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Concept
Maupertuis's principle
Summary
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.
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).
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.
Official source
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