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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.)