Mathematical Formulation
Maupertuis' principle states that the true path of a system described by generalized coordinates between two specified states and is an extremum (i.e., a stationary point, a minimum, maximum or 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 not a function, but a functional, i.e., something that takes as its input a function (in this case, the path between the two specified states) and returns a single number, a scalar.
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