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Geometrical structures related to second-order equations. (English) Zbl 0635.58017
Differential geometry and its applications, Proc. Conf., Brno/Czech. 1986, Math. Appl., East Eur. Ser. 27, 279-299 (1987).
[For the entire collection see Zbl 0624.00014.]
In an earlier paper, the author and coworkers introduced a particular set of 1-forms $$X^*_{\Gamma}$$ which can be used to define Lagrangian (second-order) vector fields $$\Gamma$$ on TM. Also, a set of vector fields $$X_{\Gamma}$$ related to $$\Gamma$$ can be identified. In this paper, module structures for $$X^*_{\Gamma}$$ and $$X_{\Gamma}$$ are introduced. It is proved that with such structures, $$X^*_{\Gamma}$$ and $$X_{\Gamma}$$ are dual modules. Moreover, derivation and other geometrical features of the associated algebra are discussed.
Reviewer: A.Bacciotti

##### MSC:
 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 58A10 Differential forms in global analysis 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)