In classical mechanics, Maupertuis's principle
(named after Pierre Louis Maupertuis), states
that the path followed by a physical system
is the one of least length (with a suitable
interpretation of path and length). It is
a special case of the more generally stated
principle of least action. Using the calculus
of variations, it results in an integral equation
formulation of the equations of motion for
the system.
== Mathematical formulation ==
Maupertuis's principle states that the true
path of a system described by
N
{\displaystyle N}
generalized coordinates
q
=
(
q
1
,
q
2
,
…
,
q
N
)
{\displaystyle \mathbf {q} =\left(q_{1},q_{2},\ldots
,q_{N}\right)}
between two specified states
q
1
{\displaystyle \mathbf {q} _{1}}
and
q
2
{\displaystyle \mathbf {q} _{2}}
is an extremum (i.e., a stationary point,
a minimum, maximum or saddle point) of the
abbreviated action functional
S
0
[
q
(
t
)
]
=
d
e
f
∫
p
⋅
d
q
{\displaystyle {\mathcal {S}}_{0}[\mathbf
{q} (t)]\ {\stackrel {\mathrm {def} }{=}}\ \int
\mathbf {p} \cdot d\mathbf {q} }
where
p
=
(
p
1
,
p
2
,
…
,
p
N
)
{\displaystyle \mathbf {p} =\left(p_{1},p_{2},\ldots
,p_{N}\right)}
are the conjugate momenta of the generalized
coordinates, defined by the equation
p
k
=
d
e
f
∂
L
∂
q
˙
k
{\displaystyle p_{k}\ {\stackrel {\mathrm
{def} }{=}}\ {\frac {\partial L}{\partial
{\dot {q}}_{k}}}}
where
L
(
q
,
q
˙
,
t
)
{\displaystyle L(\mathbf {q} ,{\dot {\mathbf
{q} }},t)}
is the Lagrangian function for the system.
In other words, any first-order perturbation
of the path results in (at most) second-order
changes in
S
0
{\displaystyle {\mathcal {S}}_{0}}
. Note that the abbreviated action
S
0
{\displaystyle {\mathcal {S}}_{0}}
is a functional (i.e. a function from a vector
space into its underlying scalar field), which
in this case takes as its input a function
(i.e. the pathes between the two specified
states).
== Jacobi's formulation ==
For many systems, the kinetic energy
T
{\displaystyle T}
is quadratic in the generalized velocities
q
˙
{\displaystyle {\dot {\mathbf {q} }}}
T
=
1
2
d
q
d
t
⋅
M
⋅
d
q
d
t
{\displaystyle T={\frac {1}{2}}{\frac {d\mathbf
{q} }{dt}}\cdot \mathbf {M} \cdot {\frac {d\mathbf
{q} }{dt}}}
although the mass tensor
M
{\displaystyle \mathbf {M} }
may be a complicated function of the generalized
coordinates
q
{\displaystyle \mathbf {q} }
. For such systems, a simple relation relates
the kinetic energy, the generalized momenta
and the generalized velocities
2
T
=
p
⋅
q
˙
{\displaystyle 2T=\mathbf {p} \cdot {\dot
{\mathbf {q} }}}
provided that the potential energy
V
(
q
)
{\displaystyle V(\mathbf {q} )}
does not involve the generalized velocities.
By defining a normalized distance or metric
d
s
{\displaystyle ds}
in the space of generalized coordinates
d
s
2
=
d
q
⋅
M
⋅
d
q
{\displaystyle ds^{2}=d\mathbf {q} \cdot \mathbf
{M} \cdot d\mathbf {q} }
one may immediately recognize the mass tensor
as a metric tensor. The kinetic energy may
be written in a massless form
T
=
1
2
(
d
s
d
t
)
2
{\displaystyle T={\frac {1}{2}}\left({\frac
{ds}{dt}}\right)^{2}}
or, equivalently,
2
T
d
t
=
p
⋅
d
q
=
2
T
d
s
.
{\displaystyle 2Tdt=\mathbf {p} \cdot d\mathbf
{q} ={\sqrt {2T}}\ ds.}
Hence, the abbreviated action can be written
S
0
=
d
e
f
∫
p
⋅
d
q
=
∫
d
s
2
E
tot
−
V
(
q
)
{\displaystyle {\mathcal {S}}_{0}\ {\stackrel
{\mathrm {def} }{=}}\ \int \mathbf {p} \cdot
d\mathbf {q} =\int ds\,{\sqrt {2}}{\sqrt {E_{\text{tot}}-V(\mathbf
{q} )}}}
since the kinetic energy
T
=
E
tot
−
V
(
q
)
{\displaystyle T=E_{\text{tot}}-V(\mathbf
{q} )}
equals the (constant) total energy
E
tot
{\displaystyle E_{\text{tot}}}
minus the potential energy
V
(
q
)
{\displaystyle V(\mathbf {q} )}
. In particular, if the potential energy is
a constant, then Jacobi's principle reduces
to minimizing the path length
s
=
∫
d
s
{\displaystyle s=\int ds}
in the space of the generalized coordinates,
which is equivalent to Hertz's principle of
least curvature.
== Comparison with Hamilton's principle ==
Hamilton's principle and Maupertuis's principle
are occasionally confused and both have been
called the principle of least action. They
differ from each other in three important
ways:
their definition of the action...Hamilton's
principle uses
S
=
d
e
f
∫
L
d
t
{\displaystyle {\mathcal {S}}\ {\stackrel
{\mathrm {def} }{=}}\ \int L\,dt}
, the integral of the Lagrangian over time,
varied between two fixed end times
t
1
{\displaystyle t_{1}}
,
t
2
{\displaystyle t_{2}}
and endpoints
q
1
{\displaystyle q_{1}}
,
q
2
{\displaystyle q_{2}}
. By contrast, Maupertuis's principle uses
the abbreviated action integral over the generalized
coordinates, varied along all constant energy
paths ending at
q
1
{\displaystyle q_{1}}
and
q
2
{\displaystyle q_{2}}
.the solution that they determine...Hamilton's
principle determines the trajectory
q
(
t
)
{\displaystyle \mathbf {q} (t)}
as a function of time, whereas Maupertuis's
principle determines only the shape of the
trajectory in the generalized coordinates.
For example, Maupertuis's principle determines
the shape of the ellipse on which a particle
moves under the influence of an inverse-square
central force such as gravity, but does not
describe per se how the particle moves along
that trajectory. (However, this time parameterization
may be determined from the trajectory itself
in subsequent calculations using the conservation
of energy.) By contrast, Hamilton's principle
directly specifies the motion along the ellipse
as a function of time....and the constraints
on the variation.Maupertuis's principle requires
that the two endpoint states
q
1
{\displaystyle q_{1}}
and
q
2
{\displaystyle q_{2}}
be given and that energy be conserved along
every trajectory. By contrast, Hamilton's
principle does not require the conservation
of energy, but does require that the endpoint
times
t
1
{\displaystyle t_{1}}
and
t
2
{\displaystyle t_{2}}
be specified as well as the endpoint states
q
1
{\displaystyle q_{1}}
and
q
2
{\displaystyle q_{2}}
.
== History ==
Maupertuis was the first to publish a principle
of least action, where he defined action as
∫
v
d
s
{\displaystyle \int v\,ds}
, which was to be minimized over all paths
connecting two specified points. However,
Maupertuis applied the principle only to light,
not matter (see the 1744 Maupertuis reference
below). He arrived at the principle by considering
Snell's law for the refraction of light, which
Fermat had explained by Fermat's principle,
that light follows the path of shortest time,
not distance. This troubled Maupertuis, since
he felt that time and distance should be on
an equal footing: "why should light prefer
the path of shortest time over that of distance?"
Accordingly, Maupertuis asserts with no further
justification the principle of least action
as equivalent but more fundamental than Fermat's
principle, and uses it to derive Snell's law.
Maupertuis specifically states that light
does not follow the same laws as material
objects.
A few months later, well before Maupertuis's
work appeared in print, Leonhard Euler independently
defined action in its modern abbreviated form
S
0
=
d
e
f
∫
m
v
d
s
=
d
e
f
∫
p
d
q
{\displaystyle {\mathcal {S}}_{0}\ {\stackrel
{\mathrm {def} }{=}}\ \int mv\,ds\ {\stackrel
{\mathrm {def} }{=}}\ \int p\,dq}
and applied it to the motion of a particle,
but not to light (see the 1744 Euler reference
below). Euler also recognized that the principle
only held when the speed was a function only
of position, i.e., when the total energy was
conserved. (The mass factor in the action
and the requirement for energy conservation
were not relevant to Maupertuis, who was concerned
only with light.) Euler used this principle
to derive the equations of motion of a particle
in uniform motion, in a uniform and non-uniform
force field, and in a central force field.
Euler's approach is entirely consistent with
the modern understanding of Maupertuis's principle
described above, except that he insisted that
the action should always be a minimum, rather
than a stationary point.
Two years later, Maupertuis cites Euler's
1744 work as a "beautiful application of my
principle to the motion of the planets" and
goes on to apply the principle of least action
to the lever problem in mechanical equilibrium
and to perfectly elastic and perfectly inelastic
collisions (see the 1746 publication below).
Thus, Maupertuis takes credit for conceiving
the principle of least action as a general
principle applicable to all physical systems
(not merely to light), whereas the historical
evidence suggests that Euler was the one to
make this intuitive leap. Notably, Maupertuis's
definitions of the action and protocols for
minimizing it in this paper are inconsistent
with the modern approach described above.
Thus, Maupertuis's published work does not
contain a single example in which he used
Maupertuis's principle (as presently understood).
In 1751, Maupertuis's priority for the principle
of least action was challenged in print (Nova
Acta Eruditorum of Leipzig) by an old acquaintance,
Johann Samuel Koenig, who quoted a 1707 letter
purportedly from Leibniz that described results
similar to those derived by Euler in 1744.
However, Maupertuis and others demanded that
Koenig produce the original of the letter
to authenticate its having been written by
Leibniz. Koenig only had a copy and no clue
as to the whereabouts of the original. Consequently,
the Berlin Academy under Euler's direction
declared the letter to be a forgery and that
its President, Maupertuis, could continue
to claim priority for having invented the
principle. Koenig continued to fight for Leibniz's
priority and soon luminaries such as Voltaire
and the King of Prussia, Frederick II were
engaged in the quarrel. However, no progress
was made until the turn of the twentieth century,
when other independent copies of Leibniz's
letter were discovered.
== See also ==
Analytical mechanics
Hamilton's principle
Gauss's principle of least constraint (also
describes Hertz's principle of least curvature)
Hamilton–Jacobi equation
