Communications in Nonlinear Science and Numerical Simulation ( IF 3.4 ) Pub Date : 2021-09-04 , DOI: 10.1016/j.cnsns.2021.106021 Daniel Reyes Nozaleda 1, 2 , Piergiulio Tempesta 1, 2 , Giorgio Tondo 3
We show that the theory of classical Hamiltonian systems admitting separating variables can be formulated in the context of () structures. They are symplectic manifolds endowed with a compatible Haantjes algebra , namely an algebra of (1,1)tensor fields with vanishing Haantjes torsion. A special class of coordinates, called Darboux-Haantjes coordinates, will be constructed from the Haantjes algebras associated with a separable system. These coordinates enable the additive separation of variables of the corresponding Hamilton-Jacobi equation.
We shall prove that a multiseparable system admits as many structures as separation coordinate systems. In particular, we will show that a large class of multiseparable, superintegrable systems, including the Smorodinsky-Winternitz systems and some physically relevant systems with three degrees of freedom, possesses multiple Haantjes structures.
中文翻译:
经典多重可分哈密顿系统、超可积性和 Haantjes 几何
我们表明,允许分离变量的经典哈密顿系统的理论可以在() 结构。它们是具有相容 Haantjes 代数的辛流形,即具有消失的 Haantjes 扭转的 (1,1) 张量场的代数。一类特殊的坐标,称为 Darboux-Haantjes 坐标,将由与可分离系统相关的 Haantjes 代数构造。这些坐标使相应的 Hamilton-Jacobi 方程的变量能够相加分离。
我们将证明一个多可分系统允许尽可能多的 结构作为分离坐标系。特别是,我们将证明一大类多重可分、超可积系统,包括 Smorodinsky-Winternitz 系统和一些具有三个自由度的物理相关系统,具有多个 Haantjes 结构。