scholarly journals Stability of the relative equilibria in the generalized J2 problem

2000 ◽  
pp. 9-13
Author(s):  
V. Mioc ◽  
M. Stavinschi
Keyword(s):  
Dynamical Systems ◽  
Small Parameter ◽  
Relative Equilibrium ◽  
Relative Equilibria ◽  
Usual Method ◽  
Real Constant ◽  
Block Diagonalization ◽  
Reduction Procedure ◽  
Celestial Bodies ◽  
Test Points

For a large class of concrete astronomical situations, the motion of celestial bodies is modelled by dynamical systems associated to a potential function ?/r + ?U (r = distance between particles, ? = real constant, ? = real small parameter, U = perturbing potential). In this paper the nonlinear stability of the relative equilibrium orbits corresponding to such a potential is being investigated using a less usual method, which combines a block diagonalization technique with the reduction procedure. The test points out certain nonlinearly stable orbits, and is inconclusive for the remaining equilibria. The latter ones are treated via linearization; all of them prove instability. The nonlinearly stable orbits remain stable under any perturbation that preserves the conserved momentum.

A GENERALIZED REDUCTION PROCEDURE FOR DYNAMICAL SYSTEMS

1991 ◽  
Vol 06 (37) ◽  
pp. 3445-3453 ◽  
Author(s):  
G. LANDI ◽  
G. MARMO ◽  
G. SPARANO ◽  
G. VILASI
Keyword(s):  
Vector Field ◽  
Dynamical Systems ◽  
Equivalence Relation ◽  
Reduction Procedure

We describe a reduction procedure for dynamical systems. If Γ is a dynamical vector field on a manifold M, a reduced system is obtained by projecting Γ to a manifold Σ/≈ where Σ is a submanifold of M invariant under Γ and ≈ is a suitable equivalence relation.

Symplectic analysis of dynamical systems with a small parameter. A new criterion for stabilization of homoclinic separatrices and its application

1992 ◽  
Vol 44 (1) ◽  
pp. 41-58
Author(s):  
Yu. O. Mitropol'skii ◽  
I. O. Antonishin ◽  
A. K. Prikarpats'kyy ◽  
V. G. Samoilenko
Keyword(s):  
Dynamical Systems ◽  
Small Parameter ◽  
New Criterion

GENERALIZED REDUCTION PROCEDURE AND NONLINEAR NONSTATIONARY DYNAMICAL SYSTEMS

1992 ◽  
Vol 07 (36) ◽  
pp. 3411-3418 ◽  
Author(s):  
V.I. MAN’KO ◽  
G. MARMO
Keyword(s):  
Dynamical Systems ◽  
Reduction Procedure

In this paper a generalized reduction procedure is used to obtain nonlinear nonstationary dynamical systems out of free ones.

scholarly journals Bifurcations of relative equilibria in the 4- and 5-body problem

1988 ◽  
Vol 8 (8) ◽  
pp. 215-225 ◽  
Keyword(s):  
Equilateral Triangle ◽  
Body Problem ◽  
Relative Equilibrium ◽  
Isosceles Triangle ◽  
Relative Equilibria ◽  
Arbitrary Mass

AbstractThe equilateral triangle family of relative equilibria of the 4-body problem consists of three particles of mass 1 at the vertices of an equilateral triangle and the fourth particle of arbitrary mass m at the centroid. For one value of the mass m this relative equilibrium is degenerate. We show that families of isosceles triangle relative equilibria bifurcate from the equilateral triangle family as m passes through the degenerate value.The square family of relative equilibria of the 5-body problem consists of four particles of mass 1 at the vertices of a square and the fifth particle of arbitrary mass m at the centroid. For one value of the mass m this relative equilibrium is degenerate. We show that families of kite and isosceles trapezoidal relative equilibria bifurcate from the square family as m passes through the degenerate value.

Euler-type Relative Equilibria and their Stability in Spaces of Constant Curvature

2018 ◽  
Vol 70 (2) ◽  
pp. 426-450 ◽  
Author(s):  
Ernesto Pérez-Chavela ◽  
Juan Manuel Sánchez-Cerritos
Keyword(s):  
Constant Curvature ◽  
Algebraic Curve ◽  
Positive Measure ◽  
Relative Equilibrium ◽  
Relative Equilibria ◽  
Spectral Stability ◽  
Spaces Of Constant Curvature ◽  
Central Mass ◽  
Elliptic Case ◽  
The Masses

AbstractWe consider three point positivemasses moving onS2andH2. An Eulerian-relative equilibrium is a relative equilibrium where the three masses are on the same geodesic. In this paper we analyze the spectral stability of these kind of orbits where the mass at the middle is arbitrary and the masses at the ends are equal and located at the same distance from the central mass. For the case of S2, we found a positive measure set in the set of parameters where the relative equilibria are spectrally stable, and we give a complete classiûcation of the spectral stability of these solutions, in the sense that, except on an algebraic curve in the space of parameters, we can determine if the corresponding relative equilibriumis spectrally stable or unstable. OnH2, in the elliptic case, we prove that generically all Eulerian-relative equilibria are unstable; in the particular degenerate case when the two equal masses are negligible, we get that the corresponding solutions are spectrally stable. For the hyperbolic case we consider the system where the mass in the middle is negligible; in this case the Eulerian-relative equilibria are unstable.

scholarly journals Collective behaviour of large number of vortices in the plane

2013 ◽  
Vol 469 (2156) ◽  
pp. 20130085 ◽  
Author(s):  
Yuxin Chen ◽  
Theodore Kolokolnikov ◽  
Daniel Zhirov
Keyword(s):  
Vortex Dynamics ◽  
Initial Conditions ◽  
Relative Equilibrium ◽  
Relative Equilibria ◽  
Point Vortices ◽  
Collective Behaviour ◽  
Aggregation Model ◽  
Equal Strength ◽  
Artificial Damping ◽  
Analytical Formulae

We investigate the dynamics of N point vortices in the plane, in the limit of large N . We consider relative equilibria , which are rigidly rotating lattice-like configurations of vortices. These configurations were observed in several recent experiments. We show that these solutions and their stability are fully characterized via a related aggregation model which was recently investigated in the context of biological swarms. By using this connection, we give explicit analytical formulae for many of the configurations that have been observed experimentally. These include configurations of vortices of equal strength; the N +1 configurations of N vortices of equal strength and one vortex of much higher strength; and more generally, N + K configurations. We also give examples of configurations that have not been studied experimentally, including N +2 configurations, where N vortices aggregate inside an ellipse. Finally, we introduce an artificial ‘damping’ to the vortex dynamics, in an attempt to explain the phenomenon of crystallization that is often observed in real experiments. The diffusion breaks the conservative structure of vortex dynamics, so that any initial conditions converge to the lattice-like relative equilibrium.

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