Impacto
Downloads
Downloads per month over past year
Ibort, A. and Marmo, G. and Rodríguez González, Miguel Ángel and Tempesta, Piergiulio (2019) Nilpotent integrability, reduction of dynamical systems and a third-order Calogero-Moser system. Annali di matematica pura ed applicata, 198 (5). pp. 1513-1540. ISSN 0373-3114
Preview |
PDF
861kB |
Official URL: http://dx.doi.org/10.1007/s10231-019-00828-x
Abstract
We present an algebraic formulation of the notion of integrability of dynamical systems, based on a nilpotency property of its flow: It can be explicitly described as a polynomial on its evolution parameter. Such a property is established in a purely geometric-algebraic language, in terms both of the algebra of all higher-order constants of the motion (named the nilpotent algebra of the dynamics) and of a maximal Abelian algebra of symmetries (called a Cartan subalgebra of the dynamics). It is shown that this notion of integrability amounts to the annihilator of the nilpotent algebra being contained in a Cartan subalgebra of the dynamics. Systems exhibiting this property will be said to be nilpotent-integrable. Our notion of nilpotent integrability offers a new insight into the intrinsic dynamical properties of a system, which is independent of any auxiliary geometric structure defined on its phase space. At the same time, it extends in a natural way the classical concept of integrability for Hamiltonian systems. An algebraic reduction procedure valid for nilpotent-integrable systems, generalizing the well-known reduction procedures for symplectic and/or Poisson systems on appropriate quotient spaces, is also discussed. In particular, it is shown that a large class of nilpotent-integrable systems can be obtained by reduction of higher-order free systems. The case of the third-order free system is analyzed and a non-trivial set of third-order Calogero-Moser-like nilpotent-integrable equations is obtained.
Item Type: | Article |
---|---|
Additional Information: | © 2019 Springer Heidelberg. |
Uncontrolled Keywords: | Dynamical systems; Integrable systems; Reduction methods; Lie algebras; 37N05; 37K10. |
Subjects: | Sciences > Physics > Physics-Mathematical models Sciences > Physics > Mathematical physics |
ID Code: | 58709 |
Deposited On: | 28 Feb 2020 17:19 |
Last Modified: | 28 Feb 2020 17:19 |
Origin of downloads
Repository Staff Only: item control page