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. This article describes the historical development of such principles.

Antiquity

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.

First variational principles

Principle of Virtual Work

In the static analysis of objects under forces but fixed at mechanical equilibrium, the principle of virtual work imagines tiny mathematical shifts away from equilibrium. Each shift does work -- energy lost or gained -- against the forces, but the sum of all these bits of virtual work must be zero. This principle was developed by Johann Bernoulli in a letter to Pierre Varignon in 1715, but never separately published. Cornelius Lanczos uses a slightly different definition as the single postulate for all analytic mechanics, showing thereby the power of energy based variational principles over Newtonian mechanics.

D'Alembert's principle

In 1743 Jean le Rond d'Alembert generalized the concept we now call virtual work to dynamical systems with rigid constraints, like rods or string under tension, a form that became known as the d'Alembert principle. In the case of static (in equilibrium) rigid bodies without friction, the principle of virtual work says the net work of all applied forces ($F_{i}^{(a)}$) under variation of positions ($r_{i}$) is zero:

$\Sigma _{i}F_{i}^{(a)}\cdot \delta r_{i}=0$

A similar condition but valid for dynamics (systems in motion) introduces, for each force, the change in momentum ${\dot {p}}_{i}$:

$\Sigma _{i}(F_{i}-{\dot {p}}_{i})\cdot \delta r_{i}=0$

which is d'Alembert's principle.

Principle of least time

The earlier geometrical ideas in optics were generalized 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. Fermat showed that principle predicts the observed law of refraction. His approach was metaphysical, arguing that Nature acts simply and economically.

The brachystochrone problem

The brachistochrone problem. The path of the least time shown in red.

Techniques based on small variations in the path of motion grew out of analysis of the brachistochrone problem.

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") 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. Thus this became the first application of the variational technique, albeit as a special-case rather than an general principle.

Principle of least action

Main article: Maupertuis principle

In 1744 and 1746, Pierre Louis Maupertuis generalized Fermat's concept to mechanics, 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.

— Pierre Louis Maupertuis

This notion of Maupertuis, although somewhat deterministic today, does capture much of the essence of variational mechanics.

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 action; his definitions of action varied with the problems he discussed. One form he used was called "vis viva",

Maupertuis' principle

$\delta \int 2T(t)dt=0$

which is the integral of twice what we now call the kinetic energy T of the system.

Euler's refinement

Leonhard Euler corresponded with Maupertuis from 1740 to 1744; in 1744 Euler proposed a refined 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 $v$ while being moved over a distance ds. The body will have a momentum $M{\sqrt {v}}$ that, when multiplied by the distance ds, will give $Mds{\sqrt {v}}$, 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 $\int Mds{\sqrt {v}}$ or, provided that M is constant, $\int ds{\sqrt {v}}$."

As Euler states, $\int Mds{\sqrt {v}}$ is the integral of the momentum over distance traveled (note that here $v$ contrary to usual notation denotes the squared velocity) which, in modern notation, equals the abbreviated action:

Euler's principle

$\delta \int p\,dq=0$

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."

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.

Lagrangian mechanics

The first use of the term "method of variations" came in 1755 through the work of a young Joseph Louis Lagrange; Euler presented Lagrange's approach to the Berlin Academy in 1756 as the "calculus of variations". Unlike Euler, Lagrange's approach was purely analytic rather than geometrical. Lagrange introduced the idea of variation of entire curves or paths between the endpoints than of individual coordinates. For this he introduced a new form of a differential, written $\delta$, that acts on integrals rather than $d$ acting on coordinates. His notation continues to be used today.

Hamilton-Jacobi mechanics

Main article: Hamilton's principle

See also: Hamilton–Jacobi equation

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 Lagrangian function $L=T-V$ (where T is the kinetic energy and V the potential energy of an object) to obtain what are now called the Euler–Lagrange equations. Hamilton believed his results were constrained by conservation of energy, which he called conservation of living force.

While few German scientists read English papers in this era, in 1836 the German mathematician Carl Gustav Jacobi read of Hamilton's work and immediately began new mathematical work, publishing ground breaking work on the variational principle in the following year. 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).

Extensions by Gauss and Hertz

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.

Modern action principles

Modern terminology

Action principles were developed by trial and error over three centuries; the names of the principles are not self-describing. Richard Feynman, through his PhD thesis and later through his reinvention of the undergraduate physics course, reinvigorated the field of variational principles in physics. In the process he upended the terminology. Feynman called Hamilton's principal function simply the "action" and Hamilton's principle he called "the principle of least action". The table below summarizes the key terminology found in modern physics literature.

Action principle terminology

action	historical name	modern name	principle	common name	Alternative name
$S=\int _{t_{1}}^{t_{2}}L\ dt$	Hamilton's principle function	action	$(\delta S)_{T}$	Hamilton's principle	Least action, Stationary action
$W=\int _{x_{1}}^{x_{2}}mv\ ds$	action	abbreviated or Maupertuis action	$(\delta W)_{E}$	Maupertuis's principle	Least action

The notation $(\delta S)_{T}$ means variations on $S$ with $T=t_{2}-t_{1}$ fixed; $(\delta W)_{E}$ means variation with constant energy. The abbreviated action is sometimes labeled $S_{0}$. Some authors use "stationary action" or "least action" to mean any variational principle involving action.

In relativity

In 1915 David Hilbert applied variational principles to derive the gravitational field equations of general relativity in agreement with Albert Einstein's independent derivation. Hilbert's approach required accepting the variational principle as "axiomatic", a broadly accepted requirement today but questionable to the physicists of 1915. Hilbert's variations were based on what became known as the Einstein–Hilbert action, given by

${\mathcal {S}}[g]={\frac {1}{2\kappa }}\int R{\sqrt {-g}}\,\mathrm {d} ^{4}x$,

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

In quantum mechanics

See also: History of quantum mechanics

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

Sommerfeld's atom

Main article: Bohr–Sommerfeld model

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, Neils 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.

Schrodinger's equation

Combining Einstein's relativity and photoelectric effect results, De Broglie suggested that Sommerfeld's quantized action may relate to quantized wave effects; Edwin Schrodinger took up this idea, applying Hamilton's optico-mechanical analogy to connect the quantized action to Hamilton-Jacobi equations for the action. Hamilton's connection between light rays and light waves now became a connection between matter trajectories and de Broglie matter waves. The resulting Schrodinger equation became the first successful quantum mechanics.

Dirac's quantum action

The work that built on Schrodinger's equation relied on analogies to Hamiltonian mechanics. In 1933 Paul Dirac published a paper seeking an alternative formulation based on Lagrangian mechanics. He was motivated by the power of the action principle and the relativistic invariance of the action itself. Dirac was able to show that the wavefunctions probability amplitude at $x_{1},t_{2}$ was related to the amplitude at $x_{2},t_{2}$ through an complex exponential function of the action.

Feynman's least action mechanics

Main article: Path integral formulation

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 $x_{1},t_{2}$ and $x_{2},t_{2}$. These possibilities interfere; constructive interference gives the paths with the most amplitude. In the classical limit with large values of action compared to $h$, the single classical path given by the action principle results.

Schwinger's quantum action principle

Main article: Schwinger's quantum action principle

In 1950, Julian Schwinger revisited Dirac's Lagrangian paper to develop the action principle in a different direction. Unlike Feynman's focus on paths, Schwinger's approach was "differential" or local.

In particle physics

The Standard Model is defined in terms of a Lagrangian density that includes all known elementary particles, the Higgs field and three of the fundamental interactions (electromagnetism, weak interaction and strong interaction, not including gravitational interaction). Its formulation started in the 1970s and has successfully explained almost all experimental results related to microscopic physics.

Teleology in action principles

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...".

Variational methods

In quantum chemistry and condensed matter physics, variational methods were developed to study atoms, molecules, nuclei and solids under a quantum mechanical framework. Some of these include 1909 Ritz method, 1930 Hartree–Fock method and 1992 density matrix renormalization group (DMRG).

In 2013, the variational quantum eigensolver was proposed for quantum computers, allowing noisy intermediate-scale quantum (NISQ) computers to exploit quantum phenomena to simulate atoms and small molecules using variational methods.

Variational methods are exploited in many modern software to simulate matter and light.