Morse Index and Linear Stability of the Lagrangian Circular Orbit in a Three-Body-Type Problem Via Index Theory

@article{Barutello2014MorseIA,
  title={Morse Index and Linear Stability of the Lagrangian Circular Orbit in a Three-Body-Type Problem Via Index Theory},
  author={Vivina L. Barutello and Riccardo D. Jadanza and Alessandro Portaluri},
  journal={Archive for Rational Mechanics and Analysis},
  year={2014},
  volume={219},
  pages={387-444}
}
It is well known that the linear stability of the Lagrangian elliptic solutions in the classical planar three-body problem depends on a mass parameter β and on the eccentricity e of the orbit. We consider only the circular case (e = 0) but under the action of a broader family of singular potentials: α-homogeneous potentials, for $$\alpha \in (0, 2)$$α∈(0,2), and the logarithmic one. It turns out indeed that the Lagrangian circular orbit persists also in this more general setting. We discover a… 
Linear stability of elliptic relative equilibria of restricted four-body problem
In this paper, we consider the linear stability of the elliptic relative equilibria of the restricted 4-body problems where the three primaries form a Lagrangian triangle. By reduction, the
Linear instability of relative equilibria for n-body problems in the plane
Abstract Following Smale, we study simple symmetric mechanical systems of n point particles in the plane. In particular, we address the question of the linear and spectral stability properties of
Morse Index and Stability of the Planar N-vortex Problem
This paper concerns the investigation of the stability properties of relative equilibria which are rigidly rotating vortex configurations sometimes called vortex crystals, in the N-vortex problem.
Spectral stability, spectral flow and circular relative equilibria for the Newtonian $n$-body problem
For the Newtonian (gravitational) n-body problem in the Euclidean d-dimensional space, d ≥ 2, the simplest possible periodic solutions are provided by circular relative equilibria, (RE) for short,
Morse index and linear stability of relative equilibria in singular mechanical systems
We have focussed on the study of the linear stability of some particular periodic orbits (called relative equilibria) in planar singular mechanical systems with SO(2)-symmetry, and we have achieved
An index theory for asymptotic motions under singular potentials
We develop an index theory for parabolic and collision solutions to the classical n-body problem and we prove sufficient conditions for the finiteness of the spectral index valid in a large class of
Trace Formula for Linear Hamiltonian Systems with its Applications to Elliptic Lagrangian Solutions
In the present paper, we build up trace formulas for both the linear Hamiltonian systems and Sturm–Liouville systems. The formula connects the monodromy matrix of a symmetric periodic orbit with the
Linear instability for periodic orbits of non-autonomous Lagrangian systems
Inspired by the classical Poincaré criterion about the instability of orientation preserving minimizing closed geodesics on surfaces, we investigate the relation intertwining the instability and the
Morse Theory and Relative Equilibria in the Planar n-Vortex Problem
Morse theoretical ideas are applied to the study of relative equilibria in the planar n-vortex problem. For the case of positive circulations, we prove that the Morse index of a critical point of the
Index theory for heteroclinic orbits of Hamiltonian systems
Index theory revealed its outstanding role in the study of periodic orbits of Hamiltonian systems and the dynamical consequences of this theory are enormous. Although the index theory in the periodic
...
1
2
3
...

References

SHOWING 1-10 OF 38 REFERENCES
Linear instability of relative equilibria for n-body problems in the plane
Abstract Following Smale, we study simple symmetric mechanical systems of n point particles in the plane. In particular, we address the question of the linear and spectral stability properties of
Index and Stability of Symmetric Periodic Orbits in Hamiltonian Systems with Application to Figure-Eight Orbit
In this paper, using the Maslov index theory in symplectic geometry, we build up some stability criteria for symmetric periodic orbits in a Hamiltonian system, which is motivated by the recent
Morse index and stability of elliptic Lagrangian solutions in the planar three-body problem
Abstract We illustrate a new way to study the stability problem in celestial mechanics. In this paper, using the variational nature of elliptic Lagrangian solutions in the planar three-body problem,
Elliptic relative equilibria in the N-body problem
A planar central configuration of the N-body problem gives rise to a solution where each particle moves on a specific Keplerian orbit while the totality of the particles move on a homothety motion.
Linear Stability of Elliptic Lagrangian Solutions of the Planar Three-Body Problem via Index Theory
It is well known that the linear stability of Lagrangian elliptic equilateral triangle homographic solutions in the classical planar three-body problem depends on the mass parameter
Computation of the Maslov index and the spectral flow via partial signatures
Given a smooth Lagrangian path, both in the finite and in the infinite di- mensional (Fredholm) case, we introduce the notion of partial signatures at each isolated intersection of the path with the
A Morse index theorem for perturbed geodesics on semi-Riemannian manifolds
Perturbed geodesics are trajectories of particles moving on a semi-Riemannian manifold in the presence of a potential. Our purpose here is to extend to perturbed geodesics on semi-Riemannian
Action minimizing orbits in the n-body problem with simple choreography constraint
In 1999 Chenciner and Montgomery found a remarkably simple choreographic motion for the planar three-body problem (see [11]). In this solution, three equal masses travel on an figure-of-eight shaped
High action orbits for Tonelli Lagrangians and superlinear Hamiltonians on compact configuration spaces
Abstract Multiplicity results for solutions of various boundary value problems are known for dynamical systems on compact configuration manifolds, given by Lagrangians or Hamiltonians which have
Spectral Flow, Maslov Index and Bifurcation of Semi-Riemannian Geodesics
We give a functional analytical proof of the equalitybetween the Maslov index of a semi-Riemannian geodesicand the spectral flow of the path of self-adjointFredholm operators obtained from the index
...
1
2
3
4
...