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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].

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