Demeter Krupka

The contact ideal

Abstract 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., (J τ γ)∗ρ = 0 for all γ. The contact forms define a subcomplex of the de Rham complex on JτY, and an ideal in the exterior algebra of forms on JτY, called the contact ideal. The contact ideal is not generated by linear forms. Together with contact forms, we consider a modified notion of a strongly contact form which leads to a modified subcomplex of the de Rham complex. The local structure of all contact, and strongly contact forms is described. Applications to the higher order variational calculus on fibered manifolds are given.

Trivial lagrangians in field theory

Abstract The paper presents a complete description of trivial lagrangians in field theory. It is shown that any higher order trivial lagrangian can be expressed as the horizontal component of the exterior derivative of a projectable form.

Lagrange theory in fibered manifolds

Abstract Some basic notions and theorems of the Lagrange theory in fibered manifolds are stated, with the stress on coordinate-free formulation of the theory.

Invariants of velocities and higher-order Grassmann bundles☆

Abstract An ( r, n )-velocity is an r -jet with source at 0 eR 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 eR n . The manifold T n r Y of ( r, n )-velocities as well as its open, L n r -invariant, dense submanifold Imm T n r Y of regular ( r, n )-velocities, are endowed with a natural action of the differential group L n r of invertible r -jets with source and target 0 eR n . In this paper, we describe all continuous, L n r -invariant, real-valued functions on T n r Y and Imm T n r Y . We find local bases of L n r -invariants Imm T n r Y in an explicit, recurrent form. To this purpose, higher-order Grassmann bundles are considered as the corresponding quotients P n r Y =Imm T n r Y/L n r , and their basic properties are studied. We show that nontrivial L n r -invariants on Imm T n r Y cannot be continuously extended onto T n r Y .

The Cartan form and its generalizations in the calculus of variations

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 Lepage form, and basic properties of the Lepage forms. Both earlier and recent examples of differential forms generalizing the Cartan form are reviewed.

Variational principles for energy-momentum tensors

Abstract 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.

Global Variational Principles: Foundations and Current Problems

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.

On generalized invariant transformations

Abstract The notion of generalized invariant transformations of variational problems in Lagrangian formulation is discussed in the framework of the variational theory in fibred manifolds. Necessary and sufficient conditions for generators of one-parameter groups of such transformations are derived, completing thus some previous results of A. Trautman on the theory of transformations.

On a bicomplex induced by the variational sequence

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 bicomplex is studied. Properties of classes entering the infinite-order bicomplex, induced from the finite-order variational sequences by means of an infinite canonical construction, are formulated as a remark, insisting further research.

Foundations of higher-order variational theory on Grassmann fibrations

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 immersions and vector fields to the Grassmann fibrations are defined as a geometric tool for the variations of immersions, and the first variation formula in the infinitesimal form is derived. Its consequences, the Euler–Lagrange equations for submanifolds and the Noether theorem on invariant variational functionals are proved. Examples clarifying the meaning of the Noether theorem in the context of variational principles for submanifolds are given.