A note to the representation of the variational sequence in mechanics

Jana Musilová and Michael Krbek

E-mail: janam@physics.muni.cz

Abstract. The problem of the representation of the physically important terms in the variational sequence in the higher order mechanics (Euler-Lagrange and Helmholtz-Sonin mappings) is completely solved in the local case. The physically adequate representation of classes of $1-$forms, $2-$forms and $3-$forms is found. Moreover, the conditions for recognizing the forms which play the role of representatives of classes in the variational sequence are given. Finally, the complete reconstruction of classes of forms based on the knowledge of the representative is presented.

AMSclassification. 49F05, 58A15, 58E99

Keywords. Fibered manifold, $r$--jet prolongation, variational sequence, Lagrange structure, Euler--Lagrange mapping, Helmholtz--Sonin mapping