Ján Brajerčík

University of Presov in Presov | UNIPO · Department of Physics, Mathematics and Techniques

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....

In this paper, we introduce the structure of a principal bundle on the r-jet prolongation J r FX of the frame bundle FX over a manifold X. Our construction reduces the well-known principal prolongation W r FX of FX with structure group G nr. For a structure group of J r FX we find a suitable subgroup of G nr. We also discuss the structure of the as...

The conditions under which a partially ordered quasigroup can be represented as sections of a sheaf space of partially ordered quasigroups are investigated.

We study the structure of second order natural Lagrangians on the bundle of linear coframes F * X over an n-dimensional manifold X. They are identified with the corresponding differential invariants which can be obtained by the factorization method. We give an explicit description of these differential invariants in terms of their bases. For constr...

Let μ: FX → X be a principal bundle of frames with the structure group Gln (ℝ). It is shown that the variational problem, defined by Gln (ℝ)-invariant Lagrangian on J r FX, can be equivalently studied on the associated space of connections with some compatibility condition, which gives us order reduction of the corresponding Euler-Lagrange equation...

Let μ:FX→X be a principal bundle of frames with the structure group Gln(R) and let λ be a Gln(R)-invariant Lagrangian on J1FX. We give an explicit expressions of reduced equations for the associated sections of the corresponding bundle of linear connections which replace the Euler–Lagrange equations for the variational problem defined by λ.

In this work, we consider variational problems defned by G-invariant Lagrangians on the r-jet prolongation of a principal bundle P, where G is the structure group of P. These problems can be also considered as defned on the associated bundle of the r-th order connections. The correspondence between the Euler-Lagrange equations for these variational...

The aim of this paper is to characterize all second order tensor-valued and scalar differential invariants of the bundle of linear frames F X over an n-dimensional manifold X. These differential invariants are ob-tained by factorization method and are described in terms of bases of invariants. Second order natural Lagrangians of frames have been ch...

Variational principles on frame bundles, given by the first and the second order Lagrangians invariant with respect to the structure group, are considered. Noether's currents, associated with the corresponding Lepage equivalents, are obtained. It is shown that for the first and the second order invariant variational problems, the system of the Eule...

We present the theory of first order local variational principles in fibered manifolds, in which the fundamental global concept is a locally variational dynamical form. The resulting theory of extremals and symmetries is also discussed. Using any Lepage form, our approach differs from Prieto, who used the Poincaré-Cartan form for definiton of a loc...

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 differ...

Variational principles on frame bundles, invariant with respect to the structure group are investigated. Explicit expressions for the first-order invariant lagrangians, the Poincaré-Cartan, and the Euler-Lagrange forms are found, and the corresponding conservation laws are obtained as a consequence of the Noether’s theorem. We show that the (second...

Variational principles on frame bundles, given by the first and the second order Lagrangians invariant with respect to the struc-ture group, are considered. Noether's currents, associated with the corre-sponding Lepage equivalents, are obtained. It is shown that for the first and the second order invariant variational problems, the system of the Eu...