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A geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of T²
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We give a proof based in geometric perturbation theory of a result proved by J.N. Mather using variational methods. Namely, the existence of orbits with unbounded energy in perturbations of a generic geodesic flows in T 2 by a generic periodic potential. amadeu@ma1.upc.es y llave@math.utexas.edu z tere@ma1.upc.es 1 1 INTRODUCTION 2 1 Introduction The goal of this paper is to give a proof, using geometric perturbation methods, of a result proved by J.N. Mather using variational methods. We will prove: Theorem 1.1 Let g be a C r generic metric on T 2 , U : T 2 Theta T ! R a generic C r function (on T 2 , and periodic in time), r sufficiently large. Consider the time dependent Lagrangian L(q; q; t) = 1 2 g q ( q; q) Gamma U(q; t): Then, the Euler-Lagrange equation of L has a solution q(t) whose energy E(t) = H( q(t); q(t); t) = 1 2 g q ( q(t); q(t)) + U(q(t); t); goes to infinity as t !1. Remark 1.2 Note that, in fact, the only unbounded part in...
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... The understanding of Arnold diffusion mechanisms has had outstanding progress in the last decades, relying on a wide variety of techniques: the original geometric approach by Arnold, which has been deeply developed in [7,13,14,15,18,26,19,27], variational methods [9,8], topological tools [29,10], the so-called separatrix map [53,54] or a combination of different approaches [6,37]. ...
... Once we have introduced the hierarchy of models in Section 4.1 and the dynamical systems tools in Section 4.2, we are ready to explain the instability mechanism that we consider to achieve drift in the eccentricity of the satellite. Such mechanism fits into what are usually called a priory chaotic Hamiltonian systems in Arnold diffusion literature [13,47,22]. ...
... We compute numerically the functions appearing in the first orders of these maps. Those of the scattering maps, F out,i M , are given by Melnikov-like integrals (similar so those in [13,21]). Note that the scattering maps are not globally defined since the invariant manifolds of the cylinder may have tangencies. ...
Space debris mitigation guidelines represent the most effective method to preserve the circumterrestrial environment. Among them, end-of-life disposal solutions play a key role. A growing effort is devoted to exploit natural perturbations to lead the satellites towards an atmospheric reentry, reducing the disposal cost, also if departing from high-altitude regions. In the case of the Medium Earth Orbit region, home of the navigation satellites (like Galileo), the main driver is the gravitational perturbation due to the Moon, that can increase the eccentricity in the long term. In this way, the pericenter altitude can get into the atmospheric drag domain and the satellite can eventually reenter.
In this work, we show how an Arnold diffusion mechanism can trigger the eccentricity growth. Focusing on the case of Galileo, we consider a hierarchy of Hamiltonian models, assuming that the main perturbations on the motion of the spacecraft are the oblateness of the Earth and the gravitational attraction of the Moon. First, the Moon is assumed to lay on the ecliptic plane and periodic orbits and associated stable and unstable invariant manifolds are computed for various energy levels, in the neighborhood of a given resonance. Along each invariant manifold, the eccentricity increases naturally, achieving its maximum at the first intersection between them. This growth is, however, not sufficient to achieve reentry. By moving to a model where the inclination of the Moon is taken into account, the problem becomes non-autonomous and the satellite is able to move along different energy levels. Under the ansatz of transversality of the manifolds in the autonomous case, checked numerically, Poincaré-Melnikov techniques are applied to show how diffusion can be attained, by constructing a sequence of homoclinic orbits that connect invariant tori at different energy levels.
... This phenomenon is a form of Arnold diffusion [2,8,17], in the a priori chaotic case [17,26], meaning that the unperturbed system (i.e. the geodesic flow on the static manifold M ) has a normally hyperbolic invariant manifold, the stable and unstable manifolds of which have a transverse homoclinic intersection. Other forms of Arnold diffusion have been studied: in the a priori unstable case [18,41], the unperturbed system has a normally hyperbolic invariant manifold, but the stable and unstable manifolds coincide; and in the a priori stable case the unperturbed system is completely integrable, and no further assumptions are made [3,4,9,33]. ...
... This phenomenon is a form of Arnold diffusion [2,8,17], in the a priori chaotic case [17,26], meaning that the unperturbed system (i.e. the geodesic flow on the static manifold M ) has a normally hyperbolic invariant manifold, the stable and unstable manifolds of which have a transverse homoclinic intersection. Other forms of Arnold diffusion have been studied: in the a priori unstable case [18,41], the unperturbed system has a normally hyperbolic invariant manifold, but the stable and unstable manifolds coincide; and in the a priori stable case the unperturbed system is completely integrable, and no further assumptions are made [3,4,9,33]. ...
... The setting of perturbations of geodesic flows in Arnold diffusion is well-studied [14,17,19,28]. However none of the previous literature applies directly to geodesic flows on fluctuating hypersurfaces and, to the best of the author's knowledge, the present article is the first in which such models are considered. ...
We construct $C^{\infty}$ time-periodic fluctuating surfaces in $\mathbb{R}^3$ such that the corresponding nonautonomous geodesic flow has orbits along which the energy, and thus the speed, goes to infinity.
... In particular, it has a first integral and a periodic orbit with transverse homoclinics at each energy level. Examples of such settings are certain geodesic flows with a time dependent potential, see [11,24,25,26,47,48]. 2 Such long term computations are checked to pass various consistency tests (e.g. the preservation of first integrals). But due to the exponential divergence of solutions, they are statistical in nature: an uncertainty of a few centimeters on the initial position of the Earth leads to an uncertainty of the size of the Solar System after a few hundred millions years. ...
... Proof. Recall from the definition of the Deprit variables in Section 2 that a j = L 2 j µ 2 j Mj , the eccentricity e j is defined by (25), and the inclination i 12 is defined via its cosine in (28). It follows from (29) that ...
... where e 2 is defined by (25), and cos i 12 is defined by (28). Therefore, using again the fact that Γ 1 |Λ 0 = L 1 , we see that the first and second partial derivatives of K 1 are ...
Poincar\'e's work more than one century ago, or Laskar's numerical simulations from the 1990's on, have irrevocably impaired the long-held belief that the Solar System should be stable. But mathematical mechanisms explaining this instability have remained mysterious. In 1968, Arnold conjectured the existence of "Arnold diffusion" in celestial mechanics. We prove Arnold's conjecture in the planetary spatial $4$-body problem as well as in the corresponding hierarchical problem (where the bodies are increasingly separated), and show that this diffusion leads, on a long time interval, to some large-scale instability. Along the diffusive orbits, the mutual inclination of the two inner planets is close to $\pi/2$, which hints at why even marginal stability in planetary systems may exist only when inner planets are not inclined. More precisely, consider the normalised angular momentum of the second planet, obtained by rescaling the angular momentum by the square root of its semimajor axis and by an adequate mass factor (its direction and norm give the plane of revolution and the eccentricity of the second planet). It is a vector of the unit $3$-ball. We show that any finite sequence in this ball may be realised, up to an arbitrary precision, as a sequence of values of the normalised angular momentum in the $4$-body problem. For example, the second planet may flip from prograde nearly horizontal revolutions to retrograde ones. As a consequence of the proof, the non-recurrent set of any finite-order secular normal form accumulates on circular motions -- a weak form of a celebrated conjecture of Herman.
... The orbits constructed in Theorem 1 rely on an Arnold diffusion mechanism [2]. Progress in the understanding of Arnold diffusion in nearly-integrable Hamiltonian systems in these last decades has been remarkable, especially for two and a half degrees of freedom (see [4,6,10,[16][17][18]21,35,43,49,55], or [5,9,15,20,33,34,36,56] for results in higher dimension). However, most of these results deal with generic nearly-integrable Hamiltonian systems in C r or C ∞ regularity whereas results in the analytic category, including results in Celestial Mechanics, are rather scarce. ...
... Indeed, recall the first order of the Hamiltonian that depends on the Poincaré variables ξ, η is of order 1 L 6 2 (i.e. the first order term H 12 0 in the expansion of the secular Hamiltonian, defined by (48)). Moreover the first order term of the graph ρ defining that depends on the variablẽ L 3 is of order L 10 2 (L * 3 ) 7 (see Lemma (16)). Since H 12 0 depends quadratically on ξ and η (see (48)), the term H 12 1 is the lowest-order term that could contain products of the formL 3 P for P ∈ {˜ 3 ,˜ 1 ,˜ 3 }; since the coefficient of H 12 1 is of order 1 L 7 2 , the order of such terms is 1 L 7 2 L 10 2 (L * 3 ) 7 = ε 4 μ 7 . ...
A longstanding belief has been that the semimajor axes, in the Newtonian planetary problem, are stable. Our the course of the XIX century, Laplace, Lagrange and others gave stronger and stronger arguments in this direction, thus culminating in what has commonly been referred to as the first Laplace–Lagrange stability theorem. In the problem with 3 planets, we prove the existence of orbits along which the semimajor axis of the outer planet undergoes large random variations thus disproving the conclusion of the Laplace–Lagrange theorem. The time of instability varies as a negative power of the masses of the planets. The orbits we have found fall outside the scope of the theory of Nekhoroshev–Niederman because they are not confined by the conservation of angular momentum and because the Hamiltonian is not (uniformly) convex with respect to the Keplerian actions.
... and can also be used to obtain Birkhoff sections [11]. They are also the basic skeleton for Mather acceleration theorems in Arnold diffusion [5,15,24,35]. ...
We show that a Kupka–Smale riemannian metric on a closed surface contains a finite primary set of closed geodesics, i.e. they intersect any other geodesic and divide the surface into simply connected regions. From them we obtain a finite set of disjoint surfaces of section of genera 0 or 1, which intersect any orbit of the geodesic flow. As an application we obtain that the geodesic flow of a Kupka–Smale riemannian metric on a closed surface has homoclinic orbits for all branches of all of its hyperbolic closed geodesics.
... and can also be used to obtain Birkhoff sections [11]. They are also the basic skeleton for Mather acceleration theorems in Arnold diffusion [35], [5], [15], [24]. ...
We show that a Kupka-Smale riemannian metric on a closed surface contains a finite primary set of closed geodesics, i.e. they intersect any other geodesic and divide the surface into simply connected regions. From them we obtain a finite set of disjoint surfaces of section of genera 0 or 1, which intersect any orbit of the geodesic flow. As an application we obtain that the geodesic flow of a Kupka-Smale riemannian metric on a closed surface has homoclinic orbits for all branches of all of its hyperbolic closed geodesics.
... The orbits constructed in Theorem 1 rely on an Arnold diffusion mechanism [2]. Progress in the understanding of Arnold diffusion in nearly-integrable Hamiltonian systems in these last decades has been remarkable, especially for two and a half degrees of freedom (see [4,6,10,16,17,18,21,35,43,49,54], or [5,9,15,20,33,34,36,55] for results in higher dimension). However, most of these results deal with generic nearly-integrable Hamiltonian systems in C r or C ∞ regularity whereas results in the analytic category, including results in Celestial Mechanics, are rather scarce. ...
A longstanding belief has been that the semimajor axes, in the Newtonian planetary problem, are stable. In the course of the XIX century, Laplace, Lagrange and others gave stronger and stronger arguments in this direction, thus culminating in what has commonly been referred to as the first Laplace-Lagrange stability theorem. In the problem with 3 planets, we prove the existence of orbits along which the semimajor axis of the outer planet undergoes large random variations thus disproving the theorem of Laplace-Lagrange. The time of instability varies as a negative power of the masses of the planets. The orbits we have found fall outside the scope of the theory of Nekhoroshev-Niederman because they are not confined by the conservation of angular momentum and because the Hamiltonian is not (uniformly) convex with respect to the Keplerian actions.
... Our goal now is to build a map which encodes the dynamics along each of the manifolds Γ ± which, following [DdlLS08], we denote as homoclinic channels . These are the so-called scattering maps introduced by Delshams, de la Llave and Seara in [DdlLS00,DdlLS06] (see also [DdlLS08], where the geometric properties of this object are thoroughly studied). Loosely speaking, given one of the channels Γ ± , at a point (ϕ u , I u ) ∈ P * ∞ , its associated scattering map gives the forward asymptotic (α, G) components along the unique heteroclinic orbit through Γ ± which is asymptotic in the past to (ϕ u , I u ). ...
A major question in dynamical systems is to understand the mechanisms driving global instability in the 3 Body Problem (3BP), which models the motion of three bodies under Newtonian gravitational interaction. The 3BP is called restricted if one of the bodies has zero mass and the other two, the primaries, have strictly positive masses $m_0,m_1$. We consider the Restricted Planar Elliptic 3 Body Problem (RPE3BP) where the primaries revolve in Keplerian ellipses. We prove that the RPE3BP exhibits topological instability: for any values of the masses $m_0,m_1$ (except $m_0=m_1$), we build orbits along which the angular momentum of the massless body experiences an arbitrarily large variation provided the eccentricity of the orbit of the primaries is positive but small enough. In order to prove this result we show that a degenerate Arnold Diffusion Mechanism, which moreover involves exponentially small phenomena, takes place in the RPE3BP. Our work extends the result obtained in \cite{MR3927089} for the a priori unstable case $m_1/m_0\ll1$, to the case of arbitrary masses $m_0,m_1>0$, where the model displays features of the so-called \textit{a priori stable} setting.
... A crucial tool to understand the dynamics close to the invariant manifolds to infinity is the so-called Scattering map. The Scattering map was introduced by Delshams, de la Llave and Seara [DdlLS00,DdlLS06,DdlLS08] to analyze the heteroclinic connections to a normally hyperbolic invariant manifold. However, as shown in [DKdlRS19] (see Section 4.3), the theory in [DdlLS08] can be adapted to the parabolic setting of the present paper. ...
Consider the planar 3 Body Problem with masses $m_0,m_1,m_2>0$. In this paper we address two fundamental questions: the existence of oscillatory motions and of chaotic hyperbolic sets. In 1922, Chazy classified the possible final motions of the three bodies, that is the behaviors the bodies may have when time tends to infinity. One of the possible behaviors are oscillatory motions, that is, solutions of the 3 Body Problem such that the positions of the bodies $q_0, q_1, q_2$ satisfy \[ \liminf_{t\to\pm\infty}\sup_{i,j=0,1,2, i\neq j}\|q_i-q_j\|<+\infty \quad \text{ and }\quad \limsup_{t\to\pm\infty}\sup_{i,j=0,1,2, i\neq j}\|q_i-q_j\|=+\infty. \] Assume that all three masses $m_0,m_1,m_2>0$ are not equal. Then, we prove that such motions exists. We also prove that one can construct solutions of the three body problem whose forward and backward final motions are of different type. This result relies on constructing invariant sets whose dynamics is conjugated to the (infinite symbols) Bernouilli shift. These sets are hyperbolic for the symplectically reduced planar 3 Body Problem. As a consequence, we obtain the existence of chaotic motions, an infinite number of periodic orbits and positive topological entropy for the 3 Body Problem.
In this paper, we prove that the nearly integrable system of the form $$H(x,y) = h(y) + \varepsilon P(x,y),\,\,\,\,\,\,\,x \in {\mathbb{T}^n},\,\,\,\,\,\,\,\,y \in {\mathbb{R}^n},\,\,\,\,\,\,n\,\geqslant\,3$$ admits orbits that pass through any finitely many prescribed small balls on the same energy level H−1(E) provided that E > min h, if h is convex, and εP is typical. This settles the Arnold diffusion conjecture for convex systems in the smooth category. We also prove the counterpart of Arnold diffusion for the Riemannian metric perturbation of the flat torus.
The splitting of separatrices of hyperbolic fixed points for exact symplectic maps of n degrees of freedom is considered. The non-degenerate critical points of a real-valued function (called the Melnikov potential) are associated to transverse homoclinic orbits and an asymptotic expression for the symplectic area between homoclinic orbits is given. Moreover, if the unperturbed invariant manifolds are completely doubled, it is shown that there exist, in general, at least 4 primary homoclinic orbits (4n in antisymmetric maps). Both lower bounds are optimal. Two examples are presented: a 2n-dimensional central standard-like map and the Hamiltonian map associated to a magnetized spherical pendulum. Several topics are studied about these examples: existence of splitting, explicit computations of Melnikov potentials, transverse homoclinic orbits, exponentially small splitting, etc.
In the past several decades many significant results in averaging for systems of ODE's have been obtained. These results have not attracted a tention in proportion to their importance, partly because they have been overshadowed by KAM theory, and partly because they remain widely scattered - and often untranslated - throughout the Russian literature. The present book seeks to remedy that situation by providing a summary, including proofs, of averaging and related techniques for single and multiphase systems of ODE's. The first part of the book surveys most of what is known in the general case and examines the role of ergodicity in averaging. Stronger stability results are then obtained for the special case of Hamiltonian systems, and the relation of these results to KAM Theory is discussed. Finally, in view of their close relation to averaging methods, both classical and quantum adiabatic theorems are considered at some length. With the inclusion of nine concise appendices, the book is very nearly self-contained, and should serve the needs of both physicists desiring an accessible summary of known results, and of mathematicians seeing an introduction to current areas of research in averaging.
We show that a nearly integrable Hamiltonian system has invariant tori of all dimensions smaller than the number of degrees of freedom provided that certain nondegeneracy conditions are met. The tori we construct are generated by the resonances of the system and are topologically different from the orbits that are present in the integrable system. We also show that the tori we construct have stable and unstable manifolds and point out how to construct other types of interesting orbits. The method of proof is a combination of different perturbation methods.
CONTENTSPart I § 1. Introduction § 2. Prerequisites from ergodic theory § 3. Basic properties of the characteristic exponents of dynamical systems § 4. Properties of local stable manifoldsPart II § 5. The entropy of smooth dynamical systems § 6. "Measurable foliations". Description of the pi-partition § 7. Ergodicity of a diffeomorphism with non-zero exponents on a set of positive measure. The K-property § 8. The Bernoullian property § 9. Flows § 10. Geodesic flows on closed Riemannian manifolds without focal pointsReferences