The Four Variational Principles of Mechanics

Abstract

We argue that there arefourbasic forms of the variational principles of mechanics: Hamilton's least action principle (HP), the generalized Maupertuis principle (MP), and their two reciprocal principles, RHP and RMP. This set is invariant under reciprocity and Legendre transformations. One of these forms (HP) is in the literature: only special cases of the other three are known. The generalized MP has a weaker constraint compared to the traditional formulation: only the mean energy Ē is kept fixed between virtual paths. This reformulation of MP alleviates several weaknesses of the old version. The reciprocal Maupertuis principle (RMP) is the classical limit of Schrödinger's variational principle of quantum mechanics, and this connection emphasizes the importance of the reciprocity transformation for variational principles. Two unconstrained formulations (UHP and UMP) of these four principles are also proposed, with completely specified Lagrange multipliers. Percival's variational principle for invariant tori and variational principles for scattering orbits are derived from the RMP. The RMP is very convenient for approximate variational solutions to problems in mechanics using Ritz type methods. Examples are provided.