Maupertuis' Principle - Comparison With Hamilton's Principle

Comparison With Hamilton's Principle

Hamilton's principle and Maupertuis' principle are occasionally confused 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, the integral of the Lagrangian over time, varied between two fixed end times, and endpoints, . By contrast, Maupertuis' principle uses the abbreviated action integral over the generalized coordinates, varied along all constant energy paths ending at and .

• the solution that they determine...

Hamilton's principle determines the trajectory as a function of time, whereas Maupertuis' principle determines only the shape of the trajectory in the generalized coordinates. For example, Maupertuis' principle determines the shape of the ellipse on which a particle moves under the influence of an inverse-square central force such as gravity, but does not describe per se how the particle moves along that trajectory. (However, this time parameterization may be determined from the trajectory itself in subsequent calculations using the conservation of energy.) By contrast, Hamilton's principle directly specifies the motion along the ellipse as a function of time.

• ...and the constraints on the variation.

Maupertuis' principle requires that the two endpoint states and be given and that energy be conserved along every trajectory. By contrast, Hamilton's principle does not require the conservation of energy, but does require that the endpoint times and be specified as well as the endpoint states and .