleological viewpoint runs from the earliest physics through Fermat, Maupertuis, and on up to Max Planck, without, however, any scientific backing.
[21]: 162 The use of colorful language continues in the modern era with phrases like "Nature's command (to) Explore all paths!
[17]: II:19 Lord Rayleigh was the first to popularly adapt the variational principles for the search of eigenvalues and eigenvectors for the study of elasticity and classical waves in his 1877 Theory of Sound.
[29] The Rayleigh method allows approximation of the fundamental frequencies without full knowledge of the material composition and without the requirement of computational power.
[29] From 1903 to 1908, Walther Ritz introduced a series of improved methods for static and free vibration problems based on the optimization of an ansatz or trial function.
[30] In Russia, physicists like Ivan Bubnov (in 1913) and Boris Galerkin (in 1915) would rediscover and popularize some of Ritz's methods from 1908.
[29] In 1911, Rayleigh complemented Ritz for his method for solving Chladni's problem, but complained for the lack of citation of his earlier work.
The theorem was first discussed by Schrödinger in 1926, the first proof was given by Paul Güttinger in 1932, and later rediscovered independently by Wolfgang Pauli and Hans Hellmann in 1933, and by Feynman in 1939.
