Hidden symmetry of hyperbolic monopole motion

Abstract

Hyperbolic monopole motion is studied for well separated monopoles. It is shown that the motion of a hyperbolic monopole in the presence of one or more fixed monopoles is equivalent to geodesic motion on a particular submanifold of the full moduli space. The metric on this submanifold is found to be a generalisation of the multi-centre Taub-NUT metric introduced by LeBrun. The one centre case is analysed in detail as a special case of a class of systems admitting a conserved Runge-Lenz vector. The two centre problem is also considered. An integrable classical string motion is exhibited.

arXiv comment39 pages, 7 figures, references added, minor changes to section 2

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• Superintegrability on N-dimensional curved spaces: Central potentials, centrifugal terms and monopoles
• Bertrand spacetimes as Kepler/oscillator potentials
• Killing-Yano tensors and some applications
• Hamiltonian systems admitting a Runge-Lenz vector and an optimal extension of Bertrand's theorem to curved manifolds
• Multi-center MICZ-Kepler system, supersymmetry and integrability
• Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform
• Relationship between quantum mechanics with and without monopoles
• Curved manifolds with conserved Runge-Lenz vectors
• An exactly solvable deformation of the Coulomb problem associated with the Taub-NUT metric
• The classical Taub-Nut System: factorization, spectrum generating algebra and solution to the equations of motion
• Generalized Kaluza-Klein monopole, quadratic algebras and ladder operators
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• Generic N=4 supersymmetric hyper-Kähler sigma models in D=1
• Superintegrable systems, polynomial algebra structures and exact derivations of spectra
• Hidden symmetries in a gauge covariant approach, Hamiltonian reduction and oxidation
• A Spacetime Geometry picture of Forest Fire Spreading and of Quantum Navigation
• Recurrence approach and higher rank polynomial algebras for superintegrable monopole systems
• N=8 supersymmetric mechanics on the sphere S^3
• Bianchi-IX, Darboux-Halphen and Chazy-Ramanujan
• The fate of Hamilton's Hodograph in Special and General Relativity
• A maximally superintegrable deformation of the N-dimensional quantum Kepler-Coulomb system
• Conformal symmetries of self-dual hyperbolic monopole metrics
• The Asymptotic Structure of the Centred Hyperbolic 2-Monopole Moduli Space
• Kepler unbound: some elegant curiosities of classical mechanics
• Exactly solvable deformations of the oscillator and Coulomb systems and their generalization
• On the non-Abelian monopoles on the background of spaces with constant curvature