Periodic orbit theory revisited in the anisotropic Kepler problem (original) (raw)
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Orbit systematics in anisotropic Kepler problem
Artificial Life and Robotics, 2008
We revisit the Anisotropic Kepler Problem (AKP), which concerns with trajectories of an electron with anisotropic mass term in a Coulomb fi eld. This is one of the most fundamental fi elds in Quantum Chaos. Nowadays various quantum systems are challenging us. Classical theories of these may have chaos. Quantum mechanics have developed from integrable cases and may have to be reformulated for such cases. AKP then serves as a suitable testing ground for quantum chaos. We fi rst review a pioneering work by Martin Gutzwiller (J Math Phys (1977) 18:106). We shall show the systematics of the trajectories using ample fi gures from an extensive numerical analysis. Then we focus on the rolê of hyperbolic singularities and we comment on the approximations in an analytic formulation.
Periodic Orbits of the Planar Anisotropic Kepler Problem
In this paper, we prove that at every energy level the anisotropic Kepler problem with small anisotropy has two periodic orbits which bifurcate from elliptic orbits of the Kepler problem with high eccentricity. Moreover we provide approximate analytic expressions for these periodic orbits. The tool for proving this result is the averaging theory.
Periodic orbits in the Chermnykh problem
Astrodynamics, 2017
Periodic orbits in irregular gravitational fields are significant for an understanding of dynamical behaviors around asteroids as well as the engineering aspect for deep space explorations. The rotating mass dipole, referred to as the Chermnykh problem, is a good alternative model to study qualitative dynamical environments near elongated asteroids, like the asteroid 1620 Geographos, 216 Kleopatra, or 25143 Itokawa. In this paper a global searching method is adopted to search for periodic orbits around the dipole model based on the concept of Poincaré section of surface. Representative families of periodic orbits are illustrated with respect to all three topological cases of the dipole model. Topological transitions of orbits during iso-energetic continuations are also presented as well as identification of new types of periodic orbits. KEYWORDS periodic orbits the Chermnykh problem topological transition iso-energetic continuation Research Article
Quantum-classical correspondence in the vicinity of periodic orbits
Physical review. E, 2018
Quantum-classical correspondence in chaotic systems is a long-standing problem. We describe a method to quantify Bohr's correspondence principle and calculate the size of quantum numbers for which we can expect to observe quantum-classical correspondence near periodic orbits of Floquet systems. Our method shows how the stability of classical periodic orbits affects quantum dynamics. We demonstrate our method by analyzing quantum-classical correspondence in the quantum kicked top (QKT), which exhibits both regular and chaotic behavior. We use our correspondence conditions to identify signatures of classical bifurcations even in a deep quantum regime. Our method can be used to explain the breakdown of quantum-classical correspondence in chaotic systems.
Periodic Orbit Quantization by Harmonic Inversion of Gutzwiller's Recurrence Function
Physical Review Letters, 1997
Semiclassical eigenenergies and resonances are obtained from classical periodic orbits by harmonic inversion of Gutzwiller's semiclassical recurrence function, i.e., the trace of the propagator. Applications to the chaotic three disk scattering system and, as a mathematical model, to the Riemann zeta function demonstrate the power of the technique. The method does not depend on the existence of a symbolic code and might be a tool for a semiclassical quantization of systems with nonhyperbolic or mixed regular-chaotic dynamics as well.
The angular momentum in the classical anisotropic Kepler problem
The behavior of the angular momentum of the two dimensional Anisotropic Kepler Problem (AKP) is addressed. We find here ourselves, from the point of view of physics didactics, with a classical mechanics ``simple" problem that should be carefully analyzed from the outset. Taking into account that the angular momentum varies with time due to an ``inertial torque", we are still allowed to restrict the problem to a two dimensional motion, and then, being the angular momentum in this restricted case, a one-dimensional variable, we study how its behavior can describe the dynamics of this chaotic system. The approach to this problem through the angular momentum, to our knowledge, has not been reported in the literature. We investigate, from a numerical solution of the equations of motion, different features of this quantity and obtain a return plot for the angular momentum, as well as some phase space diagrams for the torque vs. angular momentum, for different values of the anisotropy parameter, by using a Poincare surface section.
The amplitude distribution - a periodic orbit theory approach
Physics Letters A, 1991
Irregular eigenfunctions in chaotic Hamiltonian systems seem to follow a Gaussian amplitude distribution. However, there has been no explanation using periodic orbit theory (POT) which forms the basis ofsemiclassical mechanics.Using the ideas of POT, we shall argue that these states do have a universal limiting distribution which closely approximates a Gaussian.
Journal of physics, 2018
Using binary coding of orbit we introduce a finite level (N) surface over the initial value domain D of 2-dim AKP. It gives a tiling of D by base ribbons. The scheme of the one-time map is studied and the properness of the tiling is proved. This analysis in turn resolves the long standing puzzle in AKP-the non-uniqueness issue of a PO for a given code. We argue that the unique existence of a periodic orbit (PO) for a given binary code generally holds (for inverse anisotropy parameter γ < 8/9) but there is a remarkable exception in which a ribbon with a certain code escapes from shrinking at large N and embodies the Broucke-type stable PO (S). It comes along the bifurcation of an unstable PO (U): U (R) → S(R) + U ′ (N R) (R for retracing and NR for non-retracing). An analysis based on orbit topology clarifies the pattern of the bifurcation; we give a conjecture that it occurs among odd rank Y-symmetric POs.