For over a century, the Kepler problem and the harmonic oscillator have been known as the only central force dynamical systems, all of whose bounded motions are periodic. Two of the authors (T. I. and N. K.) have found an infinite number of dynamical systems possessing such a periodicity property, which have been called multi-fold Kepler systems or ν-fold Kepler systems, with ν a positive rational number. If ν is allowed to take the real positive numbers, say ν=α, then for the α-fold Kepler system, all the bounded motions become periodic or not, according to whether the parameter α is a rational number or not. A purpose of this paper is to quantize the α-fold Kepler system and thereby to figure out a quantum analog of the closed orbit property of the α-fold Kepler system. It will turn out that the quantized α-fold Kepler system admits accidental degeneracy in energy levels or not, according to whether α is a rational number or not.
According to the Bertrand theorem, the Kepler problem and the harmonic oscillator are the only central force dynamical systems that have closed orbits for all bounded motions. In this article, an infinite number of dynamical systems having such a closed orbit property are found on T*(R3−{0}) by applying a slightly modified Bertrand’s method to a spherical symmetric Hamiltonian with two undetermined functions of the radius. Actually, for any positive rational number ν, there exists a Hamiltonian system with the closed orbit property just mentioned, which system will be called the ν-fold Kepler system. Each of the systems is completely integrable and further allows the explicit expression of trajectories. The bounded trajectories in the configuration space R3−{0} may have self-intersection points. Moreover, the ν-fold Kepler system is reducible to a two-degrees-of-freedom system, which is completely integrable and gives rise to flows on the two-torus for bounded motions. If ν is allowed to take irrational numbers, any flow is shown to be dense in the torus. In conclusion, on the analogy of the Kepler problem, the Runge–Lenz-like vector for the ν-fold Kepler system is touched upon.
A class of integrable Hamiltonian systems whose solutions are (perhaps) all completely periodic According to the Bertrand theorem, the Kepler problem and the harmonic oscillator are the only central force dynamical systems that have closed orbits for all bounded motions. In this article, other dynamical systems having such a closed orbit property are found on T*(R3--0)).Consider a natural dynamical system on T*(R4-{0}) whose Hamiltonian function is composed of kinetic and potential energies, and invariant under a SO(2) action. Then one can reduce the system to a Hamiltonian system on T*(R3-{0}) by the use of the Kustaanheimo-Stiefel transformation. If the original potential on R"-(O) is a central one, Bertrand's method is applicable to the reduced system for determining the potential so that any bounded motions may be periodic. As a result, two types of potential functions will be found; one is linear in the radial variable and the other proportional to the inverse square root of that. The dynamical systems obtained are capable of physical interpretation. In particular, the dynamical system with the inverse square root potential may be called the twofold Kepler system, whose bounded trajectories have a selfintersection point.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.