Symbolic dynamics for the $N$-centre problem at negative energies

Soave, Nicola; Terracini, Susanna

doi:10.3934/dcds.2012.32.3245

Cited by 35 publications

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“…Note the similarity between J ν and the usual energy function H(z,ż) = |ż| 2 /2 − V (z): it results H = J 0 . In this paper we generalize the approach already developed in [11], proving the existence of infinitely many collision-free periodic solutions of Eq. (3) with negative and small (in absolute value) Jacobi constant, provided the angular velocity |ν| is sufficiently small.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…where x = x(t) ∈ R 2 denotes the position of the particle at time t ∈ R; basic references for such a problem are [3,4,[6][7][8][9]11] and the references therein. In this paper we consider α-gravitational potentials of type…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Since the terms in z andż are multiplied by powers of ν, the idea is that if |ν| is sufficiently small, then Eq. (3) can be regarded as a perturbation of the planar N -centre problem, which we dealt with in [11]. Note that, contrary to (1), Eq.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In particular, letting ν to tend to 0, the relation (4) implies that either m k → 0 for every k or |c k | → 0 for every k; as a consequence, the equation of the restricted problem in the limit case ν → 0 tends toz = 0, which has no relation with the N -centre problem or the N -body problem. As a toy model towards the real restricted (N + 1)-body problem, we introduce the rotating N -centre problem; we point out that the motivation for its study is prevalently mathematical: our goal is to understand if the techniques introduced in [11] are sufficiently robust to survive when we perturb the N -centre problem by letting the centres move; the answer is yes, but, as we will see, the extension of our broken geodesics method is not trivial and requires new ideas. Therefore, the generalization to the real restricted problem seems possible, but extremely complicated.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Symbolic dynamics: from the N-centre to the (N + 1)-body problem, a preliminary study

Soave

2013

Nonlinear Differ. Equ. Appl.

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We consider a rotating N-centre problem, with N ≥ 3 and homogeneous potentials of degree −α < 0, α ∈ [1, 2). We prove the existence of infinitely many collision-free periodic solutions with negative and small Jacobi constant and small values of the angular velocity, for any initial configuration of the centres. We will introduce a Maupertuis' type variational principle in order to apply the broken geodesics technique developed in Soave and Terracini (Discrete Contin Dyn Syst 32:3245-3301, 2012). Major difficulties arise from the fact that, contrary to the classical Jacobi length, the related functional does not come from a Riemaniann structure but from a Finslerian one. Our existence result allows us to characterize the associated dynamical system with a symbolic dynamics, where the symbols are given partitions of the centres in two non-empty sets.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

See 2 more Smart Citations

Symbolic dynamics: from the N-centre to the (N + 1)-body problem, a preliminary study

Soave

2013

Nonlinear Differ. Equ. Appl.

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“…A deeper understanding of the behaviour of near-boundary JM geodesics seems necessary to the further development of JM variational methods in case where the Hill boundary is not empty. For some results in celestial mechanics based on JM variational methods in instances where the Hill boundary is not empty see [5] and [7] whin this direction 2. Does the fact that JM curvatures tend to positive infinity imply there are conjugate points near the boundary?…”

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Who's Afraid of the Hill Boundary?

Montgomery¹

2014

SIGMA

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Abstract. The Jacobi-Maupertuis metric allows one to reformulate Newton's equations as geodesic equations for a Riemannian metric which degenerates at the Hill boundary. We prove that a JM geodesic which comes sufficiently close to a regular point of the boundary contains pairs of conjugate points close to the boundary. We prove the conjugate locus of any point near enough to the boundary is a hypersurface tangent to the boundary. Our method of proof is to reduce analysis of geodesics near the boundary to that of solutions to Newton's equations in the simplest model case: a constant force. This model case is equivalent to the beginning physics problem of throwing balls upward from a fixed point at fixed speeds and describing the resulting arcs, see Results and motivationOne constructs the Jacobi metric ds 2 JM of classical mechanics by fixing the total energy E of the system and multiplying the kinetic energy metric ds 2 K by the conformal factorwhere V is the potential energy. It is well-known that the geodesics for this Jacobi-Maupertuis metric (henceforth JM metric for short) are, up to reparameterization, exactly the solutions to Newton's equations having energy E. (See Proposition 1 below for a careful statement. See [1, Theorem 3.7.7] for another discussion and a nice proof.) The domain of the Jacobi metric is the domain in configuration space where this conformal factor is non-negative and is called the Hill region:The Hill region includes the Hill boundary (sometimes called the zero velocity surface) where the conformal factor, and hence the metric, vanishes:A "regular point" q 0 of the Hill boundary is one for which df (q 0 ) = 0. Here is our main result.Theorem 1. Any JM geodesic which comes sufficiently close to a regular point q 0 of the Hill boundary contains a pair of conjugate points close to q 0 which are conjugate along a short arc close to q 0 . In particular, such a geodesic fails to minimize JM length.This theorem is a direct consequence of a structure theorem, Theorem 2 below, regarding the conjugate locus of near-boundary points, and results from Seifert's seminal paper [6] which we recall in the next section. R. MontgomeryMotivations. Two questions motivated this paper. 1. Can the calculus of variations, applied to the JM metric reformulation of mechanics, uncover new results regarding the classical three-body problem? The direct method of the calculus of variations breaks down at the Hill boundary since curves lying in the boundary have zero JM length. A deeper understanding of the behaviour of near-boundary JM geodesics seems necessary to the further development of JM variational methods in case where the Hill boundary is not empty. For some results in celestial mechanics based on JM variational methods in instances where the Hill boundary is not empty see [5] and [7] whin this direction 2. Does the fact that JM curvatures tend to positive infinity imply there are conjugate points near the boundary? Let q be a point near a regular point of the Hill boundary and let y denote its Rieman...

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Topologically Distinct Collision-Free Periodic Solutions for the $${N}$$ N -Center Problem

Castelli¹

2016

Arch Rational Mech Anal

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This work concerns the planar N -center problem with homogeneous potential of degree −α (α ∈ [1, 2)). The existence of infinitely many, topologically distinct, non-collision periodic solutions with a prescribed energy is proved. A notion of admissibility in the space of loops on the punctured plane is introduced so that in any admissible class and for any positive h the existence of a classical periodic solution with energy h for the N -center problem with α ∈ (1, 2) is proven. In case α = 1 a slightly different result is shown: it is the case that there is either a non-collision periodic solution or a collision-reflection solution. The results hold for any position of the centres and it is possible to prescribe in advance the shape of the periodic solutions. The proof combines the topological properties of the space of loops in the punctured plane with variational and geometrical arguments.

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Symbolic dynamics for the $N$-centre problem at negative energies

Cited by 35 publications

References 19 publications

Symbolic dynamics: from the N-centre to the (N + 1)-body problem, a preliminary study

Symbolic dynamics: from the N-centre to the (N + 1)-body problem, a preliminary study

Who's Afraid of the Hill Boundary?

Topologically Distinct Collision-Free Periodic Solutions for the $${N}$$ N -Center Problem

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