Abstract
We consider the generic quadratic first integral (QFI) of the form I=Kab(t,q)q˙aq˙b+Ka(t,q)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 Kab(t,q), Ka(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.
Highlights
The precise meaning of the solution of a system of differential equations can be cast in several ways [1]
On the other hand, when we have proved the existence of a sufficient number of independent explicit first integrals and invariants for the dynamical system, we say that we have found an analytic solution of the dynamical equations
We find the solution by using the time-dependent linear FIs (LFIs) L42± = e±kt (ẋ ∓ kx) and L43± = e±kt (ẏ ∓ ky)
Summary
The precise meaning of the solution of a system of differential equations can be cast in several ways [1]. A universal method which computes the FIs for all types of dynamical equations independently of their complexity and degrees of freedom is not available For this reason, the existing studies restrict their considerations to flat spaces or spaces of constant curvature of low dimension (see, e.g., in [3,9,10,11,12,13,14,15] and references therein). Koenigs [17] used coordinate transformations in order to solve the system of equations resulting from the condition { H, I } = 0 This solution of that system of equations gives the general functional form of the QFIs and the superintegrable free Hamiltonians, that is, the ones which possess two more QFIs—in addition to the Hamiltonian—which are functionally independent. All the results listed in the review paper of [16] as well as in more recent works (see, e.g., in [14,15]) are recovered while some new ones are found which admit time-dependent LFIs and QFIs
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
