26 research outputs found
Two body problem on two point homogeneous spaces, invariant differential operators and the mass center concept
We consider the two body problem with central interaction on two point
homogeneous spaces from point of view of the invariant differential operators
theory. The representation of the two particle Hamiltonian in terms of the
radial differential operator and invariant operators on the symmetry group is
found. The connection of different mass center definitions for these spaces to
the obtained expression for Hamiltonian operator is studied.Comment: 26 pages, LaTeX, no figures, text improve
Nonintegrability of the two-body problem in constant curvature spaces
We consider the reduced two-body problem with the Newton and the oscillator
potentials on the sphere and the hyperbolic plane .
For both types of interaction we prove the nonexistence of an additional
meromorphic integral for the complexified dynamic systems.Comment: 20 pages, typos correcte
Two-body quantum mechanical problem on spheres
The quantum mechanical two-body problem with a central interaction on the
sphere is considered. Using recent results in representation
theory an ordinary differential equation for some energy levels is found. For
several interactive potentials these energy levels are calculated in explicit
form.Comment: 41 pages, no figures, typos corrected; appendix D was adde
The geometric sense of R. Sasaki connection
For the Riemannian manifold two special connections on the sum of the
tangent bundle and the trivial one-dimensional bundle are constructed.
These connections are flat if and only if the space has a constant
sectional curvature . The geometric explanation of this property is
given. This construction gives a coordinate free many-dimensional
generalization of the connection from the paper: R. Sasaki 1979 Soliton
equations and pseudospherical surfaces, Nuclear Phys., {\bf 154 B}, pp.
343-357. It is shown that these connections are in close relation with the
imbedding of into Euclidean or pseudoeuclidean -dimension
spaces.Comment: 7 pages, the key reference to the paper of Min-Oo is included in the
second versio
Hidden symmetry of hyperbolic monopole motion
Hyperbolic monopole motion is studied for well separated monopoles. It is
shown that the motion of a hyperbolic monopole in the presence of one or more
fixed monopoles is equivalent to geodesic motion on a particular submanifold of
the full moduli space. The metric on this submanifold is found to be a
generalisation of the multi-centre Taub-NUT metric introduced by LeBrun. The
one centre case is analysed in detail as a special case of a class of systems
admitting a conserved Runge-Lenz vector. The two centre problem is also
considered. An integrable classical string motion is exhibited.Comment: 39 pages, 7 figures, references added, minor changes to section
On the stability of tetrahedral relative equilibria in the positively curved 4-body problem
We consider the motion of point masses given by a natural extension of
Newtonian gravitation to spaces of constant positive curvature. Our goal is to
explore the spectral stability of tetrahedral orbits of the corresponding
4-body problem in the 2-dimensional case, a situation that can be reduced to
studying the motion of the bodies on the unit sphere. We first perform some
extensive and highly precise numerical experiments to find the likely regions
of stability and instability, relative to the values of the masses and to the
latitude of the position of three equal masses. Then we support the numerical
evidence with rigorous analytic proofs in the vicinity of some limit cases in
which certain masses are either very large or negligible, or the latitude is
close to zero.Comment: 32 pages, 6 figure
Superintegrable potentials on 3D Riemannian and Lorentzian spaces with non-constant curvature
A quantum sl(2,R) coalgebra is shown to underly the construction of a large
class of superintegrable potentials on 3D curved spaces, that include the
non-constant curvature analogues of the spherical, hyperbolic and (anti-)de
Sitter spaces. The connection and curvature tensors for these "deformed" spaces
are fully studied by working on two different phase spaces. The former directly
comes from a 3D symplectic realization of the deformed coalgebra, while the
latter is obtained through a map leading to a spherical-type phase space. In
this framework, the non-deformed limit is identified with the flat contraction
leading to the Euclidean and Minkowskian spaces/potentials. The resulting
Hamiltonians always admit, at least, three functionally independent constants
of motion coming from the coalgebra structure. Furthermore, the intrinsic
oscillator and Kepler potentials on such Riemannian and Lorentzian spaces of
non-constant curvature are identified, and several examples of them are
explicitly presented.Comment: 14 pages. Based in the contribution presented at the Group 27
conference, Yerevan, Armenia, August 13-19, 200