  Although conservative Hamiltonian systems with constraints can be formulated
in terms of Dirac structures, a more general framework is necessary to cover
also dissipative systems such as gradient and metriplectic systems with
constraints. We define Leibniz-Dirac structures which lead to a natural
generalization of Dirac and Riemannian structures, for instance. From modeling
point of view, Leibniz-Dirac structures make it easy to formulate implicit
dissipative Hamiltonian systems. We give their exact characterization in terms
of bundle maps from the tangent bundle to the cotangent bundle and vice verse.
Physical systems which can be formulated in terms of Leibniz-Dirac structures
are discussed.
