Integrable and Superintegrable 3D Newtonian Potentials Using Quadratic First Integrals: A Review

The determination of the first integrals (FIs) of a dynamical system and the subsequent assessment of their integrability or superintegrability in a systematic way is still an open subject. One method which has been developed along these lines for holonomic autonomous dynamical systems with dynamical equations q¨a=−Γbca(q)q˙bq˙c−Qa(q), where Γbca(q) are the coefficients of the Riemannian connection defined by the kinetic metric of the system and −Qa(q) are the generalized forces, is the so-called direct method. According to this method, one assumes a general functional form for the FI I and requires the condition dIdt=0 along the dynamical equations. This results in a system of partial differential equations (PDEs) to which one adds the necessary integrability conditions of the involved scalar quantities. It is found that the final system of PDEs breaks into two sets: a. One set containing geometric elements only and b. A second set with geometric and dynamical quantities. Then, provided the geometric quantities are known or can be found, one uses the second set to compute the FIs and, accordingly, assess the integrability of the dynamical system. The `solution’ of the system of PDEs for quadratic FIs (QFIs) has been given in a recent paper (M. Tsamparlis and A. Mitsopoulos, J. Math. Phys. 61, 122701 (2020)). In the present work, we consider the application of this `solution’ to Newtonian autonomous conservative dynamical systems with three degrees of freedom, and compute integrable and superintegrable potentials V(x,y,z) whose integrability is determined via autonomous and/or time-dependent QFIs. The geometric elements of these systems are the ones of the Euclidean space E3, which are known. Setting various values for the parameters determining the geometric elements, we determine in a systematic way all known integrable and superintegrable potentials in E3 together with new ones obtained in this work. For easy reference, the results are collected in tables so that the present work may act as an updated review of the QFIs of Newtonian autonomous conservative dynamical systems with three degrees of freedom. It is emphasized that, by assuming different values for the parameters, other authors may find more integrable potentials of this type of system.