Generalizations of Noether’s Theorem in Classical Mechanics

In this paper, a review is presented of various approaches to the generalization of the version of Noether’s theorem, which is presented in most textbooks on classical mechanics. Its motivation is the controversy still persisting around the possible scope of a Noether-type theorem allowing for velocity-dependent transformations. Our analysis is centered around the one factor common to all known treatments, namely the structure of the related first integral. We first discuss the most general framework, in which a function of the above-mentioned structure constitutes a first integral of a given Lagrangian system, and show that one cannot really talk about an “interrelationship” between symmetries and first integrals there. We then compare different proposed generalizations of Noether’s theorem, by describing the nature of the restrictions which characterize them, when they are situated within the broadest framework. We prove a seemingly new equivalence-result between the two main approaches: that of invariance of the action functional, and that of invariance of dθ (θ being the Cartan-form). A number of arguments are discussed in favor of this last version of a generalized Noether theorem. Throughout the analysis we pay attention to practical considerations, such as the complexity of the Killing-type partial differential equations in each approach, which must be solved in order to identify “Noether-transformations”.