Deformation of Dirac structures along isotropic subbundles

Given a Dirac subbundle and an isotropic subbundle of a Courant algebroid, we provide a canonical method to obtain a new Dirac subbundle. When the original Dirac subbundle is involutive (i.e. a Dirac structure) this construction has interesting applications, for instance to Dirac's theory of constraints and to the Marsden-Ratiu reduction in Poisson geometry.