Demeter Krupka

Demeter Krupka
Lepage Research Institute, University of Presov

Professor

About

195
Publications
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1,497
Citations

Publications

Publications (195)
Article
The class of integral variational functionals for paths in smooth manifolds, whose extremals are (nonparameterized) sets, is considered in this study. Recently, it was shown that for the functionals depending on tangent vectors, this property follows from any of the following two equivalent conditions: (a) the Lagrange function, defined on the tang...
Article
Projectability of Lepage forms, defined on higher-order jet spaces, onto the corresponding Grassmann fibrations, is a basic requirement for the extension of the theory of Lepage forms to integral variational functionals for submanifolds. In this paper, projectability of second-order Lepage forms is considered for variational functionals of the Fins...
Article
Full-text available
Systems of ordinary differential equations (or dynamical forms in Lagrangian mechanics), induced by embeddings of smooth fibered manifolds over one-dimensional basis, are considered in the class of variational equations. For a given non-variational system, conditions assuring variationality (the Helmholtz conditions) of the induced system with resp...
Article
Full-text available
Given a non-variational system of differential equations, the simplest way of turning it into a variational one is by adding a correction term. In the paper, we propose a way of obtaining such a correction term, based on the so-called Vainberg-Tonti Lagrangian, and present several applications in general relativity and classical mechanics.
Article
The construction of a finite-order bicomplex whose morphisms are the horizontal and vertical derivatives of differential forms on finite-order jet prolongations of fibered manifolds over one-dimensional bases is presented. In particular, relationship between the morphisms and classes entering the variational sequence and the associated finite-order...
Article
The aim of this paper is to present a relatively complete theory of invariance of global, higher-order integral variational functionals in fibered spaces, as developed during a few past decades. We unify and extend recent results of the geometric invariance theory; new results on deformations of extremals are also included. We show that the theory...
Article
The book is devoted to recent research in the global variational theory on smooth manifolds. Its main objective is an extension of the classical variational calculus on Euclidean spaces to (topologically nontrivial) finite-dimensional smooth manifolds; to this purpose the methods of global analysis of differential forms are used. Emphasis is placed...
Article
Simple examples of variational functionals on Grassmann fibrations are analysed on the basis of the Hilbert form. The Lagrange, Euler - Lagrange, and Noether classes, characterizing the functionals, their extremals and invariance properties are discussed. The relationship of equations for extremals and conservation law equations is established; in...
Chapter
In this chapter we consider the relationship between the classical inverse problem of the calculus of variations and the method of controlled Lagrangians. The latter is a technique for deriving stabilizing feedback controls for nonlinear controlled mechanical systems. It relies on deriving a Lagrangian which describes the feedback controlled dynami...
Chapter
The Sonin-Douglas inverse problem of the calculus of variations is considered.
Chapter
Let X be any manifold, W an open set in X, and let α: W → X be a smooth mapping. A differential form η, defined on the set α(W) in X, is said to be invariant with respect to α, if the transformed form \( \alpha \ast \eta \) coincides with η, that is, if \( \alpha \ast \eta = \eta \) on the set \( W \cap \alpha (W) \); in this case, we also say that...
Chapter
We introduced in Chap. 4 the Euler–Lagrange mapping of the calculus of variations as an \( {\mathbf{R}} \)-linear mapping, assigning to a Lagrangian \( \lambda \), defined on the r-jet prolongation \( J^{r} Y \) of a fibered manifold Y, its Euler–Lagrange form \( E_{\lambda } \). Local properties of this mapping are determined by the components of...
Chapter
Examples presented in this chapter include typical variational functionals that appear as variational principles in the theory of geometric and physical fields. We begin by the discussion of the well-known Hilbert variational functional for the metric fields, first considered in Hilbert in 1915, whose Euler–Lagrange equations are the Einstein vacuu...
Chapter
The purpose of this chapter is to explain selected topics of the sheaf theory over paracompact, Hausdorff topological spaces. The choice of questions we consider are predetermined by the global variational theory over (topologically nontrivial) fibered manifolds, namely by the problem how to characterize differences between the local and global pro...
Chapter
In this chapter, a complete treatment of the foundations of the calculus of variations on fibered manifolds is presented. Using the calculus of differential forms as the main tool, the aim is to study higher-order integral variational functionals of the orm \( \gamma \to {\int }J^{r} \gamma \ast \rho \), depending on sections γ of a fibered manifol...
Chapter
This chapter introduces fibered manifolds and their jet prolongations. First, we recall properties of differentiable mappings of constant rank and introduce, with the help of rank, the notion of a fibered manifold. Then, we define automorphisms of fibered manifolds as the mappings preserving their fibered structure. The r-jets of sections of a fibe...
Chapter
In this chapter, we present a decomposition theory of differential forms on jet prolongations of fibered manifolds; the tools inducing the decompositions are the algebraic trace decomposition theory and the canonical jet projections. Of particular interest is the structure of the contact forms, annihilating integrable sections of the jet prolongati...
Article
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...
Article
A setting for higher-order global variational analysis on Grassmann fibrations is presented. The integral variational principles for one-dimensional immersed submanifolds are introduced by means of differential 1-forms with specific properties, similar to the Lepage forms from the variational calculus on fibred manifolds. Prolongations of immersion...
Article
A well-known construction in geometric mechanics and Riemann -Finsler geometry assigns to a (first order) homogeneous Lagrangian the Hilbert form, serving as an integrand in the corresponding variational functional. Analogous constructions, needed for higher-order mechanics and Finsler-Kawaguchi geometry, have not been found yet. In this paper we c...
Article
Variationality of systems of second order ordinary differential equations is studied within the class of positive homogeneous systems. The concept of a higher-order positive homogeneous function, related to Finsler geometry, is represented by the well-known Zermelo conditions, and applied to the theory of variational equations. In particular, it is...
Article
Full-text available
A canonical vector field on the tangent bundle is a vector field defined by an invariant coordinate construction. In this paper, a complete classification of canonical vector fields on tangent bundles, depending on vector fields defined on their bases, is obtained. It is shown that every canonical vector field is a linear combination with constant...
Article
Full-text available
We study underlying geometric structures for integral variational functionals, depending on submanifolds of a given manifold. Applications include (first order) variational functionals of Finsler and areal geometries with integrand the Hilbert 1-form, and admit immediate extensions to higher-order functionals.
Article
Invariance under reparametrizations of integral curves of higher order differential equations, including variational equations related to Finsler geometry, is studied. The classical homogeneity concepts are introduced within the theory of (jet) differential groups, known in the theory of differential invariants. On this basis the well-known general...
Article
The aim of this paper is to give a survey of recent developments in global variational geometry, and in particular, to complete the results on the construction of classes (terms) in the variational sequences related to higher-order variational problems on fibred spaces. Explicit description of the first order variational sequences is given as an ex...
Article
We present variational principles for mechanics on the basis of the Cartan 1-form. We extend the theory to the Cartan–Lepage 2-forms related with dynamical forms and equations of motion. Explicit coordinate formulae for Lepage equivalents of dynamical forms are given.
Article
Full-text available
The metrizability problem for a symmetric affine connection on a manifold, invariant with respect to a group of diffeomorphisms G, is considered. We say that the connection is G-metrizable, if it is expressible as the Levi-Civita connection of a G-invariant metric field. In this paper we analyze the G-metrizability equations for the rotation group...
Article
Extension of the variational sequence theory in mechanics to the first order Grassmann fibrations of 1-dimensional submanifolds is presented. The correspondence with the variational theory of parameter-invariant problems on manifolds is discussed in terms of the theory of jets (slit tangent bundles) and contact elements. In particular, the Helmholt...
Article
In this paper, we discuss possible extensions of the concept of the Cartan form of classical mechanics to higher-order mechanics on manifolds, higher-order field theory on jet bundles and to parametric variational problems on slit tangent bundles and on bundles of nondegenerate velocities. We present a generalization of the Cartan form, known as a...
Article
Full-text available
Geometric structure of global integral variational functionals on higher order tangent bundles and Grassmann fibrations are investi-gated. The theory of Lepage forms is extended to these structures. The concept of a Lepage form allows us to introduce the Euler-Lagrange dis-tribution for variational functionals, depending on velocities, in a similar...
Article
The purpose of this research-expository work is to introduce basic concepts of the theory of jets, and to study their general properties. An r -jet of a real function of several real variables at a point is simply the collection of the coefficients of the r -th Taylor polynomial of f at this point. The concept of an r -jet is easily generalized to...
Article
The purpose of this paper is to present foundations of the theory of global, higher order variational functionals for sections of fibred manifolds, and to review recent general developments in this field. We discuss basic concepts as well as new local and global results obtained during a few last decades.
Article
We present the theory of higher order velocities and their scalar differential invariants. We consider a natural action of a differential group on manifolds of higher order velocities, and study properties of its orbits (contact elements) and orbit spaces (higher order Grassmann bundles). We show that this action defines on a manifold of regular ve...
Article
The global variational functional, defined by the Hilbert-Yang-Mills Lagrangian over a smooth manifold, is investigated within the framework of prolongation theory of principal fiber bundles, and global variational theory on fibered manifolds. The principal Lepage equivalent of this Lagrangian is constructed, and the corresponding infinitesimal fir...
Article
Symmetries of the contact ideal on the r-jet bundle over a fibred manifold are studied, and transformation properties under contact symmetries of different objects in the variational sequence related with systems of partial differential equations are investigated.
Article
This is a comprehensive exposition of topics covered by the American Mathematical Societys classification Global Analysis, dealing with modern developments in calculus expressed using abstract terminology. It will be invaluable for graduate students and researchers embarking on advanced studies in mathematics and mathematical physics. This book pro...
Article
The purpose of this paper is to review properties of the Euler-Lagrange mapping in the higher order variational theory on fibred manifolds. We present basic theorems on the kernel of the Euler-Lagrange mapping, describing variationally trivial Lagrangians, and its image, characterizing variational source forms. We discuss invariance properties of L...
Article
Full-text available
In this paper, vector fields which are symmetries of the contact ideal are studied. It is shown that contact symmetries of the Helmholtz form transform a dynamical form to a dynamical form which is variational (i.e. comes as the Euler–Lagrange form from a Lagrangian). The case of dynamical forms representing first-order classes in the variational s...
Article
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...
Article
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The trace decomposition theory of tensor spaces, based on duality, is presented. The trace decomposition equations for tensors, symmetric in some sets of superscripts, and antisymmetric in the subscripts, are derived by means of the trace operations and appropriate symmetrizations and antisymmetrizations. Commutation relations for the corresponding...
Article
In this chapter, we introduce formal divergence equations on Euclidean spaces and study their basic properties. These partial differential equations naturally appear in the variational geometry on fibered manifolds, but also have a broader meaning related to differential equations, conservation laws, and integration of forms on manifolds with bound...
Article
Full-text available
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...
Conference Paper
Full-text available
One of the results of the variational sequence theory, related to the inverse problem of the calculus of variations, states that a dynamical form ", representing a system of partial difierential equations, is locally variational if and only if the Helmholtz form H(") vanishes. In this paper, a relationship between the Lie derivatives of " and H(")...
Article
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...
Article
A survey of basic concepts of the theory of unconstrained higher order variational principles in fibered spaces is given, and selected open problems, related to the variational sequence theory, are discussed.
Article
The problem of finding all (higher order) differential invariants of immersions f : X → Y, where X and Y are manifolds endowed with metric fields, is investigated. The underlying jet manifolds are introduced, and the action of the corresponding differential groups on them are analysed. It is shown that the differential invariants can be described b...
Article
The projections of tensor spaces of types (1,3), (2,2), and (1,4) over a real, n-dimensional vector space onto their complementary subspaces of Weyl (i.e. traceless), and Kronecker tensors are considered. The corresponding trace decomposition formulas providing a basis for an algebraic classification of these tensors, are discussed.
Article
In this paper we will be considering a basic geometric problem, the extension problem of classical Hamilton-Cartan variational theory to higher jet prolongations on fibered manifolds.
Article
Full-text available
The goal of this paper is to give acomplete description of the Lagrange functions L satisfying $\cal E (L)=0$. In other words we shall study the kernel of the linear mapping $\cal L (J^1) \ni L \to \cal E (L) \in \Omega^1(J^2)$, which will be referred to as the Euler mapping.
Article
It is shown that any interaction Lagrangian, depending on a collection of fields and on the metric field on a (space-time) manifold whose energy-momentum tensor depends on at most first derivatives of the metric tensor, is of a certain polynomial character in these derivatives.
Article
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The aim of this paper is to introduce a method of invariant decompositions of the tensor space TsrRn = Rn ⊗ Rn ⊗ ⋯ ⊗ Rn ⊗ R n* ⊗ Rn* ⊗ ⋯ ⊗ Rn* (r factors Rn, s factors the dual vector space Rn*), endowed with the tensor representation of the general linear group GLn(R). The method is elementary, and is based on the concept of a natural (GLn(R)-equi...
Article
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This work contains an exposition of foundations of the variational calculus in fibered manifolds. The emphasis is laid on the geometric aspects of the theory. Especially functionals defined by real functions (Lagrange functions) or differential forms (Lagrangian forms) on the first jet prolongation of a given fibered manifold are studied. Critical...
Article
The purpose of this paper is to discuss in a consistent way global properties of higher order functionals of the calculus of variations, depending on mappings between smooth manifolds. Basic global concepts, such as the lagrangian, the Euler-Lagrange, Hamilton-deDonder, and Helmholtz-Sonin forms, are introduced. Recent general results, based on the...
Article
Full-text available
this paper is to announce some new results on the structure of the higher order Euler-Lagrange mapping of the multiple-integral variational calculus on fibered manifolds, namely a description of its kernel and its image, and an explicit characterization of the conditions under which a system of partial differential equations (of arbitrary order) is...
Article
. We compare basic features of the variational sequence, and the variational bicomplex theories on a fibered manifold. We show that these two theories, based on different underlying geometric structures, give different results. 1. Introduction The idea that there should exist a close relationship between the exterior derivative operator on one side...
Article
An (r, n)-velocity is an r-jet with source at 0 ϵ n, and target in a manifold Y. An (r, n)-velocity is said to be regular if it has a representative which is an immersion at 0 ϵ n. The manifold TnrY of (r, n)-velocities as well as its open, Lnr-invariant, dense submanifold Imm TnrY of regular (r, n)-velocities, are endowed with a natural action of...
Article
Let be a fibered manifold over a base manifold . A differential 1-form , defined on the -jet prolongation of , is said to be contact, if it vanishes along the -jet prolongation of every section of . The notion of contactness is naturally extended to -forms with . The contact forms define a subsequence of the De Rham sequence on . The corresponding...
Article
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. The problem of decomposition of mixed tensor spaces by the trace operation is considered. It is shown that a tensor A = i A i 1 i 2 :::i p k1 k2 :::k q j can always be expressed as the sum of a traceless term and a linear combination of the Kronecker's ffi-tensor, with traceless coefficients. The uniqueness of this decomposition is discussed. Par...
Article
Elementary methods are applied to determine the trace decomposition of tensors of type (1,2), and (1,3) on a real, finite-dimensional vector space. Explicit decomposition formulas are given showing the dependence of the decomposition on the dimension of the underlying vector space. The uniqueness of the decomposition is discussed.
Article
If Y is a fibered manifold over a base manifold X, a differential form ρ, defined on the (finite) τ-jet prolongation JτY of Y, is said to be contact, if it vanishes along the τ-jet prolongation Jτγ of every section γ of Y, i.e., (Jτγ)∗ρ = 0 for all γ. The contact forms define a subcomplex of the de Rham complex on JτY, and an ideal in the exterior...
Article
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: It is known that there exists a mapping assigning to a first order lagrangian its Lepagean equivalent () in such a way that d() = 0 if and only if the Euler-Lagrange form E vanishes identically, i.e., E = 0. In this paper we discuss within the theory of finite order variational sequences an analogue of for higher order lagrangians. It turns out t...

Projects

Project (1)
Project
We aim to construct solid mathematical foundations of a Finslerian extension of general relativity, together with a generalization of variational principles to higher order Grassmann fibrations.