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Vivina Barutello Riccardo D. Jadanza Alessandro Portaluri 《Archive for Rational Mechanics and Analysis》2016,219(1):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 \(\alpha \in (0, 2)\), and the logarithmic one. It turns out indeed that the Lagrangian circular orbit persists also in this more general setting. We discover a region of linear stability expressed in terms of the homogeneity parameter α and the mass parameter β, then we compute the Morse index of this orbit and of its iterates and we find that the boundary of the stability region is the envelope of a family of curves on which the Morse indices of the iterates jump. In order to conduct our analysis we rely on a Maslov-type index theory devised and developed by Y. Long, X. Hu and S. Sun; a key role is played by an appropriate index theorem and by some precise computations of suitable Maslov-type indices. 相似文献
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Vivina L. Barutello Riccardo D. Jadanza Alessandro Portaluri 《Journal of Differential Equations》2014
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 relative equilibria, which are special solutions of the equations of motion. 相似文献
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Gianni Arioli Vivina Barutello Susanna Terracini 《Communications in Mathematical Physics》2006,268(2):439-463
We prove the existence of a new branch of solutions of Mountain Pass type for the periodic 3-body problem with choreographical
constraint. At first we describe the variational structure of the action functional associated to the choreographical three
body problem in
. In the second part, using a bisection algorithm, we provide a numerical non-rigorous solution of Mountain Pass type for
this problem in a rotating frame with angular velocity 1.5. The last step consists in the rigorous computer-assisted proof
of the existence of a full branch of solutions for the problem starting from the Mountain Pass solution detected numerically.
Electronic Supplementary Material Supplementary material is available for this article at and is accessible for authorized users.
This work was supported by the MIUR project “Metodi Variazionali ed Equazioni Differenziali non Lineari.” 相似文献
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Vivina Barutello Davide L. Ferrario Susanna Terracini 《Archive for Rational Mechanics and Analysis》2008,190(2):189-226
We consider periodic and quasi-periodic solutions of the three-body problem with homogeneous potential from the point of view
of equivariant calculus of variations. First, we show that symmetry groups of the Lagrangian action functional can be reduced
to groups in a finite explicitly given list, after a suitable change of coordinates. Then, we show that local symmetric minimizers
are always collisionless, without any assumption on the group other than the fact that collisions are not forced by the group
itself. Moreover, we describe some properties of the resulting symmetric collisionless minimizers (Lagrange, Euler, Hill-type
orbits and Chenciner–Montgomery figure-eights). 相似文献
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We study a singular Hamiltonian system with an α-homogeneous potential that contains, as a particular case, the classical N-body problem. We introduce a variational Morse-like index for a class of collision solutions and, using the asymptotic estimates near collisions, we prove the non-minimality of some special classes of colliding trajectories under suitable spectral conditions provided α is sufficiently away from zero. We then prove some minimality results for small values of the parameter α. 相似文献
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Vivina Barutello Susanna Terracini Gianmaria Verzini 《Calculus of Variations and Partial Differential Equations》2014,49(1-2):391-429
For the class of anisotropic Kepler problems in $\mathbb{R }^d\setminus \{0\}$ with homogeneous potentials, we seek parabolic trajectories having prescribed asymptotic directions at infinity and which, in addition, are Morse minimizing geodesics for the Jacobi metric. Such trajectories correspond to saddle heteroclinics on the collision manifold, are structurally unstable and appear only for a codimension-one submanifold of such potentials. We give them a variational characterization in terms of the behavior of the parameter-free minimizers of an associated obstacle problem. We then give a full characterization of such a codimension-one manifold of potentials and we show how to parameterize it with respect to the degree of homogeneity. 相似文献
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Vivina Barutello Susanna Terracini Gianmaria Verzini 《Archive for Rational Mechanics and Analysis》2013,207(2):583-609
We continue the variational approach to parabolic trajectories introduced in our previous paper (Barutello et al., Entire parabolic trajectories as minimal phase transitions. arXiv:1105.3358v1, 2011), which sees parabolic orbits as minimal phase transitions. We deepen and complete the analysis in the planar case for homogeneous singular potentials. We characterize all parabolic orbits connecting two minimal central configurations as free-time Morse minimizers (in a given homotopy class of paths). These may occur for at most one value of the homogeneity exponent. In addition, we link this threshold of existence of parabolic trajectories with the absence of collisions for all the minimizers of fixed-end problems, and also with the existence of action minimizing periodic trajectories with nontrivial homotopy type. 相似文献
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Vivina Barutello Susanna Terracini 《NoDEA : Nonlinear Differential Equations and Applications》2007,14(5-6):527-539
We propose a constructive proof for the Ambrosetti-Rabinowitz Mountain Pass Theorem providing an algorithm, based on a bisection
method, for its implementation. The efficiency of our algorithm, particularly suitable for problems in high dimensions, consists
in the low number of flow lines to be computed for its convergence; for this reason it improves the one currently used and
proposed by Y.S. Choi and P.J. McKenna in [3].
Susanna Terracini: This work is partially supported by M.I.U.R. project “Metodi Variazionali ed Equazioni Differenziali Nonlineari”. 相似文献
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Vivina Barutello Enrico Serra 《Journal of Mathematical Analysis and Applications》2008,341(1):720-728
We prove the existence of a positive radial solution for the Hénon equation with arbitrary growth. The solution is found by means of a shooting method and turns out to be an increasing function of the radial variable. Some numerical experiments suggest the existence of many positive oscillating solutions. 相似文献
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