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 velocitydependent 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 dO (0 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".
As a continuation of previous papers, we study the concept of a Lie algebroid
structure on an affine bundle by means of the canonical immersion of the affine
bundle into its bidual. We pay particular attention to the prolongation and
various lifting procedures, and to the geometrical construction of
Lagrangian-type dynamics on an affine Lie algebroid.Comment: 28 pages, Late
We describe a novel approach to the study of the inverse problem of the calculus of variations, which gives new insights into Douglas's solution of the two degree of freedom case.
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