In mathematics a Dirac structure is a geometric structure generalizing both symplectic structures and Poisson structures, and having several applications to mechanics.
It is based on the notion of the Dirac bracket constraint introduced by Paul Dirac and was first introduced by Ted Courant and Alan Weinstein.
be a real vector space, and
A (linear) Dirac structure on
is a linear subspace
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
, where orthogonality is with respect to the symmetric bilinear form on
on a smooth manifold
is an assignment of a (linear) Dirac structure on the tangent space to
That is, Many authors, in particular in geometry rather than the mechanics applications, require a Dirac structure to satisfy an extra integrability condition as follows: In the mechanics literature this would be called a closed or integrable Dirac structure.