ABSTRACT. In this paper, we introduce the structure of a principal bundle on the r-jet prolongation J r F X of the frame bundle F X over a manifold X. Our construction reduces the well-known principal prolongation W r F X of F X with structure group G r n . For a structure group of J r F X we find a suitable subgroup of G r n . We also discuss the structure of the associated bundles. We show that the associated action of the structure group of J r F X corresponds with the standard actions of differential groups on tensor spaces.

We present the theory of higher order local variational principles in fibered manifolds, in which the fundamental global concept is a locally variational dynamical form. Any two Lepage forms, defining a local variational principle for this form, differ on intersection of their domains, by a variationally trivial form. In this sense, but in a different geometric setting, the local variational principles satisfy analogous properties as the variational functionals of the Chern-Simons type. The resulting theory of extremals and symmetries extends the first order theories of the Lagrange-Souriau form, presented by Grigore and Popp, and closed equivalents of the first order Euler-Lagrange forms of Haková and Krupková. Conceptually, our approach differs from Prieto, who uses the Poincaré-Cartan forms, which do not have higher order global analogues.

A setting for global variational geometry on Grassmann fibrations is presented. The integral variational functionals for finite dimensional immersed submanifolds are studied by means of the fundamental Lepage equivalent of a homogeneous Lagrangian, which can be regarded as a generalization of the well-known Hilbert form in the classical mechanics. Prolongations of immersions, diffeomorphisms and vector fields to the Grassmann fibrations are introduced as geometric tools for the variations of immersions. The first infinitesimal variation formula together with its consequences, the Euler-Lagrange equations for extremal submanifolds and the Noether theorem for invariant variational functionals are proved. The theory is illustrated on the variational functional for minimal submanifolds.2000 Mathematics Subject Classification. 58E30; 58A20; 58D19; 53A10.

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