Integrability and Non-Integrability of Planar Hamiltonian Systems of Cosmological Origin

Abstract

We study the problem of non-integrability (integrability) of cosmological dynamical systems which are given in the Hamiltonian form with indefinite kinetic energy form T = 1 2 g (v, v), where g is a two-dimensional pseudo-Riemannian metric with a Lorentzian signature (+, −), and v ∈ T x M is a tangent vector at a point x ∈ M of the configuration space M.

On marginal deformations and non-integrability

“…The above approach to declaring systems non-integrable has been successfully applied to various situations, with some interesting examples including [39][40][41]. It has also been applied to cosmological models as well as to theories arising in the context of gauge/gravity duality.…”

Chaos around holographic Regge trajectories

“…An important property of the Kovacic's algorithm is that it works if and only if the system is integrable, thus a failure of completing the algorithm equates to a proof of non-integrability. This route of declaring systems non-integrable has been successfully applied to various situations, some interesting examples include: [26,27,28,29]. See also [30] for nonintegrability of generalizations of the Hénon-Heiles system [24].…”

Analytic Non-integrability in String Theory

“…In all the previous examples in the mathematical literature homogeneity of the potential played a crucial role in the proof [20,21,22]. Another mathematical curiosity comes from the fact that traditionally due to the works of Hadamard and later of Anosov, chaos has been associated with the motion of particles in negatively curved spaces through the Jacobi equation.…”

“…An important property of the Kovacic's algorithm is that it works if and only if the system is integrable, thus a failure of completing the algorithm equates to a proof of non-integrability. This route of declaring systems non-integrable has been successfully applied to various situations, some interesting examples include: [20,22,19,21]. See also [23] for nonintegrability of generalizations of the Hénon-Heiles system [17].…”