A frame bundle generalization of multisymplectic geometries (Q5938850)

The article studies the geometric settings of classical field theories. The core of the paper is a careful analysis of the structure of first jet bundles. Given an arbitrary fibre bundle the author constructs a natural principal fibre bundle -- the ``vertically adapted frame bundle''. This principal bundle allows to represent the first jet bundle as an associated affine bundle. In addition the linear dual jet bundle and the GIMMSY multisymplectic space (the affine dual) appear as associated bundles using appropriate representations. The poly- and multisymplectic forms on these spaces are easily obtained from the soldering form on the adapted frame bundle by the association process. Poisson brackets and symplectic operations are studied in this context. The ``vertically adapted frame bundle'' turns out to be a key object, it allows a simple and elegant approach to the jet bundle and the related various objects used for Hamiltonian and Lagrangian field equations. NEWLINENEWLINENEWLINEThese results shed light on the relationship between polysymplectic and multisymplectic approaches and are a valuable contribution to the geometric approach to classical field dynamics.