Dirac-Lie systems and Schwarzian equations

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:257

Article

Carinena, J. F.^1,2  Grabowski, J.³  de Lucas, J.⁴  Sardon, C.⁵ 

[1] Univ Zaragoza, Fac Sci, E-50009 Zaragoza, Spain

[2] Univ Zaragoza, IUMA, E-50009 Zaragoza, Spain

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

[4] Univ Warsaw, Dept Math Methods Phys, PL-00682 Warsaw, Poland

[5] Univ Salamanca, Dept Fundamental Phys, E-37008 Salamanca, Spain

关键词: Dirac structure;    Lie system;    Poisson structure;    Schwarzian equation;    Superposition rule;    Vessiot-Guldberg Lie algebra;   

DOI  :  10.1016/j.jde.2014.05.040

来源: Elsevier

【 摘 要 】

A Lie system is a system of differential equations admitting a superposition rule, i.e., a function describing its general solution in terms of any generic set of particular solutions and some constants. Following ideas going back to Dirac's description of constrained systems, we introduce and analyze a particular class of Lie systems on Dirac manifolds, called Dirac Lie systems, which are associated with 'Dirac-Lie Hamiltonians'. Our results enable us to investigate constants of the motion, superposition rules, and other general properties of such systems in a more effective way. Several concepts of the theory of Lie systems are adapted to this 'Dirac setting' and new applications of Dirac geometry in differential equations are presented. As an application, we analyze solutions of several types of Schwarzian equations, but our methods can be applied also to other classes of differential equations important for Physics. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free