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Dynamical systems of Lagrangian and Hamiltonian mechanical systems. (English) Zbl 1149.53016

Sabau, Sorin V. (ed.) et al., Finsler geometry, Sapporo 2005. In memory of Makoto Matsumoto. Proceedings of the 40th Finsler symposium on Finsler geometry, Sapporo, Japan, September 6–10, 2005. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-42-6/hbk). Advanced Studies in Pure Mathematics 48, 309-340 (2007).
In the first part of this paper, the author introduces and investigates the dynamical systems of the Lagrangian mechanical system \(\Sigma_{L}=(M,L(x,y),F_{e}(x,y))\). He proves the existence of a canonical dynamical system on the phase space whose integral curves are given by the Lagrangian equations of \(\Sigma_{L}\). The interesting particular case of Finslerian mechanical systems is considered. The geometry of \(\Sigma_{L}\) on \(TM\) is also described. This part may be considered as a concise survey of various recent works of the author.
In the second part, the author considers the same problem for the Hamiltonian mechanical system \(\Sigma_{H}=(M,H(x,p),F_{e}(x,p))\). He proves the existence of a canonical dynamical system on the momenta space, whose integral curves are given by the Hamiltonian equations of \(\Sigma_{H}\). The interesting particular case of Cartan mechanical systems is examined. The second part is entirely original.
For the entire collection see [Zbl 1130.53005].

MSC:

53B50 Applications of local differential geometry to the sciences
70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems
53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
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