Tangent Dirac structures. (English) Zbl 0715.58013

Courant, Ted

J. Phys. A, Math. Gen. 23, No. 22, 5153-5168 (1990).

Summary: The lift of a closed 2-form \(\Omega\) on a manifold Q to a closed 2-form on TQ may be achieved by pulling back the canonical symplectic structure on \(T^*Q\) by the bundle map \(\Omega\) : TQ\(\to T^*Q\). It is also known how to lift a Poisson structure on Q to a Poisson structure on TQ. We call these lifted structures ‘tangent’ structures. The notion of a Dirac structure is reviewed. This is a hybrid of Poisson and pre-symplectic structures, which may be thought of as a (singular) foliation of Q by pre-symplectic leaves. The main result of this paper is a single method which achieves the lift to TQ of either a Poisson or a pre-symplectic structure on Q. Natural involution on TTQ is shown to preserve the lift to TTQ of the lifted structure on TQ. The method is then applied to Dirac structures, with the result that a Dirac structure on Q has a tangent lift to TQ generalizing the lifts of Poisson and pre-symplectic structures.

MSC:

37J99

Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems

Keywords:

lifting; tangent Poisson structures; pre-symplectic structure; Dirac structures