History of variational principles in physics

The earlier ideas of variational principles in optics were generalized to refraction by Pierre de Fermat, who, in the 17th century, refined the principle to "light travels between two given points along the path of shortest time"; now known as the principle of least time or Fermat's principle.

Its generalization to mechanics, the principle of least action is commonly attributed to Pierre Louis Maupertuis, who wrote about it in 1744 and 1746, although the true priority is less clear. In application to physics, Maupertuis suggested that the quantity to be minimized was the product of the duration (time) of movement within a system by the "vis viva", twice what we now call the kinetic energy of the system.

Leonhard Euler gave a formulation of the least action principle in 1744. He writes

"Let the mass of the projectile be M, and let its squared velocity resulting from its height be ${\displaystyle v}$ while being moved over a distance ds. The body will have a momentum ${\displaystyle M{\sqrt {v}}}$ that, when multiplied by the distance ds, will give ${\displaystyle Mds{\sqrt {v}}}$, the momentum of the body integrated over the distance ds. Now I assert that the curve thus described by the body to be the curve (from among all other curves connecting the same endpoints) that minimizes ${\displaystyle \int Mds{\sqrt {v}}}$ or, provided that M is constant, ${\displaystyle \int ds{\sqrt {v}}}$."[Note 1]

As Euler states, ${\displaystyle \int Mds{\sqrt {v}}}$ is the integral of the momentum over distance traveled (note that here ${\displaystyle v}$ contrary to usual notation denotes the squared velocity) which, in modern notation, equals the reduced action ${\displaystyle \int p\,dq}$. Thus, Euler made an equivalent and (apparently) independent statement of the variational principle in the same year as Maupertuis, albeit slightly later. In rather general terms he wrote that "Since the fabric of the Universe is most perfect and is the work of a most wise Creator, nothing whatsoever takes place in the Universe in which some relation of maximum and minimum does not appear." However, Euler did not claim any priority, as the following episode shows.

Maupertuis' priority was disputed in 1751 by the mathematician Samuel König, who claimed that it had been invented by Gottfried Leibniz in 1707. Although similar to many of Leibniz's arguments, the principle itself has not been documented in Leibniz's works. König himself showed a copy of a 1707 letter from Leibniz to Jacob Hermann with the principle, but the original letter has been lost. In contentious proceedings, König was accused of forgery, and even the King of Prussia entered the debate, defending Maupertuis, while Voltaire defended König. Euler, rather than claiming priority, was a staunch defender of Maupertuis, and Euler himself prosecuted König for forgery before the Berlin Academy on 13 April 1752. The claims of forgery were re-examined 150 years later, and archival work by C.I. Gerhardt in 1898 and W. Kabitz in 1913 uncovered other copies of the letter, and three others cited by König, in the Bernoulli archives.

Euler continued to write on the topic; in his Reflexions sur quelques loix generales de la nature (1748), he called the quantity "effort". His expression corresponds to what we would now call potential energy, so that his statement of least action in statics is equivalent to the principle that a system of bodies at rest will adopt a configuration that minimizes total potential energy.

The full importance of the principle to mechanics was stated by Joseph Louis Lagrange in 1760, although the variational principle was not used to derive the equations of motion until almost 75 years later, when William Rowan Hamilton in 1834 and 1835 applied the variational principle to the function ${\displaystyle L=T-V}$ to obtain what are now called the Euler–Lagrange equations.

In 1842, Carl Gustav Jacobi tackled the problem of whether the variational principle found minima or other extrema (e.g. a saddle point); most of his work focused on geodesics on two-dimensional surfaces. The first clear general statements were given by Marston Morse in the 1920s and 1930s, leading to what is now known as Morse theory. For example, Morse showed that the number of conjugate points in a trajectory equaled the number of negative eigenvalues in the second variation of the Lagrangian.

Other extremal principles of classical mechanics have been formulated, such as Gauss' principle of least constraint and its corollary, Hertz's principle of least curvature.

The Einstein–Hilbert action was proposed in 1915 by David Hilbert to reproduce the vacuum Einstein field equations of general relativity, given by

${\displaystyle {\mathcal {S}}[g]={\frac {c^{4}}{16\pi G}}\int _{\mathcal {M}}R{\sqrt {-g}}\,\mathrm {d} ^{4}x}$,

where ${\displaystyle g=\det(g_{\alpha \beta })}$ is the determinant of a spacetime Lorentz metric and ${\displaystyle R}$ is the scalar curvature.

Several approaches were also considered for quantum mechanics. Inspired by 1933 paper by Paul Dirac, Richard Feynman developed 1948 the path integral formulation, that linked the Lagrangian formalism with the Hamiltonian formalism of quantum mechanics.

In quantum chemistry and condensed matter physics, variational methods were developed to study atoms, molecules, nuclei and solids under a quantum mechanical framework. These include Hartree–Fock method, the density matrix renormalization group, and the Ritz method.