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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.

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