Oscillatory motions for the restricted planar circular three body problem

Guàrdia, Marcel; Martín, Pau; Seara, Tere M.

doi:10.1007/s00222-015-0591-y

Cited by 45 publications

(135 citation statements)

References 25 publications

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“…Let us suppose then that lim t→t * r (t) = +∞. Again by the second equation in (7) it follows that lim t→t * r (t) = 0. Thereforeṙ (t) remains bounded as t → t * and then r (t) remains bounded as t → t * .…”

Section: Remarkmentioning

confidence: 93%

“…There exists an integerb > 0 such that any sequence a = (a n ) with a n ≥b corresponds to a solution of (7).…”

Section: Theoremmentioning

confidence: 99%

“…We have computed numerically the first intersection of P u,s with the section , solving in backward time system (7). We denote by γ u,s this first intersection.…”

Section: Transversal Intersection Of the Invariant Manifolds P Usmentioning

confidence: 99%

“…So, using classical perturbation theory they prove the transversality of the invariant parabolic manifolds. More recently in [7] the transversality of the invariant manifolds is proved for all values of the mass ratio. They have used the Jacobi constant as a parameter since their system is conservative.…”

Section: Introductionmentioning

confidence: 98%

“…Completeness of the FlowLet us derive some general properties of the flow associated to the system(7). A straightforward computation gives us.…”

mentioning

confidence: 99%

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Dynamics and Chaos in the Elliptic Isosceles Restricted Three-Body Problem with Collision

2015

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The elliptic isosceles restricted three-body problem (EIR3BP) with collision is defined as follows: two point masses m 1 = m 2 move along a degenerate elliptic collision orbit under their gravitational attraction, then describe the motion of a third massless particle moving on a plane perpendicular to their line of motion and passing through the center of mass of the primaries. By symmetry, the component of the angular momentum of the massless particle along the direction of the line of the primaries is conserved. We fixed it to a non-zero value in order to avoid total collision, and perform the reduction to one and a half degrees of freedom. We prove that the flow defined by the EIR3BP is complete and if a solution escapes to infinity when time t → ±∞, then it is parabolic or hyperbolic. A description of the parabolic orbits is given and they are asymptotic to a degenerate periodic orbit at infinity. We verify that the unstable and stable manifolds P u,s of this periodic orbit at infinity are differentiable (in fact, C ∞ ) at the origin and analytic outside. For sufficiently large angular momentum, we prove that P u and P s intersect a surface of section in simple closed curves γ u,s having two points of intersection and we show that γ u and γ s have a transversal intersection at these points. We prove that there exists a subset of where the Poincaré map is topologically conjugate to a Bernoulli shift, in particular this shows the existence of a very complicated dynamic (chaotic dynamic) in the EIR3BP.

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Section: Remarkmentioning

confidence: 93%

“…There exists an integerb > 0 such that any sequence a = (a n ) with a n ≥b corresponds to a solution of (7).…”

Section: Theoremmentioning

confidence: 99%

“…We have computed numerically the first intersection of P u,s with the section , solving in backward time system (7). We denote by γ u,s this first intersection.…”

Section: Transversal Intersection Of the Invariant Manifolds P Usmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

“…Completeness of the FlowLet us derive some general properties of the flow associated to the system(7). A straightforward computation gives us.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Dynamics and Chaos in the Elliptic Isosceles Restricted Three-Body Problem with Collision

2015

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Invariant Manifolds of Degenerate Tori and Double Parabolic Orbits to Infinity in the $$(n+2)$$-Body Problem

Baldomá,

Fontich,

Martín

2024

Arch Rational Mech Anal

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Global Instability in the Restricted Planar Elliptic Three Body Problem

Delshams

Калошин

Rosa

et al. 2018

Commun. Math. Phys.

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The restricted planar elliptic three body problem (RPETBP) describes the motion of a massless particle (a comet or an asteroid) under the gravitational field of two massive bodies (the primaries, say the Sun and Jupiter) revolving around their center of mass on elliptic orbits with some positive eccentricity. The aim of this paper is to show the existence of orbits whose angular momentum performs arbitrary excursions in a large region. In particular, there exist diffusive orbits, that is, with a large variation of angular momentum.The leading idea of the proof consists in analyzing parabolic motions of the comet. By a well-known result of McGehee, the union of future (resp. past) parabolic orbits is an analytic manifold P + (resp. P − ). In a properly chosen coordinate system these manifolds are stable (resp. unstable) manifolds of a manifold at infinity P∞, which we call the manifold at parabolic infinity.On P∞ it is possible to define two scattering maps, which contain the map structure of the homoclinic trajectories to it, i.e. orbits parabolic both in the future and the past. Since the inner dynamics inside P∞ is trivial, two different scattering maps are used. The combination of these two scattering maps permits the design of the desired diffusive pseudo-orbits. Using shadowing techniques and these pseudo orbits we show the existence of true trajectories of the RPETBP whose angular momentum varies in any predetermined fashion.2000 Mathematics Subject Classification: Primary 37J40, 70F15. Keywords: Elliptic Restricted Three Body problem, parabolic motions, Manifold of parabolic motions at infinity, Arnold diffusion, splitting of separatrices, Melnikov integral. Main result and methodologyThe restricted planar elliptic three body problem (RPETBP) describes the motion q of a massless particle (a comet) under the gravitational field of two massive bodies (the primaries, say the Sun and Jupiter ) with mass ratio µ revolving around their center of mass on elliptic orbits with eccentricity J . In this paper we search for trajectories of motion which show a large variation of the angular momentum G = q ×q. In other words, we search for global instability ("diffusion"This is one of the reasons why we are going to restrict ourselves to the region G ≥ C large enough and J G ≤ c small enough along this paper to get the diffusive orbits.Among the harmonics L 0, of 0 order in s, by (42), the harmonic L 0,0 appears to be the dominant one, but we will also estimate L 0,1 to get information about the variable α, and bound the rest of harmonics L 0, for ≥ 2. Among the harmonics of first order L 1,k , again by (42), the and the error functions satisfyTo bound the integral (81) for m ≥ 0 we will consider two different cases: 0 ≤ q ≤ m and 0 ≤ m < q. Let us first consider the case 0 ≤ q ≤ m. By the analyticity and periodicity of the integral we change the path of integration from (E) = 0 to E = ln(2a 2 / J )so that e iE = e iu−ln(2a 2 / J) = J 2a 2 e iu and then, by (78), (79) and (83a), (83b), 23Therefore along the complex path...

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Oscillatory motions for the restricted planar circular three body problem

Cited by 45 publications

References 25 publications

Dynamics and Chaos in the Elliptic Isosceles Restricted Three-Body Problem with Collision

Dynamics and Chaos in the Elliptic Isosceles Restricted Three-Body Problem with Collision

Invariant Manifolds of Degenerate Tori and Double Parabolic Orbits to Infinity in the $$(n+2)$$-Body Problem

Global Instability in the Restricted Planar Elliptic Three Body Problem

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