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Connections deduced from mechanical system of second order. (English) Zbl 0789.53016
Szenthe, J. (ed.) et al., Differential geometry and its applications. Proceedings of a colloquium, held in Eger, Hungary, August 20-25, 1989, organized by the János Bolyai Mathematical Society. Amsterdam: North- Holland Publishing Company. Colloq. Math. Soc. János Bolyai. 56, 201-213 (1992).
Let \(\xi\) be a fourth order differential equation on a manifold \(M\), i.e., \(\xi\) is a vector field on the tangent bundle of order 3, \(T^ 3 M\), of \(M\) which takes values in \(T^ 4M\), being \(T^ 4M \to T(T^ 3M)\) the canonical embedding. There exist canonically associated with \(\xi\) three connections \(\Gamma_ 1\), \(\Gamma_ 2\) and \(\Gamma_ 3\) on the fibrations \(T^ 3M \to T^ 2M\), or \(T^ 3M\to TM\), or \(T^ 3M\to M\), respectively [see L. C. de Andrés, P. R. Rodrigues, and the reviewer, Demonstr. Math. 22, No. 3, 607-632 (1989; Zbl 0701.53050); L. C. de Andrés, Estructuras geométricas subyacentes a la mecánica clásica generalizada (Doctoral dissertation), Publ. Dept. Geometría y Topología, No. 71, Univ. Santiago (1986)]. Using the theory of natural operators the author proves in this paper that associated with \(\xi\) there exist 3-parameter, or 4-parameter, or 3- parameter families of connections on the fibrations \(T^ 3M \to T^ 2 M\), or \(T^ 3 M \to TM\), or \(T^ 3M \to M\), respectively. The connections defined by de Andrés et al. are obtained as particular cases. The results are applied to the case of the Euler-Langrange vector field \(\xi_ L\) corresponding to a regular Lagrangian \(L\) of second order. The existence of Lagrangian connections (say, a connection the horizontal distribution of which is Lagrangian with respect to the Poincaré-Cartan symplectic 2-form) is investigated.
For the entire collection see [Zbl 0764.00002].

53C05 Connections (general theory)
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
70H03 Lagrange’s equations
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