Maupertuis' principle

In classical mechanics, Maupertuis's principle (named after Pierre Louis Maupertuis, 1698 – 1759) states that the path followed by a physical system is the one of least length (with a suitable interpretation of path and length).^[1] 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 $N$ generalized coordinates $\mathbf {q} =\left(q_{1},q_{2},\ldots ,q_{N}\right)$ between two specified states $\mathbf {q} _{1}$ and $\mathbf {q} _{2}$ is a minimum or a saddle point^[2] of the abbreviated action functional,

{\mathcal {S}}_{0}\ {\stackrel {\mathrm {def} }{=}}\ \int \mathbf {p} \cdot d\mathbf {q} ,

where

\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}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\partial L}{\partial {\dot {q}}_{k}}},

where

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

{\mathcal {S}}_{0}

. Note that the abbreviated action

{\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 paths between the two specified states).

Jacobi's formulation

For many systems, the kinetic energy $T$ is quadratic in the generalized velocities ${\dot {\mathbf {q} }}$

T={\frac {1}{2}}{\dot {\mathbf {q} }}\ \mathbf {M} \ {\dot {\mathbf {q} }}^{\intercal }

although the mass tensor

\mathbf {M}

may be a complicated function of the generalized coordinates

\mathbf {q}

. For such systems, a simple relation relates the kinetic energy, the generalized momenta and the generalized velocities

2T=\mathbf {p} \cdot {\dot {\mathbf {q} }}

provided that the potential energy

V(\mathbf {q} )

does not involve the generalized velocities. By defining a normalized distance or metric

ds

in the space of generalized coordinates

ds^{2}=d\mathbf {q} \ \mathbf {M} \ d\mathbf {q^{\intercal }}

one may immediately recognize the mass tensor as a metric tensor. The kinetic energy may be written in a massless form

T={\frac {1}{2}}\left({\frac {ds}{dt}}\right)^{2}

or,

2Tdt={\sqrt {2T}}\ ds.

Therefore, the abbreviated action can be written

{\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_{\text{tot}}-V(\mathbf {q} )

equals the (constant) total energy

E_{\text{tot}}

minus the potential energy

V(\mathbf {q} )

. In particular, if the potential energy is a constant, then Jacobi's principle reduces to minimizing the path length

{\textstyle 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 with each other 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 ${\mathcal {S}}\ {\stackrel {\mathrm {def} }{=}}\ \int L\,dt$ , the integral of the Lagrangian over time, varied between two fixed end times $t_{1}$ , $t_{2}$ and endpoints $q_{1}$ , $q_{2}$

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