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On the variational calculus in fibered-fibered manifolds

W. M. Mikulski (2006)

Annales Polonici Mathematici

In this paper we extend the variational calculus to fibered-fibered manifolds. Fibered-fibered manifolds are surjective fibered submersions π:Y → X between fibered manifolds. For natural numbers s ≥ r ≤ q with r ≥ 1 we define (r,s,q)th order Lagrangians on fibered-fibered manifolds π:Y → X as base-preserving morphisms λ : J r , s , q Y → ⋀ d i m X T * X . Then similarly to the fibered manifold case we define critical fibered sections of Y. Setting p=max(q,s) we prove that there exists a canonical “Euler” morphism ( λ ) : J r + s , 2 s , r + p Y → * Y ⊗ ⋀ d i m X T * X of λ satisfying...

On the Weilian prolongations of natural bundles

Ivan Kolář (2012)

Czechoslovak Mathematical Journal

We characterize Weilian prolongations of natural bundles from the viewpoint of certain recent general results. First we describe the iteration F ( E M ) of two natural bundles E and F . Then we discuss the Weilian prolongation of an arbitrary associated bundle. These two auxiliary results enables us to solve our original problem.

On the γ -equivalence of semiholonomic jets

Miroslav Doupovec, Ivan Kolář (2019)

Archivum Mathematicum

It is well known that the concept of holonomic r -jet can be geometrically characterized in terms of the contact of individual curves. However, this is not true for the semiholonomic r -jets, [5], [8]. In the present paper, we discuss systematically the semiholonomic case.

On third order semiholonomic prolongation of a connection

Petr Vašík (2011)

Banach Center Publications

We recall several different definitions of semiholonomic jet prolongations of a fibered manifold and use them to derive some interesting properties of prolongation of a first order connection to a third order semiholonomic connection.

On Weil Bundles of the First Order

Adgam Yakhievich Sultanov (2016)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

The descriptions of Weil bundles, lifts of functions and vector fields are given. Actions of the automorphisms group of the Whitney sum of algebras of dual numbers on a Weil bundle of the first order are defined.

Orbits of families of vector fields on subcartesian spaces

Jedrzej Śniatycki (2003)

Annales de l'Institut Fourier

Orbits of complete families of vector fields on a subcartesian space are shown to be smooth manifolds. This allows a description of the structure of the reduced phase space of a Hamiltonian system in terms of the reduced Poisson algebra. Moreover, one can give a global description of smooth geometric structures on a family of manifolds, which form a singular foliation of a subcartesian space, in terms of objects defined on the corresponding family of vector fields. Stratified...

Order reduction of the Euler-Lagrange equations of higher order invariant variational problems on frame bundles

Ján Brajerčík (2011)

Czechoslovak Mathematical Journal

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

Currently displaying 761 – 780 of 1155