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].


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