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arXiv · 2012.09819

Classical Multiseparable Hamiltonian Systems, Superintegrability and Haantjes Geometry

Daniel Reyes Nozaleda·Piergiulio Tempesta·Giorgio Tondo
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Abstract

We show that the theory of classical Hamiltonian systems admitting separating variables can be formulated in the context of ($\omega, \mathscr{H}$) structures. They are symplectic manifolds endowed with a compatible Haantjes algebra $\mathscr{H}$, 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 $\omega\mathscr{H}$ 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.

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Daniel Reyes Nozaleda, Piergiulio Tempesta, Giorgio Tondo. 2020-12-17. Classical Multiseparable Hamiltonian Systems, Superintegrability and Haantjes Geometry. https://arxiv.org/abs/2012.09819

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