Generalized Stäckel Transform and Reciprocal Transformations for Finite-Dimensional Integrable Systems

Abstract

We present a multiparameter generalization of the Stäckel transform (the latter is also known as the coupling-constant metamorphosis) and show that under certain conditions this generalized Stäckel transform preserves the Liouville integrability, noncommutative integrability and superintegrability. The corresponding transformation for the equations of motion proves to be nothing but a reciprocal transformation of a special form, and we investigate the properties of this reciprocal transformation. Finally, we show that the Hamiltonians of the systems possessing separation curves of apparently very different form can be related through a suitably chosen generalized Stäckel transform.

21 pages, LaTeX 2e, no figures; major revision; Propositions 2 and 7 and several new references added

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Cited by in corpus (26)

• Classical and Quantum Superintegrability with Applications
• Superintegrability and higher order constants for classical and quantum systems
• Superintegrability and higher order constants for quantum systems
• Coupling constant metamorphosis and Nth order symmetries in classical and quantum mechanics
• A new exactly solvable quantum model in N dimensions
• Normal forms for pseudo-Riemannian 2-dimensional metrics whose geodesic flows admit integrals quadratic in momenta
• Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform
• Bi-Hamiltonian representation of Stäckel systems
• Higher-order superintegrability of a Holt related potential
• First Integrals of Extended Hamiltonians in n+1 Dimensions Generated by Powers of an Operator
• On reciprocal equivalence of Stäckel systems
• On two superintegrable nonlinear oscillators in N dimensions
• Staeckel systems generating coupled KdV hierarchies and their finite-gap and rational solutions
• Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems
• Structure results for higher order symmetry algebras of 2D classical superintegrable systems
• Comment on `conservative discretizations of the Kepler motion'
• Infinitesimal time reparametrisation and its applications
• Coupling constant metamorphosis as an integrability-preserving transformation for general finite-dimensional dynamical systems and ODEs
• Algebraic Conditions for Conformal Superintegrability in Arbitrary Dimension
• Transformation of the Stackel matrices preserving superintegrability
• Integrable quantum Stäckel systems
• Classical and Quantum Superintegrability of Stäckel Systems
• Non-Homogeneous Hydrodynamic Systems and Quasi-Stäckel Hamiltonians
• Stäckel Equivalence of Non-Degenerate Superintegrable Systems, and Invariant Quadrics
• Flat minimal quantizations of Stackel systems and quantum separability
• Algebraic curves as a source of separable multi-Hamiltonian systems