103 research outputs found
The restricted two-body problem in constant curvature spaces
We perform the bifurcation analysis of the Kepler problem on and .
An analogue of the Delaunay variables is introduced. We investigate the motion
of a point mass in the field of the Newtonian center moving along a geodesic on
and (the restricted two-body problem). When the curvature is small,
the pericenter shift is computed using the perturbation theory. We also present
the results of the numerical analysis based on the analogy with the motion of
rigid body.Comment: 29 pages, 7 figure
Non-linear stability in photogravitational non-planar restricted three body problem with oblate smaller primary
We have discussed non-linear stability in photogravitational non-planar
restricted three body problem with oblate smaller primary. By
photogravitational we mean that both primaries are radiating. We normalised the
Hamiltonian using Lie transform as in Coppola and Rand (1989). We transformed
the system into Birkhoff's normal form. Lie transforms reduce the system to an
equivalent simpler system which is immediately solvable. Applying Arnold's
theorem, we have found non-linear stability criteria. We conclude that is
stable. We plotted graphs for They are rectangular
hyperbola.Comment: Accepted for publication in Astrophysics & Space Scienc
Integrable Euler top and nonholonomic Chaplygin ball
We discuss the Poisson structures, Lax matrices, -matrices, bi-hamiltonian
structures, the variables of separation and other attributes of the modern
theory of dynamical systems in application to the integrable Euler top and to
the nonholonomic Chaplygin ball.Comment: 25 pages, LaTeX with AMS fonts, final versio
Chaplygin ball over a fixed sphere: explicit integration
We consider a nonholonomic system describing a rolling of a dynamically
non-symmetric sphere over a fixed sphere without slipping. The system
generalizes the classical nonholonomic Chaplygin sphere problem and it is shown
to be integrable for one special ratio of radii of the spheres. After a time
reparameterization the system becomes a Hamiltonian one and admits a separation
of variables and reduction to Abel--Jacobi quadratures. The separating
variables that we found appear to be a non-trivial generalization of
ellipsoidal (spheroconical) coordinates on the Poisson sphere, which can be
useful in other integrable problems.
Using the quadratures we also perform an explicit integration of the problem
in theta-functions of the new time.Comment: This is an extended version of the paper to be published in Regular
and Chaotic Dynamics, Vol. 13 (2008), No. 6. Contains 20 pages and 2 figure
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