Projectively equivalent 2-dimensional superintegrable systems with projective symmetries

Abstract

This paper combines two classical theories, namely metric projective differential geometry and superintegrability. We study superintegrable systems on 2-dimensional geometries that share the same geodesics, viewed as unparametrized curves. We give a definition of projective equivalence of such systems, which may be considered the projective analog of (conformal) Stäckel equivalence (coupling constant metamorphosis). Then, we discuss the transformation behavior for projectively equivalent superintegrable systems and find that the potential on a projectively equivalent geometry can be reconstructed from a characteristic vector field. Moreover, potentials of projectively equivalent Hamiltonians follow a linear superimposition rule. The techniques are applied to several examples. In particular, we use them to classify, up to Stäckel equivalence, the superintegrable systems on geometries with one, non-trivial projective symmetry.

arXiv comment18 pages, 2 figures, 1 table. Reorganized and further examples have been added

References in corpus (19)

• Completeness of superintegrability in two-dimensional constant curvature spaces
• Superintegrable Systems in Darboux spaces
• Superintegrability in a two-dimensional space of non-constant curvature
• A solution of a problem of Sophus Lie: Normal forms of 2-dim metrics admitting two projective vector fields
• Contractions of 2D 2nd Order Quantum Superintegrable Systems and the Askey Scheme for Hypergeometric Orthogonal Polynomials
• Geodesically equivalent metrics in general relativity
• Nondegenerate 3D complex Euclidean superintegrable systems and algebraic varieties
• Invariant classification of second-order conformally flat superintegrable systems
• Normal forms for pseudo-Riemannian 2-dimensional metrics whose geodesic flows admit integrals quadratic in momenta
• Leonard Euler: addition theorems and superintegrable systems
• A criterion for compatibility of conformal and projective structures
• Invariant Classification and Limits of Maximally Superintegrable Systems in 3D
• On Kähler metrisability of two-dimensional complex projective structures
• Weyl metrisability of two-dimensional projective structures
• Path Integral Approach for Superintegrable Potentials on Spaces of Non-constant Curvature: I. Darboux Spaces DI and DII
• Path Integral Approach for Superintegrable Potentials on Spaces of Non-constant Curvature: II. Darboux Spaces DIII and DIV
• Invariant prolongation of the Killing tensor equation
• Global structure and geodesics for Koenigs superintegrable systems
• (Super-)integrable systems associated to 2-dimensional projective connections with one projective symmetry

Cited by in corpus (3)

• Algebraic Conditions for Conformal Superintegrability in Arbitrary Dimension
• Metrisability of projective surfaces and pseudo-holomorphic curves
• Dynamical symmetry algebra of two superintegrable two-dimensional systems