Variational sequences in mechanics

Abstract

Let Y be a fibered manifold over a base manifold X . A differential 1-form ρ, defined on the r-jet prolongation of J r Y , is said to be contact, if it vanishes along the r-jet prolongation J r Y of every section γ of Y . The notion of contactness is naturally extended to k -forms with k ≥ 1. The contact forms define a subsequence of the De Rham sequence on J r Y . The corresponding quotient sequence is known as the rth order variational sequence. In this paper, the case of 1-dimensional base X is considered. A simple proof is given of the fact that the rth order variational sequence is an acyclic resolution of the constant sheaf. Then the 1st order variational sequence is studied in detail. The quotient sheaves, as well as the quotient mappings, are determined explicitly, and their relationship to the standard concepts of the 1st order calculus of variations is discussed. The following is shown: a) the lagrangians in the 1st order variational sequence (classes of 1-forms) coincide with 2nd order lagrangians, affine in the second derivative variables, b) the concept of the Euler-Lagrange form is extended to 2-forms which are not necessarily variational, c) the concept of the Helmholtz-Sonin form is introduced as the class of an arbitrary 3-form, d) the well-known fundamental notions such as the Euler-Lagrange, and Helmholtz-Sonin mappings are represented by two arrows at the beginning of the variational sequence; this relates the global structure of the Euler-Lagrange mapping to the cohomology of Y , e) all the remaining classes of k -forms with k ≥ 3, as well as the quotient mappings, are determined explicitly, f) a locally variational form is defined as a generalization of a symplectic form; locally variational forms, associated to a fixed Euler-Lagrange form, are characterized, and g) distributions associated with a locally variational form are described, and their relation to the Euler-Lagrange equations is studied. These results illustrate differences between finite order variational sequences and variational bicomplexes, based on infinite jet constructions.

On the Lagrangian structure of reduced dynamics under virtual holonomic constraints

“…There was a revival of interest in the IPLM around the 1980's thanks in part to the monograph by Santilli [31]. Using the tools of differential geometry and global analysis, researchers started to encode the Helmholtz conditions in geometric framework [1,8,9,18,19,23,31,32,38,39]. The paper by Saunders [33] reviews the contributions to IPCV since 1979 to date.…”

The Interior Euler-Lagrange Operator in Field Theory

“…Later, the variational sequence was analysed in the particular case of fibred manifolds with 1-dimensional base (fibred mechanics, Krupka [7]). Related concepts of global variational theory (especially Lepage forms and variational bicomplexes) were also studied, see Krupka [9], Vitolo [14], and references therein.…”

“…Our basic references on the variational sequence theory on finite order jet prolongations of fibred manifolds are Krupka [5,7,9] andŠeděnková-Volná [11,12].…”

A generalization of Lepage forms in mechanics

“…A complete local representation of the r-th order variational sequence in mechanics was found by Štefánek [19]. Another representation of all classes in the first order variational sequence was given by Krupka [11]. Musilová and Krbek [16] found a representation of the variational terms in higher order variational sequence in mechanics.…”