Derivations of differential forms along the tangent bundle projection II

Abstract

The study of the calculus of forms along the tangent bundle projection τ, initiated in a previous paper with the same title, is continued. The idea is to complete the basic ingredients of the theory up to a point where enough tools will be available for developing new applications in the study of second-order dynamical systems. A list of commutators of important derivations is worked out and special attention is paid to degree zero derivations having a Leibnitz-type duality property. Various ways of associating tensor fields along τ to corresponding objects on TM are investigated. When the connection coming from a given second-order system is used in this process, two important concepts present themselves: one is a degree zero derivation called the dynamical covariant derivative; the other one is a type (1, 1) tensor field along τ, called the Jacobi endomorphism. It is illustrated how these concepts play a crucial role in describing many of the interesting geometrical features of a given dynamical system, which have been dealt with in the literature.