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A new look at second-order equations and Lagrangian mechanics. (English) Zbl 0542.58011
As by now well-established, a tangent bundle TM has an intrinsic type (1,1) tensor field S with vanishing Nijenhuis tensor. In this paper, to each second-order equation field $$\Gamma$$ on TM a set of 1-forms $$\chi^*_{\Gamma}$$ is associated, containing all forms $$\phi$$ for which $${\mathfrak L}_{\Gamma}(S(\phi))=\phi$$. Lagrangian systems are then those for which $$\chi^*_{\Gamma}$$ contains a non-degenerate, exact 1- form. Type (1,1) tensor fields R are studied, which preserve $$\chi^*_{\Gamma}$$ as well as a kind of dual set of vector fields $$\chi_{\Gamma}$$. Finally, a number of general results on $$\chi^*_{\Gamma}$$ are established, which reduce to known theorems from Lagrangian mechanics in case the 1-form $$\phi$$ of interest happens to be exact.

##### MSC:
 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 70H03 Lagrange’s equations
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