A Hamilton-Jacobi Theory for general dynamical systems and integrability by quadratures in symplectic and Poisson manifolds

JOURNAL OF GEOMETRY AND PHYSICS

卷:110

Article

Grillo, Sergio^1,2  Padron, Edith³ 

[1] Univ Nacl Cuyo, Inst Balseiro, Ave Bustillo 9500,R8402AGP, San Carlos De Bariloche, Rio Negro, Argentina

[2] Consejo Nacl Invest Cient & Tecn, Ave Bustillo 9500,R8402AGP, San Carlos De Bariloche, Rio Negro, Argentina

[3] Univ La Laguna, Dept Matemat Estadist & IO, ULL CSIC Geometria Diferencial & Mecan Geometr, Tenerife, Canary Islands, Spain

关键词: Hamilton-Jacobi equations;    Integrable systems;    Poisson manifold;   

DOI  :  10.1016/j.geomphys.2016.07.010

来源: Elsevier

【 摘 要 】

In this paper we develop, in a geometric framework, a Hamilton Jacobi Theory for general dynamical systems. Such a theory contains the classical theory for Hamiltonian systems on a cotangent bundle and recent developments in the framework of general symplectic, Poisson and almost-Poisson manifolds (including some approaches to a Hamilton Jacobi Theory for nonholonomic systems). Given a dynamical system, we show that every complete solution of its related Hamilton-Jacobi Equation (HJE) gives rise to a set of first integrals, and vice versa. From that, and in the context of symplectic and Poisson manifolds, a deep connection between the HJE and the (non)commutative integrability notion, and consequently the integrability by quadratures, is established. Moreover, in the same context, we find conditions on the complete solutions of the HJE that also ensures integrability by quadratures, but they are weaker than those related to the (non)commutative integrability. Examples are developed along all the paper in order to illustrate the theoretical results. (C) 2016 Elsevier B.V. All rights reserved.

【 授权许可】

Free