Maupertuis' Principle - Mathematical Formulation

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

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

where 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 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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