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Register a new userSuperintegrability on the 3-dimensional spaces with curvature. Oscillator-related and Kepler-related systems on the Sphere $S^3$ and on the Hyperbolic space $H^3$
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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 of the Kepler problem with additional nonlinear terms in these two curved spaces, $S^3$ ($kp>0)$ and $H^3$ ($kp<0$). We prove their superintegrability and we obtain, in all the cases, the maximal number of functionally independent integrals of motion. All the mathematical expressions are presented using the curvature $kp$ as a parameter, in such a way that particularizing for $kp>0$, $kp=0$, or $kp<0$, the corresponding properties are obtained for the system on the sphere $S^3$, the Euclidean space $IE^3$, or the hyperbolic space $H^3$, respectively.
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We study four particular 3-dimensional natural Hamiltonian systems defined in conformally Euclidean spaces. We prove their superintegrability and we obtain, in the four cases, the maximal number of functionally independent integrals of motion. The two first systems are related to the 3-dimensional isotropic oscillator and the superintegrability is quadratic. The third system is obtained as a continuous deformation of an oscillator with ratio of frequencies 1:1:2 and with three additional nonlinear terms of the form $k_2/x^2$, $k_3/y^2$ and $k_4/z^2$, and the fourth system is obtained as a deformation of the Kepler Hamiltonian also with these three particular nonlinear terms. These third and fourth systems are superintegrable but with higher-order constants of motion. The four systems depend on a real parameter in such a way that they are continuous functions of the parameter (in a certain domain of the parameter) and in the limit of such parameter going to zero the Euclidean dynamics is recovered.
The isotropic harmonic oscillator in dimension 3 separates in several different coordinate systems. Separating in a particular coordinate system defines a system of three commuting operators, one of which is the Hamiltonian. We show that the joint spectrum of the Hamilton operator, the $z$ component of the angular momentum, and a quartic integral obtained from separation in prolate spheroidal coordinates has quantum monodromy for sufficiently large energies. This means that one cannot globally assign quantum numbers to the joint spectrum. The effect can be classically explained by showing that the corresponding Liouville integrable system has a non-degenerate focus-focus point, and hence Hamiltonian monodromy.
We build a family of explicit one-forms on $S^3$ which are shown to form a complete set of eigenmodes for the Laplace-de Rahm operator.
We consider several problems that involve lines in three dimensions, and present improved algorithms for solving them. The problems include (i) ray shooting amid triangles in $R^3$, (ii) reporting intersections between query lines (segments, or rays) and input triangles, as well as approximately counting the number of such intersections, (iii) computing the intersection of two nonconvex polyhedra, (iv) detecting, counting, or reporting intersections in a set of lines in $R^3$, and (v) output-sensitive construction of an arrangement of triangles in three dimensions. Our approach is based on the polynomial partitioning technique. For example, our ray-shooting algorithm processes a set of $n$ triangles in $R^3$ into a data structure for answering ray shooting queries amid the given triangles, which uses $O(n^{3/2+varepsilon})$ storage and preprocessing, and answers a query in $O(n^{1/2+varepsilon})$ time, for any $varepsilon>0$. This is a significant improvement over known results, obtained more than 25 years ago, in which, with this amount of storage, the query time bound is roughly $n^{5/8}$. The algorithms for the other problems have similar performance bounds, with similar improvements over previous results. We also derive a nontrivial improved tradeoff between storage and query time. Using it, we obtain algorithms that answer $m$ queries on $n$ objects in [ max left{ O(m^{2/3}n^{5/6+varepsilon} + n^{1+varepsilon}),; O(m^{5/6+varepsilon}n^{2/3} + m^{1+varepsilon}) right} ] time, for any $varepsilon>0$, again an improvement over the earlier bounds.
Let $H$ be the quaternion algebra. Let $g$ be a complex Lie algebra and let $U(g)$ be the enveloping algebra of $g$. We define a Lie algebra structure on the tensor product space of $H$ and $U(g)$, and obtain the quaternification $g^H$ of $g$. Let $S^3g^H$ be the set of $g^H$-valued smooth mappings over $S^3$. The Lie algebra structure on $S^3g^H$ is induced naturally from that of $g^H$. On $S^3$ exists the space of Laurent polynomial spinors spanned by a complete orthogonal system of eigen spinors of the tangential Dirac operator on $S^3$. Tensoring $U(g)$ we have the space of $U(g)$-valued Laurent polynomial spinors, which is a Lie subalgebra of $S^3g^H$. We introduce a 2-cocycle on the space of $U(g)$-valued Laurent polynomial spinors by the aid of a tangential vector field on $S^3$. Then we have the corresponding central extension $hat g(a)$ of the Lie algebra of $U(g)$-valued Laurent polynomial spinors. Finally we have the a Lie algebra $hat g=hat g(a)+Cd$ which is obtained by adding to $hat g(a)$ a derivation $d$ which acts on $hat g(a)$ as the radial derivation. When $g$ is a simple Lie algebra with its Cartan subalgebra $h$, We shall investigate the weight space decomposition of $(hat g, ad(hat h))$, where $hat h=h+Ca+Cd$ . The previo
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