Natural transformations of Lagrangians.(English)Zbl 0839.58025

Bureš, J. (ed.) et al., Proceedings of the winter school on geometry and physics, Zdíkov, Czech Republic, January 1993. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 37, 41-46 (1994).
Let $$M$$ be a differentiable manifold. The author is studying the natural transformations of Lagrangians into $$p$$-forms, where Lagrangians are defined as smooth functions on the tangent bundle $$TM$$. The set of all such transformations of order $$r$$, $$M^r_p$$, is a free module over the set of all natural transformations of order $$r$$ of Lagrangians into itself [see the author, J. Gancarzewicz, M. de León and W. Mikulski, “Natural transformations of Lagrangians into energies”, to appear, and M. de León and W. Mikulski, “Poincaré-Cartan 1-form type and Legendre type natural transformations”, to appear]. In these references are given results concerning the structure of the modules $$M^r_p$$, for $$p = 0,1$$. In the present paper the base of the free module $$M^r_2$$ is determined. The same description holds also for $$p = 3$$ (with more complicated proof).
For the entire collection see [Zbl 0823.00015].

MSC:

 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 53A55 Differential invariants (local theory), geometric objects 70H03 Lagrange’s equations