Rational first integrals of geodesic equations and generalised hidden symmetries

Abstract

We discuss novel generalisations of Killing tensors, which are introduced by considering rational first integrals of geodesic equations. We introduce the notion of inconstructible generalised Killing tensors, which cannot be constructed from ordinary Killing tensors. Moreover, we introduce inconstructible rational first integrals, which are constructed from inconstructible generalised Killing tensors, and provide a method for checking the inconstructibility of a rational first integral. Using the method, we show that the rational first integral of the Collinson-O'Donnell solution is not inconstructible. We also provide several examples of metrics admitting an inconstructible rational first integral in two and four dimensions, by using the Maciejewski-Przybylska system. Furthermore, we attempt to generalise other hidden symmetries such as Killing-Yano tensors.

arXiv comment12 pages; v2: minor modifications; v3: further improvements, refs added, matches published version in CQG

References in corpus (14)

• Higher-Dimensional Black Holes: Hidden Symmetries and Separation of Variables
• Killing and Conformal Killing tensors
• Generalized hidden symmetries and the Kerr-Sen black hole
• Generalized Killing-Yano equations in D=5 gauged supergravity
• Commuting symmetry operators of the Dirac equation, Killing-Yano and Schouten-Nijenhuis brackets
• Hidden Symmetry and Exact Solutions in Einstein Gravity
• Killing(-Yano) Tensors in String Theory
• Killing-Yano tensors and some applications
• Darboux Polynomials and First Integrals of Natural Polynomial Hamiltonian Systems
• Separability of a modified Dirac equation in a five-dimensional rotating, charged black hole in string theory
• Killing-Yano symmetry of Kaluza-Klein black holes in five dimensions
• Integrability Conditions for Killing-Yano Tensors and Maximally Symmetric Spaces in the Presence of Torsion
• Higher-dimensional lifts of Killing-Yano forms with torsion
• A common generalization of the Frölicher-Nijenhuis bracket and the Schouten bracket for symmetry multi vector fields

Cited by in corpus (5)

• Black holes, hidden symmetries, and complete integrability
• Some metrics admitting nonpolynomial first integrals of the geodesic equation
• Chaotic particle motion around a homogeneous circular ring
• Integrable magnetic geodesic flows on 2-surfaces
• Rational integrals of 2-dimensional geodesic flows: new examples