We construct and analyze a generalization of the Kepler problem. These generalized Kepler problems are parameterized by a triple $(D, κ, μ)$ where the dimension $D\ge 3$ is an integer, the curvature $κ$ is a real number, the magnetic charge $μ$ is a half integer if $D$ is odd and is 0 or 1/2 if $D$ is even. The key to construct these generalized Kepler problems is the observation that the Young powers of the fundamental spinors on a punctured space with cylindrical metric are the right analogues of the Dirac monopoles.
We discuss the generalizations of the MICZ-Kepler system (the system describing the motion of the charged particle in the field of Dirac dyon), to the curved spaces, arbitrary potentials and to the multi-dyon background.
We propose the integrable (pseudo)spherical generalization of the four-dimensional anisotropic oscillator with additional nonlinear potential. Performing its Kustaanheimo-Stiefel transformation we then obtain the pseudospherical generalization of the MICZ-Kepler system with linear and $\cosθ$ potential terms. We also present the generalization of the parabolic coordinates, in which this system admits the separation of variables. Finally, we get the spherical analog of the presented MICZ-Kepler-like system.
We study the regularized $(2n-1)$-Kepler problem and other Hamiltonian systems which are related to the nilpotent coadjoint orbits of $U(n,n)$. The Kustaanheimo-Stiefel and Cayley regularization procedures are discussed and their equivalence is shown. Some integrable generalization (perturbation) of $(2n-1)$-Kepler problem is proposed.
We present some generalization of 16D oscillator by anisotropic and nonlinear inharmonic terms and its dual analog for 9D related MICZ-Kepler systems by generalized version of the Kustaanheimo-Stiefel transformation. The solvability of the Schrödinger equation of the these problems by the variables separation method are discussed in different coordinates.
The Kepler problem is a physical problem about two bodies which attract each other by a force proportional to the inverse square of the distance. The MICZ-Kepler problems are its natural cousins and have been previously generalized from dimension three to dimension five. In this paper, we construct and analyze the (quantum) MICZ-Kepler problems in all dimensions higher than two.
It is shown that Kepler problem in deformed (quantum) four-dimensional space in non relativistic limit is integrable in quadratures. In non relativistic limit group of motion of quantum space coincide with Galilei one.
For each simple euclidean Jordan algebra $V$, we introduce the analogue of hamiltonian, angular momentum and Laplace-Runge-Lenz vector in the Kepler problem. Being referred to as the universal hamiltonian, universal angular momentum and universal Laplace-Runge-Lenz vector respectively, they are elements in (essentially) the TKK (Tits-Kantor-Koecher) algebra of $V$ and satisfy commutation relations similar to the ones for the hamiltonian, angular momentum and Laplace-Runge-Lenz vector in the Kepler problem. We also give some examples of Poisson realization of the TKK algebra, along with the res…
We construct the system of generalized coherent states for the quantum Kepler problem corresponds to the homogeneous domain $SU(2,2)/S(U(2)\times U(2))$. We show that the SU(2,2)-equivariant momentum map for this domain yields the momentum map for the classical Kepler problem via appropriate limiting passage. We also show that under this passage the average values of quantum observables in this system of coherent states pass into the functions on classical phase space and $-i/h$ times commutator pass into the Poisson bracket.
For a {\em simple Euclidean Jordan algebra}, let $\mathfrak{co}$ be its conformal algebra, $\mathscr P$ be the manifold consisting of its semi-positive rank-one elements, $C^\infty(\mathscr P)$ be the space of complex-valued smooth functions on $\mathscr P$. An explicit action of $\mathfrak{co}$ on $C^\infty(\mathscr P)$, referred to as the {\em hidden action} of $\mathfrak{co}$ on $\mathscr P$, is exhibited. This hidden action turns out to be mathematically responsible for the existence of the Kepler problem and its recently-discovered vast generalizations, referred to as $J$-Kepler problems.…
The trajectories of the $\mathrm{O}(1)$-Kepler problem at level $n\ge 2$ are completely determined. It is found in particular that a non-colliding trajectory is an ellipse, a parabola or a branch of hyperbola according as the total energy is negative, zero or positive. Moreover, it is shown that the group $\mathrm{GL}(n, \mathbb R)/\mathrm{O}(1)$ acts transitively on both the set of oriented elliptic trajectories and the set of oriented parabolic trajectories. The method employed here is similar to the one used by Levi-Civita in the study of planar Kepler problem in 1920.
This work addresses the Hamiltonian dynamics of the Kepler problem in a deformed phase space, by considering the equatorial orbit. The recursion operators are constructed and used to compute the integrals of motion. The same investigation is performed with the introduction of the Laplace-Runge-Lenz vector. The existence of quasi-bi-Hamiltonian structures is also elucidated. Related properties are studied.
Bound and scattering states of a non-central extension of the three-dimensional Kepler-Coulomb Hamiltonian are worked out analytically within the framework of the potential groups of the problem, SO(7) for bound states and SO(6,1) for scattering states. In the latter case, the S matrix is calculated by the method of intertwining operators.
We introduce the regularization Levi-Civita parameter for the `curved Kepler', i.e., motion under the `Kepler-Coulomb' potential in a configuration space with any constant curvature and metric of any signature type. Consistent use of this parameter allows to solve the problem of motion (orbit shape and time evolution along the orbit), thereby extending the use of the Levi-Civita parameter beyond the usual Kepler problem in a flat Euclidean configuration space. A `universal' description, where all relations are applicable to the motions in any space and with any energy follow from our approach,…
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.
In this article we explain that several integrable mechanical billiards in the plane are connected via conformal transformations. We first remark that the free billiard in the plane are conformal equivalent to infinitely many billiard systems defined in central force problems on a particular fixed energy level. We then explain that the classical Hooke-Kepler correspondence can be carried over to a correspondence between integrable Hooke-Kepler billiards. As part of the conclusion we show that any focused conic section gives rise to integrable Kepler billiards, which brings generalizations to a…
We will make the case that \textit{pedal coordinates} (instead of polar or Cartesian coordinates) are more natural settings in which to study force problems of classical mechanics in the plane. We will show that the trajectory of a test particle under the influence of central and Lorentz-like forces can be translated into pedal coordinates at once without the need of solving any differential equation. This will allow us to generalize Newton theorem of revolving orbits to include nonlocal transforms of curves. Finally, we apply developed methods to solve the "dark Kepler problem", i.e. central …
In this paper we consider a manifold with a dynamical vector field and inquire about the possible tangent bundle structures which would turn the starting vector field into a second order one. The analysis is restricted to manifolds which are diffeomorphic with affine spaces. In particular, we consider the problem in connection with conformal vector fields of second order and apply the procedure to vector fields conformally related with the harmonic oscillator (f-oscillators) . We select one which covers the vector field describing the Kepler problem.
It is demonstrated that, for the recently introduced classical magnetized Kepler problems in dimension $2k+1$, the non-colliding orbits in the "external configuration space" $\mathbb R^{2k+1}\setminus\{\mathbf 0\}$ are all conics, moreover, a conic orbit is an ellipse, a parabola, and a branch of a hyperbola according as the total energy is negative, zero, and positive. It is also demonstrated that the Lie group ${\mr {SO}}^+(1,2k+1)\times {\bb R}_+$ acts transitively on both the set of oriented elliptic orbits and the set of oriented parabolic orbits.
For each integer $n\ge 2$, we demonstrate that a 2n-dimensional generalized MICZ-Kepler problem has an $\widetilde{\mr{Spin}}(2, 2n+1)$ dynamical symmetry which extends the manifest $\mr{Spin}(2n)$ symmetry. The Hilbert space of bound states is shown to form a unitary highest weight $\widetilde{\mr{Spin}}(2, 2n+1)$-module which occurs at the first reduction point in the Enright-Howe-Wallach classification diagram for the unitary highest weight modules. As a byproduct, we get a simple geometric realization for such a unitary highest weight $\widetilde{\mr{Spin}}(2, 2n+1)$-module.
The existence of quasi-bi-Hamiltonian structures for the Kepler problem is studied. We first relate the superintegrability of the system with the existence of two complex functions endowed with very interesting Poisson bracket properties and then we prove the existence of a quasi-bi-Hamiltonian structure by making use of these two functions. The paper can be considered as divided in two parts. In the first part a quasi-bi-Hamiltonian structure is obtained by making use of polar coordinates and in the second part a new quasi-bi-Hamiltonian structure is obtained by making use of the separability…
Let $n\ge 2$ be a positive integer. To each irreducible representation $σ$ of $\mathrm{Sp}(1)$, an $\mathrm{Sp}(1)$-Kepler problem in dimension $(4n-3)$ is constructed and analyzed. This system is super integrable and when $n=2$ it is equivalent to a generalized MICZ-Kepler problem in dimension five. The dynamical symmetry group of this system is $\widetilde {\mathrm O}^*(4n)$ with the Hilbert space of bound states ${\mathscr H}(σ)$ being the unitary highest weight representation of $\widetilde {\mathrm {O}^*}(4n)$ with highest weight $$(\underbrace{-1, ..., -1}_{2n-1}, -(1+\barσ)),$$ which oc…
We show that for the Kepler problem the canonical Ligon-Schaaf regularization map can be understood in a straightforward manner as an adaptation of the Moser regularization. In turn this explains the hidden symmetry in a geometric way.
Let $n\ge 2$ be an integer. To each irreducible representation $σ$ of $\mathrm O(1)$, an $\mathrm {O}(1)$-Kepler problem in dimension $n$ is constructed and analyzed. This system is super integrable and when $n=2$ it is equivalent to a generalized MICZ-Kepler problem in dimension two. The dynamical symmetry group of this system is $\widetilde {\mathrm{Sp}}_{2n}(\mathbb R)$ with the Hilbert space of bound states ${\mathscr H}(σ)$ being the unitary highest weight representation of $\widetilde {\mathrm{Sp}}_{2n}(\mathbb R)$ with highest weight $$(\underbrace{-1/2, ..., -1/2}_{n-1}, -(1/2+|σ|)),$$…
We study a 2-body problem given by the sum of the Newtonian potential and an anisotropic perturbation that is a homogeneous function of degree $-β$, $β\ge 2$. For $β>2$, the sets of initial conditions leading to collisions/ejections and the one leading to escapes/captures have positive measure. For $β>2$ and $β\ne 3$, the flow on the zero-energy manifold is chaotic. For $β=2$, a case we prove integrable, the infinity manifold of the zero-energy level is a disconnected set, which has heteroclinic connections with the collision manifold.
Let $n\ge 2$ be a positive integer. To each irreducible representation $σ$ of $\mr U(1)$, a $\mr U(1)$-Kepler problem in dimension $(2n-1)$ is constructed and analyzed. This system is super integrable and when $n=2$ it is equivalent to a MICZ-Kepler problem. The dynamical symmetry group of this system is $\widetilde {\mr U}(n, n)$, and the Hilbert space of bound states ${\ms H}(σ)$ is the unitary highest weight representation of $\widetilde {\mr U}(n, n)$ with highest weight $$(\underbrace{-1/2, ..., -1/2}_n, 1/2+\bar σ, \underbrace{1/2, ..., 1/2}_{n-1})$$ when $\bar σ\ge 0$ or $$(\underbrace{…
The classical Kepler-Coulomb problem on the single-sheeted hyperboloid $H^{3}_1$ is solved in the framework of the Hamilton--Jacobi equation. We have proven that all the bounded orbits are closed and periodic. The paths are ellipses or circles for finite motion.
In this paper we show, in a systematic way, how to relate the Kepler problem to the isotropic harmonic oscillator. Unlike previous approaches, our constructions are carried over in the Lagrangian formalism dealing with second order vector fields. We therefore provide a tangent bundle version of the Kustaahneimo-Stiefel map.
We prove that the minimum-time controlled Kepler problem is not Liouville integrable in the class of meromorphic functions, via the Moralès-Ramis theory.
The linearized Kepler problem is considered, as obtained from the Kustaanheimo-Stiefel (K-S)transformation, both for negative and positive energies. The symmetry group for the Kepler problem turns out to be SU(2,2). For negative energies, the Hamiltonian of Kepler problem can be realized as the sum of the energies of four harmonic oscillator with the same frequency, with a certain constrain. For positive energies, it can be realized as the sum of the energies of four repulsive oscillator with the same (imaginary) frequency, with the same constrain. The quantization for the two cases, negative …
We show that the Kepler problem is projectively equivalent to null geodesic motion on the conformal compactification of Minkowski-4 space. This space realises the conformal triality of Minkwoski, dS and AdS spaces.
The superposition of the Kepler-Coulomb potential on the 3D Euclidean space with three centrifugal terms has recently been shown to be maximally superintegrable [Verrier P E and Evans N W 2008 J. Math. Phys. 49 022902] by finding an additional (hidden) integral of motion which is quartic in the momenta. In this paper we present the generalization of this result to the ND spherical, hyperbolic and Euclidean spaces by making use of a unified symmetry approach that makes use of the curvature parameter. The resulting Hamiltonian, formed by the (curved) Kepler-Coulomb potential together with N cent…
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.
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 stud…
We study the radial part of the MICZ-Kepler problem in an algebraic way by using the $su(1,1)$ Lie algebra. We obtain the energy spectrum and the eigenfunctions of this problem from the $su(1,1)$ theory of unitary representations and the tilting transformation to the stationary Schrödinger equation. We construct the physical Perelomov number coherent states for this problem and compute some expectation values. Also, we obtain the time evolution of these coherent states.
For each simple euclidean Jordan algebra $V$ of rank $ρ$ and degree $δ$, we introduce a family of classical dynamic problems. These dynamical problems all share the characteristic features of the Kepler problem for planetary motions, such as existence of Laplace-Runge-Lenz vector and hidden symmetry. After suitable quantizations, a family of quantum dynamic problems, parametrized by the nontrivial Wallach parameter $ν$, is obtained. Here, $ν\in{\mathcal W}(V):=\{k {δ\over 2}\mid k=1, ..., (ρ-1)\}\cup((ρ-1){δ\over 2}, \infty)$ and was introduced by N. Wallach to parametrize the set of nontrivia…
The technique of reduction of order developed by Nucci ({\it J Math Phys} {\bf 37} (1996) 1772-1775) is used to produce nonlocal symmetries additional to those reported by Krause ({\it J Math Phys} {\bf 35} (1994) 5734-5748) in his study of the complete symmetry group of the Kepler Problem. The technique is shown to be applicable to related problems containing a drag term which have been used to model the motion of low altitude satellites in the Earth's atmosphere and further generalisations. A consequence of the application of this technique is the demonstration of the group theoretical relat…
The description of number of dual (quasy)-exactly solvable models with its hidden symmetry algebra has been given at different levels of analysis within the framework of generalized Kustaanheimo-Stiefel (KS)-transformations. It's shown that N-dimensionall singular oscillator and (n+1)-dimensional generalized MICZ-Kepler system are dual to each other and the duality transformation is the generalized version of the KS transformation. The solvability of the Schrodinger equation of these problems by the variables separation method is given in a double, spherical and parabolic coordinates. The quad…
A quantum sl(2,R) coalgebra (with deformation parameter z) is shown to underly the construction of superintegrable Kepler potentials on 3D spaces of variable and constant curvature, that include the classical spherical, hyperbolic and (anti-)de Sitter spaces as well as their non-constant curvature analogues. In this context, the non-deformed limit z = 0 is identified with the flat contraction leading to the proper Euclidean and Minkowskian spaces/potentials. The corresponding Hamiltonians admit three constants of the motion coming from the coalgebra structure. Furthermore, maximal superintegra…
In this paper we regularize the Kepler problem on $κ$-spacetime in several different ways. First, we perform a Moser-type regularization and then we proceed for the Ligon-Schaaf regularization to our problem. In particular, generalizing Heckman-de Laat (J. Symplectic Geom. 10, (2012), 463-473) in the noncommutative context we show that the Ligon-Schaaf regularization map follows from an adaptation of the Moser regularization can be generalized to the Kepler problem on $κ$-spacetime.
The MICZ-Kepler orbits are the non-colliding orbits of the MICZ Kepler problems (the magnetized versions of the Kepler problem). The oriented MICZ-Kepler orbits can be parametrized by the canonical angular momentum $\mathbf L$ and the Lenz vector $\mathbf A$, with the parameter space consisting of the pairs of 3D vectors $(\mathbf A, \mathbf L)$ with ${\mathbf L}\cdot {\mathbf L} > (\mathbf L\cdot \mathbf A)^2$. The recent 4D perspective of the Kepler problem yields a new parametrization, with the parameter space consisting of the pairs of Minkowski vectors $(a,l)$ with $l\cdot l =-1$, $a\cdot…
The existence of quasi-bi-Hamiltonian structures for a two-dimensional superintegrable $(k_1,k_2,k_3)$-dependent Kepler-related problem is studied. We make use of an approach that is related with the existence of some complex functions which satisfy interesting Poisson bracket relations and that was previously applied to the standard Kepler problem as well as to some particular superintegrable systems as the Smorodinsky-Winternitz (SW) system, the Tremblay-Turbiner-Winternitz (TTW) and Post-Winternitz (PW) systems. We prove that these complex functions are important for two reasons: first, the…
While extending a famous problem asked and solved by Bertrand in 1873, Darboux found in 1877 a family of abstract surfaces of revolution, each endowed with a force function, with the striking property that all the orbits are periodic on open sets of the phase space. We give a description of this family which explains why they have this property: they are the Darboux inverses of the Kepler problem on constant curvature surfaces. What we call the Darboux inverse was briefly introduced by Darboux in 1889 as an alternative approach to the conformal maps that Goursat had just described.
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…
A Poisson realization of the simple real Lie algebra $\mathfrak {so}^*(4n)$ on the phase space of each $\mathrm {Sp}(1)$-Kepler problem is exhibited. As a consequence one obtains the Laplace-Runge-Lenz vector for each classical $\mathrm{Sp}(1)$-Kepler problem. The verification of these Poisson realizations is greatly simplified via an idea due to A. Weinstein. The totality of these Poisson realizations is shown to be equivalent to the canonical Poisson realization of $\mathfrak {so}^*(4n)$ on the Poisson manifold $T^*\mathbb H_*^n/\mathrm{Sp}(1)$. (Here $\mathbb H_*^n:=\mathbb H^n\backslash \{…
For each integer $n\ge 1$, we demonstrate that a $(2n+1)$-dimensional generalized MICZ-Kepler problem has an $\mr{Spin}(2, 2n+2)$ dynamical symmetry which extends the manifest $\mr{Spin}(2n+1)$ symmetry. The Hilbert space of bound states is shown to form a unitary highest weight $\mr{Spin}(2, 2n+2)$-module which occurs at the first reduction point in the Enright-Howe-Wallach classification diagram for the unitary highest weight modules. As a byproduct, we get a simple geometric realization for such a unitary highest weight $\mr{Spin}(2, 2n+2)$-module.
We apply the Schrödinger factorization method to the radial second-order equation for the relativistic Kepler-Coulomb problem. From these operators we construct two sets of one-variable radial operators which are realizations for the $su(1,1)$ Lie algebra. We use this algebraic structure to obtain the energy spectrum and the supersymmetric ground state for this system.
We consider the Kepler problem on surfaces of revolution that are homeomorphic to $S^2$ and have constant Gaussian curvature. We show that the system is maximally superintegrable, finding constants of motion that generalize the Runge-Lentz vector. Then, using such first integrals, we determine the class of surfaces that lead to block-regularizable collision singularities. In particular we show that the singularities are always regularizable if the surfaces are spherical orbifolds of revolution with constant curvature.
It is shown that by an appropriate canonical transformation Kepler dynamics can be put in the form which allows to exhibit the structure of the symmetry transformations related to the superintegrability. They appear to fit nicely into general scheme of nonlinear realizations. In new coordiantes the Kepler dynamics results from dimensional reduction of that describing low energy mesons with spontaneously broken chiral symmetry.
The Ligon-Schaaf regularization (LS mapping) was introduced in 1976 and has been used several times. However, we are not aware of any direct usage of the inverse mapping, perhaps since it appears at first sight to be quite complex and involves the use of a transcendental equation (referred to as the generalized Kepler equation) that cannot be solved in closed form. Here, we provide some insight into the significance of this equation, along with a very simple derivation and confirmation of the inverse LS mapping. Then we use numerical methods to investigate three applications: 1) solutions of t…
Here we provide an overview of what is known, and what is not known, about an interesting dynamical system known as the Kepler-Heisenberg problem. The main idea is to pose a version of the classical Kepler problem of planetary motion, but in a sub-Riemannian setting. The result is system which is surprisingly rich and beautiful, mysterious in some ways but tame in others, offering a substantial number of questions which seem non-trivial yet tractable.
The sum of elliptic integrals simultaneously determines orbits in thr Kepler problem and the addition of divisors on elliptic curves. Periodic motion of a body in physical space is defined by symmetries, whereas periodic motion of divisors is defined by a fixed point on the curve. Algebra of the first integrals associated with symmetries is a well-known mathematical object, whereas algebra of the first integrals associated with coordinates of fixed points is unknown. In this paper, we discuss polynomial algebras of non-polynomial first integrals of superintegrable systems associated with ellip…
The analog of the Kepler system defined on the Heisenberg group introduced by Montgomery and Shanbrom in [Fields Inst. Commun., Vol. 73, Springer, New York, 2015, 319-342, arXiv:1212.2713] is integrable on the zero level of the Hamiltonian. We show that in all other cases the system is not Liouville integrable due to the lack of additional meromorphic first integrals. We prove that the analog of the two-body problem on the Heisenberg group is not integrable in the Liouville sense.
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 …
