The Lazy Universe: An Introduction to the Principle of Least Action

Abstract

The second major version of the action is Maupertuis' action I W i which can be defined for a conceivable trajectory by the spatial integral HT W = q A q B p d q ht , with I p i being the canonical momentum (which equal to the ordinary momentum is the simplest cases). Maupertuis' principle states that with prescribed end points I q i SB A sb and I q i SB B sb and prescribed trajectory energy I E i , I W i is stationary ( I W i = 0) for true trajectories. She uses the modern notations used above for the actions, I S i (Hamilton) and I W i (Maupertuis), on. p. 250, but mostly (e.g., pp. 223, 241) she uses Lanczos's notation I A i for the Hamilton action and I S i for the Maupertuis action. (2) The action principles that have been developed in recent years, including the extension of the Maupertuis principle to nonconservative systems, reciprocal Hamilton and reciprocal Maupertuis principles, and principles which allow end-point variations and relaxation of other constraints. [Extracted from the article]