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 generalized coordinates between two specified states and is a stationary point (i.e., an extremum (minimum or maximum) or a saddle point) of the abbreviated action functional
where are the conjugate momenta of the generalized coordinates, defined by the equation
where 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 . Note that the abbreviated action 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 is quadratic in the generalized velocities
although the mass tensor may be a complicated function of the generalized coordinates . For such systems, a simple relation relates the kinetic energy, the generalized momenta and the generalized velocities
provided that the potential energy does not involve the generalized velocities. By defining a normalized distance or metric in the space of generalized coordinates
one may immediately recognize the mass tensor as a metric tensor. The kinetic energy may be written in a massless form
or,
Therefore, the abbreviated action can be written
since the kinetic energy equals the (constant) total energy minus the potential energy . In particular, if the potential energy is a constant, then Jacobi's principle reduces to minimizing the path length 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... the solution that they determine... ...and the constraints on the variation.
History
Maupertuis was the first to publish a principle of least action, where he defined action as , 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 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.