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Dirac algebroids in Lagrangian and Hamiltonian mechanics. (English) Zbl 1223.37064
Summary: We present a unified approach to constrained implicit Lagrangian and Hamiltonian systems based on the introduced concept of a Dirac algebroid. The latter is a certain almost Dirac structure associated with the Courant algebroid T$$E^* \oplus_M \text T^* E^*$$ on the dual $$E^{\ast }$$ to a vector bundle $$\tau :E\to M$$. If this almost Dirac structure is integrable (Dirac), we speak of a Dirac-Lie algebroid. The bundle $$E$$ plays the role of the bundle of kinematic configurations (quasi-velocities), while the bundle $$E^{\ast }$$ plays the role of the phase space. This setting is totally intrinsic and does not distinguish between regular and singular Lagrangians. The constraints are part of the framework, so the general approach does not change when nonholonomic constraints are imposed, and produces the (implicit) Euler-Lagrange and Hamilton equations in an elegant geometric way. The scheme includes all important cases of Lagrangian and Hamiltonian systems, no matter if they are with or without constraints, autonomous or non-autonomous etc., as well as their reductions; in particular, constrained systems on Lie algebroids. We prove also some basic facts about the geometry of Dirac and Dirac-Lie algebroids.

##### MSC:
 37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010) 70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics 70F25 Nonholonomic systems related to the dynamics of a system of particles 70H45 Constrained dynamics, Dirac’s theory of constraints 70H03 Lagrange’s equations 70H25 Hamilton’s principle 17B66 Lie algebras of vector fields and related (super) algebras
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