The Perlick system type I: from the algebra of symmetries to the geometry of the trajectories

Abstract

In this paper, we investigate the main algebraic properties of the maximally superintegrable system known as "Perlick system type I". All possible values of the relevant parameters, $K$ and $\beta$, are considered. In particular, depending on the sign of the parameter $K$ entering in the metrics, the motion will take place on compact or non compact Riemannian manifolds. To perform our analysis we follow a classical variant of the so called factorization method. Accordingly, we derive the full set of constants of motion and construct their Poisson algebra. As it is expected for maximally superintegrable systems, the algebraic structure will actually shed light also on the geometric features of the trajectories, that will be depicted for different values of the initial data and of the parameters. Especially, the crucial role played by the rational parameter $\beta$ will be seen "in action".

arXiv comment16 pages, 7 figures

References in corpus (25)

• Deformed algebras, position-dependent effective masses and curved spaces: An exactly solvable Coulomb problem
• Classical and Quantum Superintegrability with Applications
• Central potentials on spaces of constant curvature: The Kepler problem on the two-dimensional sphere $S^2$ and the hyperbolic plane $H^2$
• Quantum mechanics on spaces of nonconstant curvature: the oscillator problem and superintegrability
• Superintegrability on N-dimensional curved spaces: Central potentials, centrifugal terms and monopoles
• Factorizations of one dimensional classical systems
• A maximally superintegrable system on an n-dimensional space of nonconstant curvature
• Conformal compactification and cycle-preserving symmetries of spacetimes
• Coulomb-oscillator duality in spaces of constant curvature
• Periodic orbits for an infinite family of classical superintegrable systems
• Bertrand spacetimes as Kepler/oscillator potentials
• Hamiltonian systems admitting a Runge-Lenz vector and an optimal extension of Bertrand's theorem to curved manifolds
• Polynomial Poisson Algebras for Classical Superintegrable Systems with a Third Order Integral of Motion
• The anisotropic oscillator on curved spaces: A new exactly solvable model
• Integrable generalizations of oscillator and Coulomb systems via action-angle variables
• A new exactly solvable quantum model in N dimensions
• On Hamiltonians with position-dependent mass from Kaluza-Klein compactifications
• Classical and quantum higher order superintegrable systems from coalgebra symmetry
• Quantum Integrals from Coalgebra Structure
• The classical Taub-Nut System: factorization, spectrum generating algebra and solution to the equations of motion
• Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems
• Generalized Heisenberg Algebras, SUSYQM and Degeneracies: Infinite Well and Morse Potential
• Factorization approach to superintegrable systems: Formalism and applications
• New superintegrable models with position-dependent mass from Bertrand's Theorem on curved spaces
• The classical Darboux III oscillator: factorization, Spectrum Generating Algebra and solution to the equations of motion

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