Lie-algebraic structure of Lax-Sato integrable heavenly equations and the Lagrange-d'Alembert principle

JOURNAL OF GEOMETRY AND PHYSICS

卷:120

Article

Hentosh, Oksana E.¹  Prykarpatsky, Yarema A.²  Blackrnore, Denis³  Prykarpatski, Anatolij K.⁴ 

[1] NAS, Inst Appl Problems Mech & Math, UA-79060 Lvov, Ukraine

[2] Agr Univ Krakow, Dept Appl Math, PL-30059 Krakow, Poland

[3] NJIT, Dept Math Sci, Newark, NJ 07102 USA

[4] AGH Univ Sci & Technol, Dept Appl Math, PL-30059 Krakow, Poland

关键词: Lax-Sato integrability;    Heavenly equations;    Torus diffeomorphisms;    Lie-Poisson structure;    Lagrange-d'Alembert principle;   

DOI  :  10.1016/j.geomphys.2017.06.003

来源: Elsevier

【 摘 要 】

The work is devoted to recent investigations of the Lax-Sato compatible linear vector field equations, especially to the related Lie-algebraic structures and integrability properties of a very interesting class of nonlinear dynamical systems called the dispersionless heavenly type equations, which were initiated by Plebanski and later analyzed in a series of articles. The AKS-algebraic and related R-structure schemes are used to study the orbits of the corresponding co-adjoint actions, which are intimately related to the classical Lie-Poisson structures on them. It is demonstrated that their compatibility condition coincides with the corresponding heavenly equation being considered. It is shown that all these equations originate in this way and can be represented as a Lax compatibility condition for specially constructed loop vector fields on the torus. The infinite hierarchy of conservations laws related to the heavenly equations is described, and its analytical structure connected with the Casimir invariants is mentioned. In addition, typical examples of such equations, demonstrating in detail their integrability via the scheme devised herein, are presented. The relationship of the very interesting Lagrange-d'Alembert type mechanical interpretation of the devised integrability scheme with the Lax-Sato equations is also discussed. (C) 2017 Elsevier B.V. All rights reserved.

【 授权许可】

Free