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Principle of least action

Maupertuis Fermat Lagrange Euler Leibniz Hamilton Morse

In physics, the principle
of least action or principle of stationary action
is a variational principle by which, when applied to the
action of a mechanical
system, can be used to obtain the equations of motion
for that system. The principle led to the development of the Lagrangian and Hamiltonian
formulations of classical mechanics. The principle
remains central in modern physics and mathematics, being
applied in the theory of relativity, quantum mechanics and quantum field theory, and a focus of modern
mathematical investigation in Morse theory. This article
deals primarily with the historical development of the idea; a treatment of the
mathematical description and derivation can be found in the article on the action.

The action principle is preceded by earlier ideas in optics,
dating back to Ancient Greece. The credit for the
formulation of the principle as it applies to the action is often given to Pierre-Louis Moreau de Maupertuis, who wrote
about it in 1744[1] and 1746[2].
However, scholarship indicates that this claim of priority is not so clear; Leonhard Euler discussed the principle in 1744[3], and there is evidence that Gottfried
preceded both by 39 years[4][5][6].

Origins, statement, and dispute

The earliest precedents of the principle of least
can be found in studies of the specular reflection and refraction
of light. Hero of Alexandria
noted that the law of reflection
?i = ?r follows from
the assumption that light travels along the shortest distance between two given points. This was
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. Over the centuries, other natural
philosophers, ranging from Socrates to Leibniz have suggested that Nature
acts by the most economical means, although this concept was not defined mathematically.

Credit for the formulation of the principle of
least action
is commonly given to Pierre Louis Maupertuis, who wrote about it in
1744[1] and 1746[2], although the
true priority is less clear, as discussed below.

Maupertuis 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 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 “vis viva“, twice what we now call the kinetic energy of the

Euler’s formulation

Leonhard Euler gave a formulation of the action
principle in 1744, in very recognizable terms, in the Additamentum 2 to his
“Methodus Inveniendi Lineas Curvas Maximi Minive Proprietate Gaudentes”[3]. He begins the second paragraph [8]:

“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 that, when multiplied by the distance ds, will give , 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 or, provided that M is constant, .”

As Euler states, is the integral of the momentum over distance travelled which, in
modern notation, equals the reduced action . Thus,
Euler made an equivalent and (apparently) independent statement of the variational
principle in the same year as Maupertuis, albeit slightly later. Curiously, 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[4], and even the King of Prussia entered the debate, defending
Mauperius, 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.[4]. The claims of forgery were re-examined 150 years later, and
archival work by C.I. Gerhardt
in 1898[5] and W. Kabitz in 1913[6] uncovered other copies of the letter, and three others cited
by König, in the Bernoulli archives.

Further development

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 (need ref), 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 [9]applied the variational principle to the function L
= TV to obtain what are now called the Lagrangian equations of motion.

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 [10]. The first clear general statements
were given by Marston Morse in the 1920’s and 1930’s, [11] leading to what is now known as Morse
. For example, Morse showed that the number of conjugate points in a trajectory
equalled 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.

Apparent teleology

The mathematical equivalence of the differential
equations of motion and their integral counterpart has important philosophical
implications. The differential equations are statements about quantities localized to a
single point in space or single moment of time. For example, Newton’s second law F = ma states
that the instantaneous force F applied to a mass m produces an
acceleration a at the same instant. By contrast, the action principle is not
localized to a point; rather, it involves integrals over an interval of time and (for
fields) extended region of space. Moreover, in the usual formulation of classical action principles, the initial and final
states of the system are fixed, e.g.,

Given that the particle begins at position x1 at time t1 and ends
at position x2 at time t2, the physical trajectory that connects
these two endpoints is an extremum of the action integral.

In particular, the fixing of the final state appears to give the action
principle a teleological character which has been
controversial historically. This apparent teleology is
eliminated in the quantum mechanical version of the
action principle.


  1. P.L.M. de Maupertuis, Accord
    de différentes lois de la nature qui avaient jusqu’ici paru incompatibles.
    Mém. As. Sc. Paris p. 417. (English
  2. P.L.M. de Maupertuis, Le
    lois de mouvement et du repos, déduites d’un principe de métaphysique.
    Mém. Ac. Berlin, p. 267.(English
  3. Leonhard Euler, Methodus Inveniendi Lineas Curvas Maximi Minive Proprietate
    (1744) Bousquet, Lausanne & Geneva. 320 pages. Reprinted in Leonhardi
    Euleri Opera Omnia: Series I vol 24.
    (1952) C. Cartheodory (ed.) Orell Fuessli,
    Zurich. scanned copy of
    complete text
    at The Euler Archive,
  4. J J O’Connor and E F Robertson, “The
    Berlin Academy and forgery
    “, (2003), at The MacTutor History of
    Mathematics archive
  5. Gerhardt CI. (1898) “Über die vier Briefe von Leibniz, die Samuel König in dem
    Appel au public, Leide MDCCLIII, veröffentlicht hat”, Sitzungsberichte der
    Königlich Preussischen Akademie der Wissenschaften
    I, 419-427.
  6. Kabitz W. (1913) “Über eine in Gotha aufgefundene Abschrift des von S. König in
    seinem Streite mit Maupertuis und der Akademie veröffentlichten, seinerzeit für unecht
    erklärten Leibnizbriefes”, Sitzungsberichte der Königlich Preussischen Akademie
    der Wissenschaften
    II, 632-638.
  7. Chris Davis. Idle
  8. Euler, Additamentum
    (external link),
    ibid. (English
  9. W.R. Hamilton, “On a General Method in Dynamics.”, Philosophical
    Transaction of the Royal Society
    Part I (1834) p.247-308; Part II (1835) p. 95-144. (From
    the collection Sir William Rowan Hamilton
    (1805-1865): Mathematical Papers
    edited by David R. Wilkins, School of Mathematics,
    Trinity College, Dublin 2, Ireland. (2000); also reviewed as On a General Method
    in Dynamics

G.C.J. Jacobi, Vorlesungen über Dynamik, gehalten an der Universität
Königsberg im Wintersemester 1842-1843
. A. Clebsch (ed.) (1866); Reimer; Berlin. 290
pages, available online Œuvres
complètes volume
8 at Gallica-Math
from the Gallica Bibliothèque nationale de France.

10. Marston Morse (1934). “The Calculus of Variations in the Large”, American
Mathematical Society Colloquium Publication
18; New York.

Action (physics)

In physics, the action is an integral quantity that is used
to determine the evolution of a physical system between two defined states using the calculus of variations. Several different
definitions of the action are in common use in physics, as described below. The action is
usually an integral over time, but may be integrated over spatial variables as well (for
action pertaining to fields); in still other cases,
the action is integrated along the path followed by the physical system.

The evolution of a physical system between two states is determined by requiring the
action be minimized or, more generally, be stationary
for small perturbations about the true evolution. This requirement leads to differential
equations that describe the true evolution. Conversely, an
action principle is a
method for reformulating differential equations of motion for a physical system as an
equivalent integral equation. Although
several variants have been defined (see below), the most commonly used action principle is
Hamilton’s principle. An earlier, less
informative action principle is Maupertuis’
, which is sometimes called by its (less correct) historical name, the principle of least action.

The differential equations of motion for any physical system can be
re-formulated as an equivalent integral equation. Hence, Hamilton’s principle applies not
only to the classical mechanics of a single
particle, but also to classical fields such as the electromagnetic and gravitational
fields. Hamilton’s principle has also been extended to quantum
and quantum field theory.

“action” in classical physics

In classical physics, the term action has at
least eight distinct meanings.

The most common meaning of action in classical physics is a functional of the physical system. A functional
is a mathematical entity that returns a single number (i.e., a scalar) from an input function that describes the system completely as a
function of time and (for fields) space. For example,
in classical mechanics, the input function is the
evolution of the system between two time points t1 and t2,
where represent the generalized coordinates.
The action is defined as the integral of the Lagrangian L for an input evolution between the two
time points where the endpoints and of the evolution are fixed. According to Hamilton’s principle, the true evolution is an evolution for which the action is stationary (a
minimum, maximum, or a saddle point). This principle results in the equations of motion in Lagrangian mechanics.

A second meaning of action in classical physics is the abbreviated action
(usually denoted as ), which is also a functional. Here the
input function is the path followed by the physical system without regard to its
parameterization by time. For example, the path of a planetary orbit is an ellipse, and
the path of a particle in a uniform gravitational field is a parabola; in both cases, the
path does not depend on how fast the particle traverses the path. The abbreviated action
is defined as the integral of the generalized momenta along a path in the generalized coordinates

According to Maupertuis’ principle, the true
path is a path for which the abbreviated action is stationary.

A third, much less common meaning of action in classical physics is the function
(not functional) S that is defined by the Hamilton-Jacobi equations, another alternative
formulation of classical mechanics. This function
S (Hamilton’s principal function) is related to the functional by fixing the
initial time t1 and endpoint and allowing the upper limits t2
and the second endpoint to vary; these variables are the arguments of the function S. In other words,
the action function S is the indefinite integral
of the Lagrangian with respect to time. For physical systems that conserve the total
energy E, another action function is often defined: Hamilton’s characteristic
function . The Hamilton-Jacobi equations are
often solved by additive separability; in some cases, the individual terms of the
solution, e.g., Sk(qk), are also called an
“action”. These three definitions of “action” are relatively rare
compared to the first two definitions above. More details on these functions can be found
under Hamilton-Jacobi equations.

A sixth meaning of action in classical physics is a single variable Jk
in the action-angle coordinates, defined by
integrating a single generalized momentum around a closed path in phase space, corresponding to rotating or oscillating motion.

The variable Jk is called the “action” of the generalized
coordinate qk; the corresponding canonical variable conjugate to Jk
is its “angle” wk, for reasons described more fully under action-angle coordinates. NB! The integration is
only over a single variable qk and, therefore, unlike the integrated dot
product in the abbreviated action integral above. The Jk variable equals
the change in Sk(qk) as qk is varied
around the closed path. For several physical systems of interest, Jk is
either a constant or varies very slowly; hence, the variable Jk is often
used in perturbation calculations and in determining adiabatic

The seventh meaning of “action” involves the tautological one-form.

Finally, “action” was defined in several obsolete ways during its
development. Gottfried Leibniz, Johann Bernoulli and Pierre Louis Maupertuis defined the
“action” for light as the integral of its speed (or
inverse speed) along its path length. Leonhard Euler
(and, possibly, Leibniz) defined it for a material particle as the integral of the
particle speed along its path through space. Maupertuis introduced several ad hoc
and contradictory definitions of “action” within a single article,
defining action as potential energy, as virtual kinetic energy, and as a strange hybrid
that ensured conservation of momentum in collisions.

Euler-Lagrange equations for the action integral

As noted above, the requirement that the action integral be stationary under small perturbations of the evolution is
equivalent to a set of differential equations
(called the Euler-Lagrange equations) that may be determined using the calculus of variations. We illustrate this
derivation here using only one coordinate, x; the extension to multiple coordinates
is straightforward.

Adopting Hamilton’s principle, we assume that
the Lagrangian L (the integrand of the action integral) depends only on the
coordinate x(t) and its time derivative dx(t)/dt, and
does not depend on time explicitly. In that case, the action integral can be written

where the initial and final times (t1 and t2) and
the final and initial positions, and , are specified in advance. Let xtrue(t)
represent the true evolution that we seek, and let xper(t) be a
slightly perturbed version of it, albeit with the same endpoints, xper(t1)
= x1 and xper(t2) = x2.
The difference between these two evolutions is infinitesimally small at all times

At the endpoints, the difference vanishes, i.e., .

Expanded to first order, the difference between the actions integrals for the two
evolutions is

Integration by parts of the last term,
together with the boundary conditions , yields the equation

The requirement that be stationary implies that
the first-order change must be zero for any possible perturbation about the true action evolution. This can be true only if

Euler-Lagrange equation

Those familiar with functional analysis will
note that the Euler-Lagrange equations simplify to

The quantity is called the conjugate momentum for the coordinate x. An
important consequence of the Euler-Lagrange eqations is that if L does not
explicitly contain coordinate x, i.e.

if , then is constant.

In such cases, the coordinate x is called a cyclic coordinate, and its
conjugate momentum is conserved.

Example: Free particle in polar coordinates

Simple examples help to appreciate the use of the action principle via the
Euler-Lagrangian equations. A free particle (mass m and velocity v) in
Euclidean space moves in a straight line. Using the Euler-Lagrange equations, this can be
shown in polar coordinates as follows. In the
absence of a potential, the Lagrangian is simply equal to the kinetic energy

in orthonormal (x,y) coordinates, where the dot represents
differentiation with respect to the curve parameter (usually the time, t). In polar
coordinates (r
, f) the kinetic energy
and hence the Lagrangian becomes

The radial r and f components of
the Euler-Lagrangian equations become, respectively

The solution of these two equations is given by

for a set of constants a, b, c, d determined by initial
conditions. Thus, indeed, the solution is a straight line given in polar

Action principle for classical fields

The action principle can be
extended to obtain the equations of motion for
fields, such as the electromagnetic field or gravity.

The Einstein equation utilizes the Einstein-Hilbert action as constrained by
a variational principle.

The path of a body in a gravitational field (i.e. free fall in space time, a so called
geodesic) can be found using the action principle.

Action principle in quantum mechanics and quantum field theory

In quantum mechanics, the system does not follow a single path whose action is
stationary, but the behavior of the system depends on all imaginable paths and the value
of their action. The action corresponding to the various paths is used to calculate the path integral, that gives the probability amplitudes of the various outcomes.

Although equivalent in classical mechanics with Newton’s
, the
action principle is better suited for generalizations and plays an
important role in modern physics. Indeed, this principle is one of the great
generalizations in physical science. In particular, it is fully appreciated and best
understood within quantum mechanics. Richard Feynman‘s
path integral formulation of quantum
mechanics is based on a stationary-action principle, using path integrals. Maxwell’s equations can be derived as conditions of
stationary action.

Action principle and conservation laws

Symmetries in a physical situation can better be treated with the action principle,
together with the Euler-Lagrange equations,
which are derived from the action principle. An example is Noether’s theorem, which states that to every continuous symmetry in a physical situation there
corresponds a conservation law (and conversely). This
deep connection requires that the action principle be assumed.

Modern extensions of the action principle

The action principle can be generalized still further. For example, the action need not be an integral because nonlocal actions are possible. The configuration space need not even be a functional space
given certain features such as noncommutative
. However, a physical basis for these mathematical extensions remains to be established experimentally.

See also


For an annotated bibliography, see Edwin F. Taylor [1] who lists, among other
things, the following books

  1. Cornelius Lanczos,
    The Variational Principles of Mechanics (Dover Publications, New York, 1986). ISBN 0-486-65067-7. The
    reference most quoted by all those who explore this field.

  2. L. D. Landau and E. M.
    Lifshitz, Mechanics, Course of Theoretical Physics (Butterworth-Heinenann, 1976), 3rd ed.,
    Vol. 1. ISBN
    . Begins with the principle of least action.

  3. Thomas A. Moore “Least-Action Principle” in Macmillan Encyclopedia of Physics
    (Simon & Schuster Macmillan, 1996), Volume 2, ISBN 0-0286457-1,
    pages 840 – 842.
  4. David Morin introduces Lagrange’s equations in Chapter 5 of his honors introductory
    physics text. Concludes with a wonderful set of 27 problems with solutions. A draft of is
    available at [2]
  5. Gerald Jay Sussman and Jack Wisdom, Structure and Interpretation of Classical Mechanics
    (MIT Press, 2001). Begins with the principle of least action, uses modern mathematical
    notation, and checks the clarity and consistency of procedures by programming them in
    computer language.
  6. Dare A. Wells, Lagrangian Dynamics, Schaum’s Outline Series (McGraw-Hill, 1967) ISBN 007-069258-0, A 350 page comprehensive “outline” of the subject.
  7. Robert Weinstock, Calculus of Variations, with Applications to Physics and Engineering
    (Dover Publications, 1974). ISBN 0-486-63069-2.
    An oldie but goodie, with the formalism carefully defined before use in physics and
  8. Wolfgang Yourgrau and Stanley Mandelstam, Variational Principles in Dynamics and Quantum
    Theory (Dover Publications, 1979). A nice treatment that does not avoid the philosophical
    implications of the theory and lauds the Feynman treatment of quantum mechanics that
    reduces to the principle of least action in the limit of large mass.
  9. Edwin F. Taylor’s page [3]
  10. Principle of least
    action interactive
    Excellent interactive explanation/webpage

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