The Restricted Three-Body Problem and Holomorphic Curves

Frauenfelder, Urs; Koert, Otto van

doi:10.1007/978-3-319-72278-8

Cited by 59 publications

(62 citation statements)

References 31 publications

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“…A final remark concerning the Hamiltonian: for the lunar problem we use the regularization of H µ , the Hamiltonian given in (8), but some care has to be taken to deal with the catastrophic cancellation in the final term.…”

Section: Integration Schemementioning

confidence: 99%

“…We also exploit symmetry properties of the lunar problem.The resulting family of orbits is numerically investigated and has interesting properties. The properties are analogous to those in [10] for very negative energy, but differ markedly for larger energy.Geometric and global properties of the larger family of periodic orbits connecting those found here and those in [10] are described using ideas from symplectic geometry [8].In particular, we study the extension of F(h, µ) as a function of µ by using a construction due to P. Rabinowitz [9].The main theorem for this paper is stated as Theorem 1 in Section 2. The proof is done in Section 3.…”

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confidence: 92%

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A family of periodic orbits in the three-dimensional lunar problem

Belbruno

Frauenfelder

Koert

2019

Celest Mech Dyn Astr

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A family of periodic orbits is proven to exist in the spatial lunar problem that are continuations of a family of consecutive collision orbits, perpendicular to the primary orbit plane. This family emanates from all but two energy values. The orbits are numerically explored. The global properties and geometry of the family is studied. IntroductionWe consider the three-dimensional circular restricted three-body problem. This models the three-dimensional motion of a particle, P 0 , of zero mass in the Newtonian gravitational field generated by two particles, P 1 , P 2 of respective positive masses, m 1 , m 2 , in a mutual uniform circular motion. It is assumed that m 1 is much larger than m 2 . This problem is studied in a rotating coordinate system that rotates with the same constant frequency, ω of the circular motion of P 1 , P 2 , so that in this system P 1 and P 2 are fixed. Because m 1 is much larger than m 2 , we refer to P 1 as the Earth and P 2 as the Moon, for convenience.When P 0 moves about the larger particle, P 1 , the motion of P 0 can be completely understood if, for example, P 0 is restricted to the two-dimensional plane of motion of P 1 , P 2 . In this case, with m 2 = 0, assume that P 0 has precessing elliptic motion, of elliptic frequency ω * about P 1 , precessing with frequency ω. Then the Kolmogorov-Arnold-Moser(KAM) Theorem proves that this precessing motion persists if m 2 is sufficiently small and if ω and ω * are sufficiently noncommensurate. Otherwise, the motion is chaotic due to heteroclinic dynamics. That is, invariant KAM tori foliate the phase space. The motion of P 2 is proven to be stable [1]. When the initial elliptic motion of P 0 is not in the same plane as the P 0 , P 1 then under similar assumptions although KAM tori can be proven to exist, but stability cannot be guaranteed.A Family of Periodic Orbits in the Three-Dimensional Lunar Problem 2 In this paper we study the three-dimensional motion of P 0 about P 2 . This is referred to as the three-dimensional, or spatial, lunar problem. Relatively little is proven in general about the motion of P 0 unless the initial motion starts infinitely close to P 2 . The proof of existence of KAM tori in the three-dimensional lunar problem was obtained by M. Kummer under the assumption that the initial motion of P 0 lies infinitely near to P 2 [2]. ‡The main result of this paper is to prove the existence of a special family of periodic orbits about P 2 , nearly perpendicular to the primary orbit plane. More precisely, if we normalize m 1 = 1 − µ, m 2 = µ, then in the case of µ = 0 there exists a family of periodic orbits on the z−axis through P 2 , so perpendicular to the P 1 , P 2 plane, parameterized by their energy h. This family consists of consecutive collision orbits: Starting at collision at P 2 , they extend up the z−axis to a maximal distance d = d(h), then fall back to P 2 , and periodically repeat this oscillation, where d can have any positive value. We label these as φ * (t, h). These orbits have period T * (h). We prove that ...

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Section: Integration Schemementioning

confidence: 99%

mentioning

confidence: 92%

A family of periodic orbits in the three-dimensional lunar problem

Belbruno

Frauenfelder

Koert

2019

Celest Mech Dyn Astr

Self Cite

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“…Note that y ± := (−(1 − 2µ) ± c 2 + 2c + (1 − 2µ) 2 )/c, which are the boundary values for y in the Earth and the Moon components, see (10), are roots of h . Using −cy 2 ± − 2(1 − 2µ)y ± + c + 2 = 0 we compute that which follows that…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

“…Since convexity is preserved under chart transition, which is a linear transformation, to show that Σ E c is fiberwise convex it suffices to show that q ∈ R 2 : H(q, p) = c bounds a strictly convex domain in R 2 for each p (cf. [10]).…”

Section: Introductionmentioning

confidence: 99%

On a Convex Embedding of the Euler Problem of Two Fixed Centers

Kim

2018

Regul. Chaot. Dyn.

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In this article, we study a convex embedding for the Euler problem of two fixed centers for energies below the critical energy level. We prove that the doubly-covered elliptic coordinates provide a 2-to-1 symplectic embedding such that the image of the bounded component near the lighter primary of the regularized Euler problem is convex for any energy below the critical Jacobi energy. This holds true if the two primaries have the equal mass, but does not holds near the heavier body.

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“…A similar construction can be done at any other negative energy, corresponding to elliptical orbits. The annoying collision orbits of the Kepler flow are thereby included into a smooth flow, and (at least the existence of) the Runge-Lenz vector becomes clear, since the geodesic flow is invariant under the action of the 3-dimensional group SO(3); see [41,86].…”

Section: Example 34 (Moser Regularization)mentioning

confidence: 99%

Untitled

2018

Bull. Amer. Math. Soc.

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We describe old and new motivations to study symplectic embedding problems, and we discuss a few of the many old and the many new results on symplectic embeddings. Contents 1. Introduction 2. Meanings of "symplectic" 3. From Newtonian mechanics to symplectic geometry 4. Why study symplectic embedding problems 5. Euclidean symplectic volume preserving 6. Symplectic ellipsoids 7. The role of J-holomorphic curves 8. The fine structure of symplectic rigidity 9. Packing flexibility for linear tori 10. Intermediate symplectic capacities or shadows do not exist 1 2 3

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The Restricted Three-Body Problem and Holomorphic Curves

Cited by 59 publications

References 31 publications

A family of periodic orbits in the three-dimensional lunar problem

A family of periodic orbits in the three-dimensional lunar problem

On a Convex Embedding of the Euler Problem of Two Fixed Centers

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