Dirac structure
In mathematics a Dirac structure is a geometric construction generalizing both symplectic structures and Poisson structures, and having several applications to mechanics. It is based on the notion of constraint introduced by Paul Dirac and was first introduced by Ted Courant and Alan Weinstein. In more detail, let V be a real vector space, and V* its dual. A (linear) Dirac structure on V is a linear subspace D of satisfying for all one has , D is maximal with respect to this property. In particular, if V is finite dimensional then the second criterion is satisfied if . (Similar definitions can be made for vector spaces over other fields.) An alternative (equivalent) definition often used is that satisfies , where orthogonality is with respect to the symmetric bilinear form on given by Examples If is a vector subspace, then is a Dirac structure on , where is the annihilator of ; that is, . Let be a skew-symmetric linear map, then the graph of is a Dirac structure. Similarly, if is a skew-symmetric linear map, then its graph is a Dirac structure. A Dirac structure on a manifold M is an assignment of a (linear) Dirac structure on the