This article discusses the history of the
principle of least action. For the application,
please refer to action (physics).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 and Hamiltonian
equations of motion, and even general relativity
(see Einstein–Hilbert action). The physicist
Paul Dirac, and after him Julian Schwinger
and Richard Feynman, demonstrated how this
principle can also be used in quantum calculations.
It was historically called "least" because
its solution requires finding the path that
has the least value. 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 thermodynamics, fluid mechanics, the theory
of relativity, quantum mechanics, particle
physics, and string theory and is 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 optics. 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 1933, Paul Dirac discerned
the quantum mechanical underpinning of the
principle in the quantum interference of amplitudes.
== General statement ==
The starting point is the action, denoted
S
{\displaystyle {\mathcal {S}}}
(calligraphic S), 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 = (q1, q2, ... , qN) which define the configuration
of the system:
q
:
R
→
R
N
{\displaystyle \mathbf {q} :\mathbf {R} \to
\mathbf {R} ^{N}}
S
[
q
,
t
1
,
t
2
]
=
∫
t
1
t
2
L
(
q
(
t
)
,
q
˙
(
t
)
,
t
)
d
t
{\displaystyle {\mathcal {S}}[\mathbf {q}
,t_{1},t_{2}]=\int _{t_{1}}^{t_{2}}L(\mathbf
{q} (t),\mathbf {\dot {q}} (t),t)dt}
where the dot denotes the time derivative,
and t is time.
Mathematically the principle is
δ
S
=
0
,
{\displaystyle \delta {\mathcal {S}}=0,}
where δ (lowercase Greek delta) means a small
change. In words this reads:
The path taken by the system between times
t1 and t2 and configurations q1 and q2 is
the one for which the action is stationary
(no change) to first order.In applications
the statement and definition of action are
taken together:
δ
∫
t
1
t
2
L
(
q
,
q
˙
,
t
)
d
t
=
0.
{\displaystyle \delta \int _{t_{1}}^{t_{2}}L(\mathbf
{q} ,\mathbf {\dot {q}} ,t)dt=0.}
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 (see also parametric
equation for this concept).
== 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 (time) 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 abbreviated
or 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.
=== 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 (the head of his Academy),
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 (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.
=== 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 (1788) 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
L
=
T
−
V
{\displaystyle L=T-V}
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 (maxima or stationary saddle 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
F
=
m
a
{\displaystyle \mathbf {F} =m\mathbf {a} }
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) 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 (the
positions but not the velocities) 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.
== 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
Georgiev, Georgi Yordanov (2012). "A Quantitative
Measure, Mechanism and Attractor for Self-Organization
in Networked Complex Systems". Self-Organizing
Systems. Lecture Notes in Computer Science.
7166. pp. 90–5. doi:10.1007/978-3-642-28583-7_9.
ISBN 978-3-642-28582-0.
Georgiev, Georgi; Georgiev, Iskren (2002).
"The Least Action and the Metric of an Organized
System". Open Systems and Information Dynamics.
9 (4): 371–380. arXiv:1004.3518. doi:10.1023/a:1021858318296.
Terekhovich, Vladislav (2015). "Metaphysics
of 
the Principle of Least Action". arXiv:1511.03429
[physics.hist-ph].
