Dirac algebroids in Lagrangian and Hamiltonian mechanics

JOURNAL OF GEOMETRY AND PHYSICS

卷:61

Article

Grabowska, Katarzyna²  Grabowski, Janusz¹ 

[1] Polish Acad Sci, Inst Math, PL-00956 Warsaw, Poland

[2] Univ Warsaw, Fac Phys, PL-00681 Warsaw, Poland

关键词: Variational calculus;    Geometrical mechanics;    Nonholonomic constraint;    Euler Lagrange equation;    Dirac structure;    Lie algebroid;   

DOI  :  10.1016/j.geomphys.2011.06.018

来源: Elsevier

【 摘 要 】

We present a unified approach to constrained implicit Lagrangian and Hamiltonian systems based on the introduced concept of Dirac algebroid. The latter is a certain almost Dirac structure associated with the Courant algebroid TE* circle plus(M) T*E* on the dual E* to a vector bundle tau : E -> M. If this almost Dirac structure is integrable (Dirac), we speak about a Dirac-Lie algebroid. The bundle E plays the role of the bundle of kinematic configurations (quasi-velocities), while the bundle E* - 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. (C) 2011 Elsevier B.V. All rights reserved.

【 授权许可】

Free