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.PACS numbers: 02.40.Ky,04.20.-q ‡ It should be remarked that the idea of considering connections with totally skew-symmetric torsion was already proposed by Strominger [21] in the context of string theories, where such a torsion is identified with the 3-form flux livinging in the theories.

On First Integrals of Two-Dimensional Geodesic Flows

“…Depending on the parameter α, the first integral F can be polynomial, rational, algebraic, or even transcendental. As observed in [19], if we put f (p 1 , p 2 ) = p 1 + p 2 , α = − s r , where r and s are coprimes; then we obtain the first integral of the geodesic flow rational in momenta:…”

“…It was proved in [18] that on two-dimensional surfaces there exist analytic Riemannian metrics whose geodesic flow has (locally) irreducible first integrals rational in momenta with the numerator and denominator of arbitrarily high degree. Note however that rather few explicit examples of these metrics are available (see [19][20][21]). Moreover, little is known about whether these metrics (and the corresponding integrals) can exist in the whole phase space.…”

“…The proof is based on applying the Cauchy-Kovalevskaya theorem. However, the explicit construction of such metrics and rational integrals (as well as polynomial integrals of high degrees) faces with various difficulties: one can find few known examples in [17]- [19] (see also [14]). In this paper we investigate this problem (partially following the ideas outlined in [15]) in the simplest case when the degrees of a numerator and a denominator of the rational integral are equal to 1.…”

Black holes, hidden symmetries, and complete integrability

“…Remark: An interesting generalization of Killing tensors has been recently proposed in Aoki et al (2016). It follows from considering 'inconstructible rational first integrals' of the type C = A/B, where A and B are monomials of arbitrary orders.…”

О Первых Интегралах Двумерных Геодезических Потоков

“…Отметим, однако, что явных примеров таких метрик известно крайне мало (см. [19][20][21]). Более того, мало что известно о том, могут ли эти метрики (и соответствующие интегралы) существовать во всем фазовом пространстве.…”