155 research outputs found
Families of periodic orbits for the spatial isosceles 3-body problem
We study the families of periodic orbits of the spatial isosceles 3-body problem (for
small enough values of the mass lying on the symmetry axis) coming via the analytic continuation
method from periodic orbits of the circular Sitnikov problem. Using the first integral of the angular
momentum, we reduce the dimension of the phase space of the problem by two units. Since periodic
orbits of the reduced isosceles problem generate invariant two-dimensional tori of the nonreduced
problem, the analytic continuation of periodic orbits of the (reduced) circular Sitnikov problem
at this level becomes the continuation of invariant two-dimensional tori from the circular Sitnikov
problem to the nonreduced isosceles problem, each one filled with periodic or quasi-periodic orbits.
These tori are not KAM tori but just isotropic, since we are dealing with a three-degrees-of-freedom
system. The continuation of periodic orbits is done in two different ways, the first going directly from
the reduced circular Sitnikov problem to the reduced isosceles problem, and the second one using
two steps: first we continue the periodic orbits from the reduced circular Sitnikov problem to the
reduced elliptic Sitnikov problem, and then we continue those periodic orbits of the reduced elliptic
Sitnikov problem to the reduced isosceles problem. The continuation in one or two steps produces
different results. This work is merely analytic and uses the variational equations in order to apply
PoincarÂŽeâs continuation method
On the relationship between instability and Lyapunov times for the 3-body problem
In this study we consider the relationship between the survival time and the
Lyapunov time for 3-body systems. It is shown that the Sitnikov problem
exhibits a two-part power law relationship as demonstrated previously for the
general 3-body problem. Using an approximate Poincare map on an appropriate
surface of section, we delineate escape regions in a domain of initial
conditions and use these regions to analytically obtain a new functional
relationship between the Lyapunov time and the survival time for the 3-body
problem. The marginal probability distributions of the Lyapunov and survival
times are discussed and we show that the probability density function of
Lyapunov times for the Sitnikov problem is similar to that for the general
3-body problem.Comment: 9 pages, 19 figures, accepted for publication in MNRA
Relativistic effects in the chaotic Sitnikov problem
We investigate the phase space structure of the relativistic Sitnikov problem
in the first post-Newtonian approximation. The phase space portraits show a
strong dependence on the gravitational radius which describes the strength of
the relativistic pericentre advance. Bifurcations appearing at increasing the
gravitational radius are presented. Transient chaotic behavior related to
escapes from the primaries are also studied. Finally, the numerically
determined chaotic saddle is investigated in the context of hyperbolic and
non-hyperbolic dynamics as a function of the gravitational radius.Comment: 8 pages, 11 figure
On the periodic orbits of the circular double Sitnikov problem
We introduce a restricted four body problem in a 2+2 configuration extending
the classical Sitnikov problem to the Double Sitnikov problem. The secondary
bodies are moving on the same perpendicular line to the planewhere the
primaries evolve, so almost every solution is a collision orbit. We extend the
solutions beyond collisions with a symplectic regularization and study the set
of energy surfaces that contain periodic orbits.Comment: 4 pages. Accepted in C. R. Acad. Sci. de Paris, Serie I. This is not
the final versio
Variational Construction of Orbits Realizing Symbolic Sequences in the Planar Sitnikov Problem
Using the variational method, Chenciner and Montgomery (2000 Ann. Math. 52 881â901) proved the existence of an eight-shaped orbit of the planar three-body problem with equal masses. Since then a number of solutions to the N-body problem have been discovered. On the other hand, symbolic dynamics is one of the most useful methods for understanding chaotic dynamics. The Sitnikov problem is a special case of the three-body problem. The system is known to be chaotic and was studied by using symbolic dynamics (J. Moser, Stable and random motions in dynamical systems, Princeton University Press, 1973). In this paper, we study the limiting case of the Sitnikov problem. By using the variational method, we show the existence of various kinds of solutions in the planar Sitnikov problem. For a given symbolic sequence, we show the existence of orbits realizing it. We also prove the existence of periodic orbits
Symplectic Regularization of Binary Collisions in the Circular N+2 Sitnikov Problem
We present a brief overview of the regularizing transformations of the Kepler
problem and we relate the Euler transformation with the symplectic structure of
the phase space of the N-body problem. We show that any particular solution of
the N-body problem where two bodies have rectilinear dynamics can be
regularized by a linear symplectic transformation and the inclusion of the
Euler transformation into the group of symplectic local diffeomorphisms over
the phase space. As an application we regularize a particular configuration of
the circular N+2 Sitnikov problem.Comment: 23 pages, 5 figures. References to algorithmic regularization
included, changes in References and small typographic corrections. Accepted
in J. of Phys. A: Math. Theor 44 (2011) 265204
http://stacks.iop.org/1751-8121/44/26520
- âŠ