Integrable and Superintegrable Potentials of 2d Autonomous Conservative Dynamical Systems

Abstract

We consider the generic quadratic first integral (QFI) of the form $I=K_{ab}(t,q){\dot{q}}^a{\dot{q}}^b+K_a(t,q){\dot{q}}^a+K(t,q)$ and require the condition $dI/dt=0$. The latter results in a system of partial differential equations which involve the tensors $K_{ab}(t,q)$, $K_a(t,q)$, $K(t,q)$ and the dynamical quantities of the dynamical equations. These equations divide in two sets. The first set involves only geometric quantities of the configuration space and the second set contains the interaction of these quantities with the dynamical fields. A theorem is presented which provides a systematic solution of the system of equations in terms of the collineations of the kinetic metric in the configuration space. This solution being geometric and covariant, applies to higher dimensions and curved spaces. The results are applied to the simple but interesting case of two-dimensional (2d) autonomous conservative Newtonian potentials. It is found that there are two classes of 2d integrable potentials and that superintegrable potentials exist in both classes. We recover most main previous results, which have been obtained by various methods, in a single and systematic way.

References in corpus (11)

• Invariant Variation Problems
• Completeness of superintegrability in two-dimensional constant curvature spaces
• Two dimensional dynamical systems which admit Lie and Noether symmetries
• Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two dimensional manifold
• Killing Tensors and Conformal Killing Tensors from Conformal Killing Vectors
• On the geometry of Killing and conformal tensors
• Nonlinear Ordinary Differential Equations: A discussion on Symmetries and Singularities
• Quadratic first integrals of autonomous conservative dynamical systems
• Constants of motion in stationary axisymmetric gravitational fields
• Killing Tensors and Symmetries
• Quadratic conservation laws and collineations: a discussion

Cited by in corpus (5)

• Higher order first integrals of autonomous dynamical systems
• The generalized Ermakov conservative system: A discussion
• Quadratic first integrals of constrained autonomous conservative dynamical systems with fixed energy
• Integrable and superintegrable 3d Newtonian potentials using quadratic first integrals: A review
• Cubic first integrals of autonomous dynamical systems in $E^2$ by an algorithmic approach