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 Dirac structure on V is a linear subspace D of $V\times V^{*}$ satisfying

• for all $(v,\alpha )\in D$ one has $\left\langle \alpha ,\,v\right\rangle =0$,
• D is maximal with respect to this property.

In particular, if V is finite dimensional then the second criterion is satisfied if $\dim D=\dim V$. (Similar definitions can be made for vector spaces over other fields.)

An alternative (equivalent) definition often used is that $D$ satisfies $D=D^{\perp }$, where orthogonality is with respect to the symmetric bilinear form on $V\times V^{*}$ given by ${\bigl \langle }(u,\alpha ),\,(v,\beta ){\bigr \rangle }=\left\langle \alpha ,v\right\rangle +\left\langle \beta ,u\right\rangle .$

Class of examples: If $U\subset V$ is a vector subspace, then $D=U\times U^{\circ }$ is a Dirac structure on $V$, where $U^{\circ }$ is the annihilator of $U$; that is, $U^{\circ }=\left\{\alpha \in V^{*}\mid \alpha _{\vert U}=0\right\}$.

A Dirac structure on a manifold M is an assignment of a Dirac structure on the tangent space to M at m, for each $m\in M$. That is,

• for each $m\in M$, a Dirac subspace $D_{m}$ of the space $T_{m}M\times T_{m}^{*}M$.

Many authors, in particular in geometry rather than the mechanics applications, require a Dirac structure to satisfy an extra integrability condition

• [give details ...]

In the mechanics literature this would be called a closed or integrable Dirac structure.