The Kepler problem is a dynamical system that is well defined not only on the Euclidean plane but also on the sphere and on the Hyperbolic plane. First, the theory of central potentials on spaces of constant curvature is studied. All the mathematical expressions are presented using the curvature $\k$ as a parameter, in such a way that they reduce to the appropriate property for the system on the sphere $S^2$, or on the hyperbolic plane $H^2$, when particularized for $\k>0$, or $\k<0$, respectively; in addition, the Euclidean case arises as the particular case $\k=0$. In the second part we study the main properties of the Kepler problem on spaces with curvature, we solve the equations and we obtain the explicit expressions of the orbits by using two different methods: first by direct integration and second by obtaining the $\k$-dependent version of the Binet's equation. The final part of the article, that has a more geometric character, is devoted to the study of the theory of conics on spaces of constant curvature.
Generalized MICZ-Kepler systems on three-dimensional sphere and hyperboloid
Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research（博戈留博夫理论物理实验室，俄罗斯杜布纳联合核研究所）；Yerevan State University（埃里温国立大学）；Yerevan Physics Institute（埃里温物理研究所）；Institute of Radiophysics and Electronics（辐射物理与电子研究所）
The superintegrability of several Hamiltonian systems defined on three-dimensional configuration spaces of constant curvature is studied. We first analyze the properties of the Killing vector fields, Noether symmetries and Noether momenta. Then we study the superintegrability of the Harmonic Oscillator, the Smorodinsky-Winternitz (S-W) system and the Harmonic Oscillator with ratio of frequencies 1:1:2 and additional nonlinear terms on the 3-dimensional sphere $S^3$ ($\kp>0)$ and on the hyperbolic space $H^3$ ($\kp<0$). In the second part we present a study first of the Kepler problem and then …
In this paper we regularize the Kepler problem on $S^3$ in several different ways. First, we perform a Moser-type regularization. Then, we adapt the Ligon-Schaaf regularization to our problem. Finally, we show that the Moser regularization and the Ligon-Schaaf map we obtained can be understood as the composition of the corresponding maps for the Kepler problem in Euclidean space and the gnomonic transformation.
In this work we study the minimization problem for the total distance in a cloud computing network on the sphere. We give a solution to this problem in terms of hyperbolic Voronoi diagrams on the sphere. We present results of computer simulations illustrating the solution.
Transition to a nonrelativistic Pauli equation in Riemann space of constant positive curvature for a Dirac particle in presence of the Coulomb field is performed in the system of radial equations, exact solutions are constructed in terms of Hein functions, the energy spectrum is derived. The same is done for the Kepler quantum problem in hyperbolic Lobachevsky space, solutions are constructed in terms of Hein functions, the energy spectrum is obtained.
Two dimensional massless Dirac Hamiltonian under the influence of hyperbolic magnetic fields is mentioned in curved space. Using a spherical surface parametrization, the Dirac operator on the sphere is presented and the system is given as two supersymmetric partner Hamiltonians which coincides with the position dependent mass Hamiltonians. We introduce two ansatzes for the component of the vector potential to acquire effective solvable models, which are Rosen Morse II potential and the model given in \cite{mid} whose bound states are Jacobi $X_1$ type polynomials, and we adapt our work to thes…
Hodographs for the Kepler problem are circles. This fact, known since almost two centuries ago, still provides the simplest path to derive the Kepler first law. Through Feynman `lost lecture', this derivation has now reached to a wider audience. Here we look again at Feynman's approach to this problem as well as at the recently suggested modification by van Haandel and Heckman (vHH), with two aims in view, both of which extend the scope of the approach. First we review the geometric constructions of the Feynman and vHH approaches (that prove the existence of {\itshape elliptic} orbits without …
The 2-body problem on the sphere and hyperbolic space are both real forms of holomorphic Hamiltonian systems defined on the complex sphere. This admits a natural description in terms of biquaternions and allows us to address questions concerning the hyperbolic system by complexifying it and treating it as the complexification of a spherical system. In this way, results for the 2-body problem on the sphere are readily translated to the hyperbolic case. For instance, we implement this idea to completely classify the relative equilibria for the 2-body problem on hyperbolic 3-space for a strictly …
The classical and quantum solutions of a nonlinear model describing harmonic oscillators on the sphere and the hyperbolic plane, derived in polar coordinates in a recent paper [Phys.\ Lett.\ A 379 (2015) 1589], are extended by the inclusion of an isotonic term.
A simple derivation of the classical solutions of a nonlinear model describing a harmonic oscillator on the sphere and the hyperbolic plane is presented in polar coordinates. These solutions are then related to those in cartesian coordinates, whose form was previously guessed. In addition, the nature of the classical orthogonal polynomials entering the bound-state radial wavefunctions of the corresponding quantum model is identified.
A new method for obtaining approximate solutions of the hyperbolic Kepler's equation
Centro Universitario de la Defensa Academia General Militar（国防大学中心军事综合学院）；IUMA, Universidad de Zaragoza（萨拉戈萨大学数学统一建模与应用研究所）；Departamento de Matemáticas and IUMA Facultad de Educación. Universidad de Zaragoza（萨拉戈萨大学教育学院数学系及数学统一建模与应用研究所）；Instituto de Educación Secundaria Élaios（Élaios中学）；Universidad de Zaragoza（萨拉戈萨大学）
This paper defines a class of variational problems on Lie groups that admit involutive automorphisms. The maximum Principle of optimal control then identifies the appropriate left invariant Hamiltonians on the Lie algebra of the group. The corresponding Hamiltonian equations admit a spectral representation with important applications to the theory of integrable systems. Particular cases make a link to Moser's work relating the solutions of Kepler's problem to the geodesics on spaces of constant curvature, Newmann's mechanical problem on the sphere and related geodesic problem of Jacobi on an e…
The Stäckel transform is applied to the geodesic motion on Euclidean space, through the harmonic oscillator and Kepler-Coloumb potentials, in order to obtain maximally superintegrable classical systems on N-dimensional Riemannian spaces of nonconstant curvature. By one hand, the harmonic oscillator potential leads to two families of superintegrable systems which are interpreted as an intrinsic Kepler-Coloumb system on a hyperbolic curved space and as the so-called Darboux III oscillator. On the other, the Kepler-Coloumb potential gives rise to an oscillator system on a spherical curved space a…
We elucidate the large-N dynamics of one-dimensional sigma models with spherical and hyperbolic target spaces and find a duality between the Lagrange multiplier and the angular momentum. In the hyperbolic model we propose a new class of operators based on the irreducible representations of hyperbolic space. We also uncover unexpected zero modes which lead to the double scaling of the 1/N expansion and explore these modes using Gelfand-Dikiy equations.
Maximal superintegrability of the generalized Kepler--Coulomb system on N-dimensional curved spaces
最近，通过找到一个附加的（隐藏的）动量四次积分，在三维欧几里得空间上叠加有三个离心项的开普勒-库仑势被证明是最大超可积的[Verrier P E and Evans N W 2008 J. Math. Phys. 49 022902]。在本文中，我们提出的推广，这一结果的ND球面，双曲和欧几里德空间，利用统一的对称性的方法，利用曲率参数。由此产生的哈密顿量，形成的（弯曲）开普勒-库仑势连同N离心条款，被赋予（2N-1）功能独立的运动积分：其中一个是四次和其余的是二次。从适当的开普勒-库仑势，其相关的二次Laplace-Runge-Lenz N-向量，广义系统的过渡得到了充分的描述。在所有这些系统中的球形，非线性（立方），和余代数对称性的作用是突出的。
The quantum oscillator and Kepler-Coulomb problems in $d$-dimensional spaces with constant curvature are analyzed from several viewpoints. In a deformed supersymmetric framework, the corresponding nonlinear potentials are shown to exhibit a deformed shape invariance property. By using the point canonical transformation method, the two deformed Schrödinger equations are mapped onto conventional ones corresponding to some shape-invariant potentials, whose rational extensions are well known. The inverse point canonical transformations then provide some rational extensions of the oscillator and Ke…
