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Alternative Lagrangians in particle dynamics. (English) Zbl 0666.58022
Differential geometry and its applications, Proc. Conf. Brno/Czech. 1986, Math. Appl., East. Eur. Ser. 27, 1-12 (1987).
[For the entire collection see Zbl 0624.00014.]
This is a review of properties of dynamical systems which admit two not trivially equivalent Lagrangians. There is a (1,1) tensor field which relates the Cartan 2-forms of the two Lagrangians, and which leads to non-Noether constants of motion, and to a Lax equation for the system. Under additional assumptions (e.g. the Nijenhuis tensor of the (1,1)- tensor field should vanish) the system is completely integrable.
Reviewer: P.Michor

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
70H03 Lagrange’s equations