A variational principle in physics 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.
== Variational principles before modern times
==
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.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.
== Principle of extremal action ==
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." [3]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 action
principle in 1744, in very recognizable terms,
in the Additamentum 2 to his "Methodus Inveniendi
Lineas Curvas Maximi Minive Proprietate Gaudentes".[4]
He begins the second paragraph:[5]
"Sit massa corporis projecti ==M, ejusque,
dum spatiolum == ds emetitur, celeritas debita
altitudini == v; erit quantitas motus corporis
in hoc loco ==
M
v
{\displaystyle M{\sqrt {v}}}
; quae per ipsum spatiolum ds multiplicata,
dabit
M
d
s
v
{\displaystyle M\,ds{\sqrt {v}}}
motum corporis collectivum per spatiolum ds.
Iam dico lineam a corpore descriptam ita fore
comparatam, ut, inter omnes alias lineas iisdem
terminis contentas, sit
∫
M
d
s
v
{\displaystyle \int Mds{\sqrt {v}}}
, seu, ob M constans,
∫
d
s
v
{\displaystyle \int ds{\sqrt {v}}}
minimum."A translation of this passage reads:
"Let the mass of the projectile be M, and
let its squared velocity resulting from its
height be
v
{\displaystyle v}
while being moved over a distance ds. The
body will have a momentum
M
v
{\displaystyle M{\sqrt {v}}}
that, when multiplied by the distance ds,
will give
M
d
s
v
{\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
∫
M
d
s
v
{\displaystyle \int Mds{\sqrt {v}}}
or, provided that M is constant,
∫
d
s
v
{\displaystyle \int ds{\sqrt {v}}}
."As Euler states,
∫
M
d
s
v
{\displaystyle \int Mds{\sqrt {v}}}
is the integral of the momentum over distance
traveled (note that here
v
{\displaystyle v}
contrary to usual notation denotes the squared
velocity) which, in modern notation, equals
the reduced action
∫
p
d
q
{\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,[6]
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.[7]
The claims of forgery were re-examined 150
years later, and archival work by C.I. Gerhardt
in 1898[8] and W. Kabitz in 1913[9] uncovered
other copies of the letter, and three others
cited by König, in the Bernoulli archives.
== Further developments of the extremal-action
principle ==
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 [10] applied the
variational principle to the function
L
=
T
−
V
{\displaystyle L=T-V}
to obtain what are now called the Lagrangian
equations of motion.
== Other formulations of the extremal-action
principle ==
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. [11] The first clear general statements
were given by Marston Morse in the 1920s and
1930s, [12] 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.
== Variational principles in electromagnetism
==
The action for electromagnetism is:
S
=
−
∫
1
4
μ
0
d
4
x
F
α
β
F
α
β
−
∫
d
4
x
j
α
A
α
{\displaystyle {\mathcal {S}}=-\int {\frac
{1}{4\mu _{0}}}\,\mathrm {d} ^{4}x\,F^{\alpha
\beta }F_{\alpha \beta }-\int \mathrm {d}
^{4}x\,j^{\alpha }A_{\alpha }}
== Variational principles in relativity theory
==
The Einstein–Hilbert action which gives
rise to the vacuum Einstein field equations
is
S
[
g
]
=
c
4
16
π
G
∫
M
R
−
g
d
4
x
{\displaystyle {\mathcal {S}}[g]={\frac {c^{4}}{16\pi
G}}\int _{\mathcal {M}}R{\sqrt {-g}}\,\mathrm
{d} ^{4}x}
,where
g
=
det
(
g
α
β
)
{\displaystyle g=\det(g_{\alpha \beta })}
is the determinant of a spacetime Lorentz
metric and
R
{\displaystyle R}
is the scalar curvature.
== Variational principles in quantum mechanics
==
Sum over possible paths, Feynman approach.
See Path integral formulation
Dirac-Frenkel Variational Principle
== Apparent teleology? ==
Although equivalent mathematically, there
is an important philosophical difference between
the differential equations of motion and their
integral counterpart. 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 F=ma}
states that the instantaneous force
F
{\displaystyle F}
applied to a mass
m
{\displaystyle m}
produces an acceleration
a
{\displaystyle 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
x
1
{\displaystyle x_{1}}
at time
t
1
{\displaystyle t_{1}}
and ends at position
x
2
{\displaystyle x_{2}}
at time
t
2
{\displaystyle t_{2}}
, 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
