This article discusses the history of the principle of least action. For the
application, please refer to action. The principle of least action – or, more
accurately, the principle of stationary action – is a variational principle
that, when applied to the action of a mechanical system, can be used to obtain
the equations of motion for that system. In relativity, a different action must
be minimized or maximized. The principle can be used to derive Newtonian,
Lagrangian, Hamiltonian equations of motion, and even General Relativity. It
was historically called "least" because its solution requires finding the path
that has the least change from nearby paths. Its classical mechanics and
electromagnetic expressions are a consequence of quantum mechanics, but
the stationary action method helped in the development of quantum 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.
Maupertuis' principle and Hamilton's principle exemplify the principle of
stationary action. The action principle is preceded by
earlier ideas in surveying and optics. Rope stretchers in ancient Egypt
stretched corded ropes to measure the distance between two points. Ptolemy, in
his Geography, emphasized that one must correct for "deviations from a straight
course". In ancient Greece, Euclid wrote in his Catoptrica that, for the path of
light reflecting from a mirror, the angle of incidence equals the angle of
reflection. Hero of Alexandria later showed that this path was the shortest
length and least time. Scholars often credit Pierre Louis
Maupertuis for formulating the principle of least action because he wrote about
it in 1744 and 1746. However, Leonhard Euler discussed the principle in 1744,
and evidence shows that Gottfried Leibniz preceded both by 39 years.
In 1932, Paul Dirac discerned the quantum mechanical underpinning of the
principle in the quantum interference of amplitudes:For macroscopic systems, the
dominant contribution to the apparent path is the classical path, while any
other path is possible in the quantum realm.
General statement The starting point is the action,
denoted , of a physical system. It is defined as the integral of the
Lagrangian L between two instants of time t1 and t2 - technically a
functional of the N generalized coordinates q = which define the
configuration of the system: where the dot denotes the time
derivative, and t is time. Mathematically the principle is
where δ means a small change. In words this reads:
The path taken by the system between times t1 and t2 is the one for which the
action is stationary to first order. In applications the statement and
definition of action are taken together: The action and Lagrangian both contain
the dynamics of the system for all times. The term "path" simply refers to
a curve traced out by the system in terms of the coordinates in the
configuration space, i.e. the curve q(t), parameterized by time.
Origins, statements, and controversy = Fermat =
In the 1600s, Pierre de Fermat postulated that "light travels between
two given points along the path of shortest time," which is known as the
principle of least time or Fermat's principle.
= Maupertuis = Credit for the formulation of the
principle of least action is commonly given to Pierre Louis Maupertuis, who
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 of movement within a system by the "vis viva",
which is the integral of twice what we now call the kinetic energy T of the
system. = Euler =
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. Beginning with the second
paragraph: As Euler states, ∫Mvds is the integral
of the momentum over distance travelled, which, in modern notation, equals the
reduced action Thus, Euler made an equivalent and
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.
= Disputed priority = 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. Further development
Euler continued to write on the topic; in his Reflexions sur quelques loix
generales de la nature, 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.
= Lagrange and Hamilton = Much of the calculus of variations was
stated by Joseph-Louis Lagrange in 1760 and he proceeded to apply this to
problems in dynamics. In Méchanique Analytique Lagrange derived the general
equations of motion of a mechanical body. William Rowan Hamilton in 1834 and
1835 applied the variational principle to the classical Lagrangian function
to obtain the Euler–Lagrange equations in their present form.
= Jacobi and Morse = In 1842, Carl Gustav Jacobi tackled the
problem of whether the variational principle always found minima as opposed
to other stationary points; 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 equalled the number of negative eigenvalues in the second
variation of the Lagrangian. = Gauss and Hertz =
Other extremal principles of classical mechanics have been formulated, such as
Gauss's principle of least constraint and its corollary, Hertz's principle of
least curvature. Disputes about possible teleological
aspects 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 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 an 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 has been interpreted as giving the
action principle a teleological character which has been controversial
historically. However, according to W. Yourgrau and S. Mandelstam, the
teleological approach... presupposes that the variational principles
themselves have mathematical characteristics which they de facto do
not possess In addition, some critics maintain this apparent teleology occurs
because of the way in which the question was asked. By specifying some but not
all aspects of both the initial and final conditions we are making some
inferences about the initial conditions from the final conditions, and it is
this "backward" inference that can be seen as a teleological explanation.
Teleology can also be overcome if we consider the classical description as a
limiting case of the quantum formalism of path integration, in which stationary
paths are obtained as a result of interference of amplitudes along all
possible paths. The short story Story of Your Life by
the speculative fiction writer Ted Chiang contains visual depictions of
Fermat's Principle along with a discussion of its teleological
dimension. Keith Devlin's The Math Instinct contains a chapter, "Elvis the
Welsh Corgi Who Can Do Calculus" that discusses the calculus "embedded" in
some animals as they solve the "least time" problem in actual situations.
More Fundamental Than Newton's 2nd Law According to Richard Feynman, the
principle of least action is mathematically more specific than
Newton's 2nd law and more fundamental in theoretical physics because it explains
a wider range of physical law. You can derive Newton's 2nd law from least
action, but the converse is not true without also applying Newton's 1st and
3rd laws and disallowing non-conservative forces like friction.
By being more specific and thereby explaining only conservative forces, the
principle of least action is able to solve problems Newton's 2nd law can't,
but the converse is not true. The principle of least action can be used to
derive the conservation of momentum and energy if its symmetry in space and time
are assumed. It correctly does not allow non-conservative potential fields, but
Newton's 2nd law allows for them by allowing for non-conservative momenta
and forces which are not fundamental forces. The mathematical basis for the
difference is that Newton's 2nd law allows for momentums p(t)=q(t)+C where
q(t) are conserved momenta allowed by least action and C is a constant that
can be non-zero in Newton's 2nd law but not in least action. The constant allows
for non-conservative momenta and therefore non-conservative forces and
potentials in Newton's 2nd law. Newton's 2nd law explains conservation of energy
and momentum and can be used to show equivalency with least action when
forces are properly conserved, e.g. when forces are summed to zero in accordance
with Newton's 1st and 3rd laws and when accounting for heat generated by
friction. Derivations of Lagrangian and Hamiltonian methods do not begin with
Newton's 2nd law, but with a more modern mathematical formulation of it that
requires forces to be conservative. See also
Notes and references External links
Interactive explanation of the principle of least action
Interactive applet to construct trajectories using principle of least
action Georgi Yordanov Georgiev 2012 [1], A
quantitative measure, mechanism and attractor for self-organization in
networked complex systems, in Lecture Notes in Computer Science, F.A. Kuipers
and P.E. Heegaard: IFIP International Federation for Information Processing,
Proceedings of the Sixth International Workshop on Self-Organizing Systems, pp.
90–95, Springer-Verlag. Georgi Yordanov Georgiev and Iskren
Yordanov Georgiev 2002 [2], The least action and the metric of an organized
system, in Open Systems and Information Dynamics, 9(4), p. 371-380