Arnold diffusion

For integrable systems, one has the conservation of the action variables. According to the KAM theorem if we perturb an integrable system slightly, then many, though certainly not all, of the solutions of the perturbed system stay close, for all time, to the unperturbed system. In particular, since the action variables were originally conserved, the theorem tells us that there is only a small change in action for many solutions of the perturbed system.

However, as first noted in Arnold's paper, there are nearly integrable systems for which there exist solutions that exhibit arbitrarily large growth in the action variables. More precisely, Arnold considered the example of nearly integrable Hamiltonian system with Hamiltonian

: $H(I,\; \backslash phi,\; p,\; q,\; t)\; =\; \{1\; \backslash over\; 2\; \}I^2\; +\; \{1\; \backslash over\; 2\; \}p^2\; +\; \backslash epsilon\; (\backslash cos\{\; q\}\; -\; 1)\; +\; \backslash mu(\backslash cos\{q\}\; -\; 1)(\backslash sin\{\backslash phi\; +\; \backslash cos\; t)\}$

The first three terms of this Hamiltonian describe a rotator-pendulum system.

Arnold showed that for this system, for any choice of $I\_+>\; I\_-\; >\; 0$, and for , there is a solution to the system for which

:

for some time $T\; \backslash gg\; 0.$

His proof relies on the existence of 'transition chains' of 'whiskered' tori, that is, sequences of tori with transitive dynamics such that the unstable manifold(whisker) of one of these tori intersects transversally the stable manifold (whisker) of the next one. Arnold conjectured that "the mechanism of 'transition chains' which guarantees that nonstability in our example is also applicable to the general case (for example, to the problem of three bodies)."

The KAM theorem and Arnold diffusion has led to a compendium of rigorous mathematical results, with insights from physics.

Pierre Lochak, (1999) [https://webusers.imj-prg.fr/~pierre.lochak/textes/compendium.pdf Arnold diffusion; a compendium of remarks and questions] In "Hamiltonian Systems with Three or More Degrees of Freedom" (S’Agar´o, 1995), C. Sim´o, ed, NATO ASI Series C: Math. Phys. Sci., Vol. 533, Kluwer Academic, Dordrecht (1999), 168–183.

Henk W. Broer, Mikhail B. Sevryuk (2007) [http://www.math.rug.nl/~broer/pdf/hdbk.pdf KAM Theory: quasi-periodicity in dynamical systems] In: H.W. Broer, B. Hasselblatt and F. Takens (eds.), Handbook of Dynamical Systems Vol. 3, North-Holland, 2010

In Arnold's model the perturbation term is of a special type. The general case of Arnold's diffusion problem concerns Hamiltonian systems of one of the forms

:{{EquationRef|(1)}} $H\_\backslash epsilon(I,\backslash phi,p,q)=H\_0(I,p,q)+\backslash epsilon\; H\_1(I,\backslash phi,\; p,\; q,t)$

where $(I,\backslash phi,p,q,t)\backslash in\backslash mathbb\{R\}^m\backslash times\backslash mathbb\{T\}^m\backslash times\backslash mathbb\{R\}^n\backslash times\backslash mathbb\{T\}^n\backslash times\backslash mathbb\{T\}^1$, $m,n\backslash geq\; 1$, and $H\_0(I,p,q)$ describes a rotator-pendulum system, or

:{{EquationRef|(2)}} $H\_\backslash epsilon(I,\backslash phi)=H\_0(I)+\backslash epsilon\; H\_1(I,\backslash phi,t)$

where $(I,\backslash phi,t)\backslash in\backslash mathbb\{R\}^N\backslash times\backslash mathbb\{T\}^N\backslash times\backslash mathbb\{T\}^1$, $N\backslash geq\; 2.$

For systems as in {{EquationNote|(1)}}, the unperturbed Hamiltonian possesses smooth families of invariant tori that have hyperbolic stable and unstable manifolds; such systems are referred to as a priori unstable. For system as in {{EquationNote|(2)}}, the phase space of the unperturbed Hamiltonian is foliated by Lagrangian invariant tori; such systems are referred to as a priori stable.{{cite journal|last1=Chierchia|first1=Luigi|last2=Gallavotti |first2=Giovanni|volume=60|journal=Annales de l'I.H.P.: Physique Théorique|pages= 1–144|title=Drift and diffusion in phase space|url=http://www.numdam.org/item/AIHPA_1994__60_1_1_0/|year = 1994 |mr=1259103 |zbl=1010.37039}} {{Erratum|[http://www.numdam.org/item/AIHPA_1998__68_1_135_0/ Annales de l'I.H.P.: Physique Théorique. 68: 135 (1998)]}} In either case, the Arnold diffusion problem asserts that, for `generic' systems, there exists $\backslash rho>0$ such that for every $\backslash epsilon>0$ sufficiently small there exist solution curves for which

:$\backslash |I(T)-I(0)\backslash |\backslash geq\; \backslash rho$

for some time $T\backslash gg\; 0.$ Precise formulations of possible genericity conditions in the context of a priori unstable and a priori stable system can be found in,{{cite journal | last1=Chen | first1=Qinbo | last2=de la Llave | first2=Rafael | title=Analytic genericity of diffusing orbits in a priori unstable Hamiltonian systems | journal=Nonlinearity | publisher=IOP Publishing | volume=35 | issue=4 | date=2022-03-09 | issn=0951-7715 | doi=10.1088/1361-6544/ac50bb | pages=1986–2019| arxiv=2103.03847 | bibcode=2022Nonli..35.1986C | doi-access=free }}{{cite book | last=Mather | first=John N. | title=Essays in Mathematics and its Applications | chapter=Arnold Diffusion by Variational Methods | publisher=Springer Berlin Heidelberg | publication-place=Berlin, Heidelberg | year=2012 | pages=271–285 | isbn=978-3-642-28820-3 | doi=10.1007/978-3-642-28821-0_11}} respectively. Informally, the Arnold diffusion problem says that small perturbations can accumulate to large effects.

Recent results in the a priori unstable case include,{{cite journal | last1=Bolotin | first1=S | last2=Treschev | first2=D | title=Unbounded growth of energy in nonautonomous Hamiltonian systems | journal=Nonlinearity | publisher=IOP Publishing | volume=12 | issue=2 | date=1999-01-01 | issn=0951-7715 | doi=10.1088/0951-7715/12/2/013 | pages=365–388| bibcode=1999Nonli..12..365B | s2cid=250852828 }}{{cite journal | last1=Cheng | first1=Chong-Qing | last2=Yan | first2=Jun | title=Existence of Diffusion Orbits in a priori Unstable Hamiltonian Systems | journal=Journal of Differential Geometry | publisher=International Press of Boston | volume=67 | issue=3 | date=2004-07-01 | issn=0022-040X | doi=10.4310/jdg/1102091356| doi-access=free }}

{{cite journal | last1=Delshams | first1=Amadeu

| last2=de la Llave | first2=Rafael

|last3=M-Seara|first3=Tere

|year=2006

|volume=179

|doi=10.1090/memo/0844

| journal= Mem. Am. Math. Soc.

| title=A geometric mechanism for diffusion in Hamiltonian systems overcoming in the large gap problem: Heuristics and rigorous verification on a model

| issue=844

| hdl=2117/872| hdl-access=free}}{{cite journal | last1=Gelfreich | first1=Vassili | last2=Turaev | first2=Dmitry | title=Arnold Diffusion in A Priori Chaotic Symplectic Maps | journal=Communications in Mathematical Physics | publisher=Springer Science and Business Media LLC | volume=353 | issue=2 | date=2017-04-24 | issn=0010-3616 | doi=10.1007/s00220-017-2867-0 | pages=507–547| bibcode=2017CMaPh.353..507G | hdl=10044/1/44044 | s2cid=253744630 | url=http://wrap.warwick.ac.uk/86190/7/WRAP-Arnold-diffusion-priory-chaotic-maps-Gelfreich-2018.pdf | hdl-access=free }}{{cite journal | last1=Gidea | first1=Marian | last2=Llave | first2=Rafael | last3=M-Seara | first3=Tere | title=A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results | journal=Communications on Pure and Applied Mathematics | publisher=Wiley | volume=73 | issue=1 | date=2019-07-24 | issn=0010-3640 | doi=10.1002/cpa.21856 | pages=150–209| hdl=2117/188944 | s2cid=119150120 | hdl-access=free }} and in the a priori stable case.{{cite journal | last=Cheng | first=Chong-Qing | title=The genericity of Arnold diffusion in nearly integrable Hamiltonian systems | journal=Asian Journal of Mathematics | publisher=International Press of Boston | volume=23 | issue=3 | year=2019 | issn=1093-6106 | doi=10.4310/ajm.2019.v23.n3.a3 | pages=401–438| arxiv=1801.02921 | s2cid=119677951 }}{{cite book | last1=Kaloshin | first1=Vadim | last2=Zhang | first2=Ke | title=Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom | publisher=Princeton University Press | date=2020-11-12 | isbn=978-0-691-20493-2 | doi=10.1515/9780691204932| url=http://digitale-objekte.hbz-nrw.de/storage2/2021/01/12/file_11/8980974.pdf }}

In the context of the restricted three-body problem, Arnold diffusion can be interpreted in the sense that, for all sufficiently small, non-zero values of the eccentricity of the elliptic orbits of the massive bodies, there are solutions along which the energy of the negligible mass changes by a quantity that is independent of eccentricity.{{cite journal | last=Xia | first=Zhihong | title=Arnold diffusion in the elliptic restricted three-body problem | journal=Journal of Dynamics and Differential Equations | publisher=Springer Science and Business Media LLC | volume=5 | issue=2 | year=1993 | issn=1040-7294 | doi=10.1007/bf01053161 | pages=219–240| bibcode=1993JDDE....5..219X | s2cid=121370238 }}{{cite journal | last1=Delshams | first1=Amadeu | last2=Kaloshin | first2=Vadim | last3=de la Rosa | first3=Abraham | last4=Seara | first4=Tere M. | title=Global Instability in the Restricted Planar Elliptic Three Body Problem | journal=Communications in Mathematical Physics | publisher=Springer Science and Business Media LLC | volume=366 | issue=3 | date=2018-09-05 | issn=0010-3616 | doi=10.1007/s00220-018-3248-z | pages=1173–1228| hdl=2117/123338 | s2cid=253752812 | hdl-access=free }}{{cite journal | last1=Capiński | first1=Maciej | last2=Gidea | first2=Marian | journal=Communications on Pure and Applied Mathematics| doi=10.1002/cpa.22014|title=A general mechanism of instability in Hamiltonian systems: skipping along a normally hyperbolic invariant manifold| year=2021| doi-access=free}}

• KAM theorem
• Nekhoroshev estimates

{{Reflist}}

{{DEFAULTSORT:Arnold Diffusion}}

Category:Dynamical systems

Category:Classical mechanics