3,683 research outputs found
Orbital stability: analysis meets geometry
We present an introduction to the orbital stability of relative equilibria of
Hamiltonian dynamical systems on (finite and infinite dimensional) Banach
spaces. A convenient formulation of the theory of Hamiltonian dynamics with
symmetry and the corresponding momentum maps is proposed that allows us to
highlight the interplay between (symplectic) geometry and (functional) analysis
in the proofs of orbital stability of relative equilibria via the so-called
energy-momentum method. The theory is illustrated with examples from finite
dimensional systems, as well as from Hamiltonian PDE's, such as solitons,
standing and plane waves for the nonlinear Schr{\"o}dinger equation, for the
wave equation, and for the Manakov system
Orbital stability via the energy-momentum method: the case of higher dimensional symmetry groups
We consider the orbital stability of relative equilibria of Hamiltonian
dynamical systems on Banach spaces, in the presence of a multi-dimensional
invariance group for the dynamics. We prove a persistence result for such
relative equilibria, present a generalization of the Vakhitov-Kolokolov slope
condition to this higher dimensional setting, and show how it allows to prove
the local coercivity of the Lyapunov function, which in turn implies orbital
stability. The method is applied to study the orbital stability of relative
equilibria of nonlinear Schr{\"o}dinger and Manakov equations. We provide a
comparison of our approach to the one by Grillakis-Shatah-Strauss
The Instability Transition for the Restricted 3-Body Problem. III. The Lyapunov Exponent Criterion
We establish a criterion for the stability of planetary orbits in stellar
binary systems by using Lyapunov exponents and power spectra for the special
case of the circular restricted 3-body problem (CR3BP). The centerpiece of our
method is the concept of Lyapunov exponents, which are incorporated into the
analysis of orbital stability by integrating the Jacobian of the CR3BP and
orthogonalizing the tangent vectors via a well-established algorithm originally
developed by Wolf et al. The criterion for orbital stability based on the
Lyapunov exponents is independently verified by using power spectra. The
obtained results are compared to results presented in the two previous papers
of this series. It is shown that the maximum Lyapunov exponent can be used as
an indicator for chaotic behaviour of planetary orbits, which is consistent
with previous applications of this method, particularly studies for the Solar
System. The chaotic behaviour corresponds to either orbital stability or
instability, and it depends solely on the mass ratio of the binary components
and the initial distance ratio of the planet relative to the stellar separation
distance. Our theoretical results allow us to link the study of planetary
orbital stability to chaos theory noting that there is a large array of
literature on the properties and significance of Lyapunov exponents. Although
our results are given for the special case of the CR3BP, we expect that it may
be possible to augment the proposed Lyapunov exponent criterion to studies of
planets in generalized stellar binary systems, which is strongly motivated by
existing observational results as well as results expected from ongoing and
future planet search missions.Comment: 10 pages, 8 figures, 3 tables; accepted by Astronomy and Astrophysic
On the stability of standing waves of Klein-Gordon equations in a semiclassical regime
We investigate the orbital stability and instability of standing waves for
two classes of Klein-Gordon equations in the semi-classical regime.Comment: 9 page
Orbital Stability of Planets in Binary Systems: A New Look at Old Results
About half of all known stellar systems with Sun-like stars consist of two or
more stars, significantly affecting the orbital stability of any planet in
these systems. This observational evidence has prompted a large array of
theoretical research, including the derivation of mathematically stringent
criteria for the orbital stability of planets in stellar binary systems, valid
for the "coplanar circular restricted three-body problem". In the following, we
use these criteria to explore the validity of results from previous theoretical
studies.Comment: 3 pages, 1 figure; submitted to: Exoplanets: Detection, Formation and
Dynamics, IAU Symposium 249, eds. Y.-S. Sun, S. Ferraz-Mello, and J.-L. Zhou
(Cambridge: Cambridge University Press
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