Conformal Killing Tensors and covariant Hamiltonian Dynamics

Abstract

A covariant algorithm for deriving the conserved quantities for natural Hamiltonian systems is combined with the non-relativistic framework of Eisenhart, and of Duval, in which the classical trajectories arise as geodesics in a higher dimensional space-time, realized by Brinkmann manifolds. Conserved quantities which are polynomial in the momenta can be built using time-dependent conformal Killing tensors with flux. The latter are associated with terms proportional to the Hamiltonian in the lower dimensional theory and with spectrum generating algebras for higher dimensional quantities of order $1$ and $2$ in the momenta. Illustrations of the general theory include the Runge-Lenz vector for planetary motion with a time-dependent gravitational constant $G(t)$, motion in a time-dependent electromagnetic field of a certain form, quantum dots, the Hénon-Heiles and Holt systems, respectively, providing us with Killing tensors of rank that ranges from one to six.

arXiv commentSome details clarified. 30 pages, no figures

References in corpus (30)

• Celestial Mechanics, Conformal Structures, and Gravitational Waves
• SUSY in the sky
• The geometry of Schrödinger symmetry in non-relativistic CFT
• Field-dependent symmetries of a non-relativistic fluid model
• Effects of symmetry breaking in finite quantum systems
• Conformal Properties of Chern-Simons Vortices in External Fields
• Conformal Properties of Chern-Simons Vortices in External Fields
• Spinors in non-relativistic Chern-Simons electrodynamics
• Covariant hamiltonian dynamics
• Conformal symmetry of the coupled Chern-Simons and gauged nonlinear Schrödinger equations
• Some Spacetimes with Higher Rank Killing-Stackel Tensors
• Quantum integrability of quadratic Killing tensors
• Explicit integration of the Hénon-Heiles Hamiltonians
• Hidden symmetries of two-electron quantum dots in a magnetic field
• On Integrability of spinning particle motion in higher-dimensional black hole spacetimes
• Generalised Eisenhart lift of the Toda chain
• Killing tensors and canonical geometry
• Higher rank Killing tensors and Calogero model
• Hidden symmetries of Eisenhart-Duval lift metrics and the Dirac equation with flux
• Constants of Motion for Constrained Hamiltonian Systems: A Particle around a Charged Rotating Black Hole
• Curved manifolds with conserved Runge-Lenz vectors
• Conserved quantities in non-abelian monopole fields
• Kohn's theorem and Galilean symmetry
• Schrödinger Manifolds
• Higher order first integrals of motion in a gauge covariant Hamiltonian framework
• Goryachev-Chaplygin, Kovalevskaya, and Brdička-Eardley-Nappi-Witten pp-waves spacetimes with higher rank Stäckel-Killing tensors
• Geometry of Lax pairs: particle motion and Killing-Yano tensors
• Separability and Dynamical Symmetry of Quantum Dots
• Dynamical supersymmetry of spin particle-magnetic field interaction
• Covariant Approach of the Dynamics of Particles in External Gauge Fields, Killing Tensors and Quantum Gravitational Anomalies

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• Ricci-flat spacetimes admitting higher rank Killing tensors
• Eisenhart lift for higher derivative systems
• Schrödinger Symmetry in Gravitational Mini-Superspaces
• Eisenhart Lift of $2$--Dimensional Mechanics
• On the Schr{ö}dinger-Newton equation and its symmetries: a geometric view
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• More on Jacobi metric: Randers-Finsler metrics, frame dragging and geometrisation techniques
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