Periodic solutions and regularization of a Kepler problem with time-dependent perturbation

Boscaggin, Alberto; Ortega, Rafael; Zhao, Lei

doi:10.1090/tran/7589

Cited by 19 publications

(46 citation statements)

References 24 publications

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“…Recently, a different point of view has been proposed in [12], where a suitable definition of generalized solution to (1) was given. We recall it below for the reader's convenience.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…However, while in these papers a generalized solution is just meant as an H 1 -function attaining the value x = 0 on a zero-measure set (and solving the equation on the complementary set), Definition 1.1 requires a precise behavior at the collisions instants: that is, both the collision direction x(t) |x(t)| and the collision energy 1 2 |ẋ(t)| 2 − 1 |x(t)| are continuous functions. As shown in [12], this is a very natural definition of solution for equation (1), since it corresponds to the notion of solution provided by the well known Levi-Civita regularization for the planar Kepler problem (see [34] for some basic references about the theory of regularization in Celestial Mechanics and [24] for an application of regularization techniques to a Kepler problem with linear drag).…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

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Periodic solutions to a forced Kepler problem in the plane

Boscaggin

Dambrosio

Papini

2019

Proc. Amer. Math. Soc.

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Given a smooth function U (t, x), T -periodic in the first variable and satisfying U (t, x) = O(|x| α ) for some α ∈ (0, 2) as |x| → ∞, we prove that the forced Kepler problemẍhas a generalized T -periodic solution, according to the definition given in the paper [Boscaggin, Ortega, Zhao, Periodic solutions and regularization of a Kepler problem with time-dependent perturbation, Trans. Amer. Math. Soc, 2018]. The proof relies on variational arguments. Projects Dinamiche complesse per il problema degli N -centri and Proprietà qualitative di alcuni problemi ai limiti and by the project PRID SiDiA Sistemi Dinamici e Applicazioni of the DMIF -Università di Udine.

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“…Recently, a different point of view has been proposed in [12], where a suitable definition of generalized solution to (1) was given. We recall it below for the reader's convenience.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Periodic solutions to a forced Kepler problem in the plane

Boscaggin

Dambrosio

Papini

2019

Proc. Amer. Math. Soc.

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“…For the purposes of this work, we need to take into account different definitions of solutions, allowing for collisions (cf. also [14,16]). Definition 1.3.…”

Section: Definementioning

confidence: 88%

“…From the energy equation in (16), the boundary of the Hill's region for this problem is the closed curve parametrized by polar coordinates in this way…”

Section: Homothetic Solutions For the Anisotropic Kepler Problemmentioning

confidence: 99%

Symbolic Dynamics for the Anisotropic N-Centre Problem at Negative Energies

Barutello

Canneori

Terracini

2021

Arch Rational Mech Anal

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The planar N-centre problem describes the motion of a particle moving in the plane under the action of the force fields of N fixed attractive centres: $$\begin{aligned} \ddot{x}(t)=\sum _{j=1}^N\nabla V_j(x(t)-c_j). \end{aligned}$$ x ¨ ( t ) = ∑ j = 1 N ∇ V j ( x ( t ) - c j ) . In this paper we prove symbolic dynamics at slightly negative energy for an N-centre problem where the potentials $$V_j$$ V j are positive, anisotropic and homogeneous of degree $$-\alpha _j$$ - α j : $$\begin{aligned} V_j(x)=|x|^{-\alpha _j}V_j\left( \frac{x}{|x|}\right) . \end{aligned}$$ V j ( x ) = | x | - α j V j x | x | . The proof is based on a broken geodesics argument and trajectories are extremals of the Maupertuis’ functional. Compared with the classical N-centre problem with Kepler potentials, a major difficulty arises from the lack of a regularization of the singularities. We will consider both the collisional dynamics and the non collision one. Symbols describe geometric and topological features of the associated trajectory.

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“…We hope that the extended LLC variables can be useful in the studies of periodically perturbed motion, where radial orbits are of interest, forming the collision manifold (e.g. Boscaggin et al 2017). In particular, this formulation may be beneficial for planar, elliptic restricted three body problem and its special cases, like the Hill problem.…”

Section: Discussionmentioning

confidence: 99%

The extended Lissajous–Levi-Civita transformation

Breiter

Langner

2018

Celest Mech Dyn Astr

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Action-angle variables for the Levi-Civita regularized planar Kepler problem were introduced independently first by Chenciner and then by Deprit and Williams. The latter used explicitly the so-called Lissajous variables. When applied to the transformed Keplerian Hamiltonian, the Lissajous transformation encounters the difficulty of being defined in terms of the constant frequency parameter, whereas the Kepler problem transformed into a harmonic oscillator involves the frequency as a function of an energy-related canonical variable. A simple canonical transformation is proposed as a remedy for this inconvenience. The problem is circumvented by adding to the physical time a correcting term, which occurs to be a generalized Kepler's equation. Unlike previous versions, the transformation is symplectic in the extended phase space and allows the treatment of time-dependent perturbations. The relation of the extended Lissajous-Levi-Civita variables to the classical Delaunay angles and actions is given, and it turns out to be a straightforward generalization of the results published by Deprit and Williams.

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Periodic solutions and regularization of a Kepler problem with time-dependent perturbation

Cited by 19 publications

References 24 publications

Periodic solutions to a forced Kepler problem in the plane

Periodic solutions to a forced Kepler problem in the plane

Symbolic Dynamics for the Anisotropic N-Centre Problem at Negative Energies

The extended Lissajous–Levi-Civita transformation

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