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Variational metrics on \(\mathbb{R}\times TM\) and the geometry of nonconservative mechanics. (English) Zbl 0818.53026
The paper under review presents a nice and compact exposition of several mathematical aspects of the time-dependent non-conservative classical mechanical systems. The recent extensive research in this area has shown that the setting of the variational calculus on fibered manifolds, together with certain generalizations of the concept of a connection, provide the mathematical tools which clarify and simplify the theory. Based on the results by J. Grifone, D. Krupka, M. de Léon, the author herself, and others, the paper offers a unified and clear picture of the relations between the variationality of metrics on \(\mathbb{R}\times TM\), the variationality and metrizability of semispray connections on \(\mathbb{R}\times TM\), and the global dynamics of a non-conservative system. The exposition starts with a detailed survey of the basic concepts, then several original results are achieved, and finally examples are presented which show in particular that a series of classical situations is covered, inclusive manifolds with Riemannian and Finslerian metrics, and manifolds with time-dependent variational metrics, manifolds with metrizable linear connections on \(TM\).
Reviewer: J.Slovák (Brno)
MSC:
53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
70H03 Lagrange’s equations
53C05 Connections (general theory)
53Z05 Applications of differential geometry to physics
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
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References:
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