KAM theorem, and its application to the stability of the Solar system. Why Phi?

Posted: May 10, 2017 by tallbloke in Astrophysics, solar system dynamics

I’m not expecting much discussion of this post, I don’t understand it either, though I have run across KAM before in another context, and eventually some light may dawn that illuminates some relationship with our phi-solar system dynamics work. I’ll just leave it here for now so it doesn’t get lost. One random synchrony is that Gabriella Pinzari is at the same university as Nicola Scafetta. Maybe we can get him interested enough to talk to her about our theory.

The following message is a guest post by Boris Khesin. Boris summarizes the wonderful talk given by Gabriella Pinzari at the workshop.  –Jim Colliander

Ecliptic_plane_3d_viewGabriella Pinzari (30min talk) described her joint result with her advisor Luigi Chierchia on a recently found fix for the famous KAM theorem, or rather for its application to the stability of the Solar system.

Namely, the original KAM theorem in Arnold’s 1963 paper claimed the persistence of the Liouville tori for perturbations of integrable systems under some nondegeneracy assumption – some determinant must be nonzero. This was a perfectly correct statement proved in the paper. But Arnold applied it to the Solar system without properly checking that for that system the determinant is indeed nonzero. (More precisely, Arnold checked the non-degeneracy condition for the first nontrivial case, the planar three-body problem, and claimed that this could extend to the general case: spatial, arbitrary n.) However, it turned out to be identically zero in the spatial case.

So later M.Herman developed a theory for how to deal with such a degeneracy in the KAM theory. Essentially it shows how to use a slighter nondegeneracy “in the next term”, which worked for the application to the Solar system. (It was published by J.Fejoz.)

But now Gabriella Pinzari explained that the source of the degeneracy in the application of the KAM theorem to the Solar system was the necessity to mod out rotations!! Once one mods out rotational symmetry in the Laplace plane, the system becomes nondegenerate (i.e. Arnold’s determinant is nondegenerate) on the quotient! This is somewhat similar to the relation of Morse and Morse-Bott functions, as far as I understand.

Quoting Pinzari, “Arnold gave only some ideas on how to construct (by series) such a reduction, but did not develop these ideas. What we have done was in essence to construct explicitly such a reduction. The miracle is that you can do it without singularities in the transformation (in a sense this is needed to control the convergence of the series that Arnold had in mind).”

[Note that since the determinant is identically zero, no higher order terms are nonzero. So M.Herman introduced a modification of the Hamiltonian which broke the rotation invariance of the modified system and computed a modified torsion with nondegeneracy in higher orders and this worked for the application to the Solar system. On the contrary, keeping the rotation invariance allows one to stay in the nondegenerate setting on the quotient.]

I am really shocked that this unavoidable degeneracy had such a simple explanation, which came unnoticed for 40 years. So all one needed was to develop a rotation-invariant version of the KAM theory. (As far I as understand, an equivariant KAM still does not exist beyond this rotation case.) And Arnold, who developed both KAM and group actions was in the best position to marry these two domains, but somehow it did not happen then!  –Boris Khesin

Here are some references from G.Pinzari:

  1. tallbloke says:



    Non-degenerate Liouville tori are KAM stable

    Abed Bounemoura (CEREMADE)
    (Submitted on 1 Dec 2014)
    In this short note, we prove that a quasi-periodic torus, with a non-resonant frequency (that can be Diophantine or Liouville) and which is invariant by a sufficiently regular Hamiltonian flow, is KAM stable provided it is Kolmogorov non-degenerate. When the Hamiltonian is smooth (respectively Gevrey-smooth, respectively real-analytic), the in-variant tori are smooth (respectively Gevrey-smooth, respectively real-analytic). This answers a question raised in a recent work by Eliasson, Fayad and Krikorian ([EFK]). We also take the opportunity to ask other questions concerning the stability of non-resonant invariant quasi-periodic tori in (analytic or smooth) Hamiltonian systems.

  2. oldbrew says:

    An extract from ‘KAM Theory and Celestial Mechanics’:

    Here are three applications of KAM theory in Celestial Mechanics which yield realistic estimates. The extension to more significant models is often limited by the computer capabilities.

    — A three-body problem for the Sun, Jupiter and the asteroid Victoria was investigated in [CC]. Careful analytical estimates were combined with a Fortran code implementing long computations using interval arithmetic. The results show that in such a model the motion of the asteroid Victoria is stable for the realistic Jupiter-Sun mass-ratio.

    — In the framework of planetary problems, the Sun-Jupiter-Saturn system was studied in [LG]. A bound was obtained on the secular motion of the planets for the observed values of the parameters. (The proof is based on the algebraic manipulation of series, analytic estimates and interval arithmetic.)

    — A third application concerns the rotational motion of the Moon in the so-called spin-orbit resonance, which is responsible for the well-known fact that the Moon always points the same face to the Earth. Here, a computer-assisted KAM proof yielded the stability of the Moon in the actual state for the true values of the parameters [C].

    Although it is clear that these models provide an (often crude) approximation of reality, they were analyzed through a rigorous method to establish the stability of objects in the solar system. The incredible effort by Kolmogorov, Arnold and Moser is starting to yield new results for concrete applications. Faster computational tools, combined with refined KAM estimates, will probably enable us to obtain good results also for more realistic models. Proving a theorem for the stability of the Earth or the motion of the Moon will definitely let us sleep more soundly!

    – – –

    From an article: Strange Stars Pulse to the Golden Mean

    “All this begs the question,” Ditto said, “what is fundamentally going on with these stars that they end up with a ratio near the golden mean?”

    The new paper offers one hypothesis. The KAM theorem, named for Andrey Kolmogorov, Vladimir Arnold and Jürgen Moser, holds that systems driven by frequencies in an irrational ratio tend to be the most stable; that is, they can’t easily be knocked off-kilter into a new state of motion. In that case, it might be the fate of unstable stars to evolve until they arrive at a number like the golden mean. “It’s the most robust number to perturbations, which means these stars may select it out,” Ditto explained. [bold added]


    This was discussed in Talkshop post: ‘Why Phi? – Fractals seen in throbs of pulsating golden stars’

    KAM theory gets a mention in the comments 😎
    William Ditto (one of the authors of the paper) also commented.

  3. Konrad says:

    “A Klemperer rosette is a gravitational system of heavier and lighter bodies orbiting in a regular repeating pattern around a common barycentre. It was first described by W. B. Klemperer in 1962”

    Note: it is not required that there should be a major gravity source at the barycentre, but it adds to stability. Ie: there could be multiple Earths at 1 AU orbiting the point at which our sun currently is, in the absence of the sun, if the speed of their “orbit” were slower. All that is needed to hold them in a circle is their gravitational attraction to each other.

    When the conjecture of “dark matter” was contrived, “particle animation” was but a gleam in the computer animators eye. The computational power was beyond us, but only the ultra-fast iterative calculations of computers could solve the problem. There is no “dark matter”. Galaxy formation is not just governed by star’s attraction to a centre of mass, but by star’s attraction to nearby stars.

    So too the motion of planets and asteroids in our solar system. Their orbits are not just governed by their speed and gravity of the sun, but by mass and their influence on each other. Observe the oscillation of the sun. From mild chaotic variance around the barycentre to a more stable tri-fold pattern nearing one solar diameter perturbation. This perturbation exceeds what could be expected of direct planetary influence, indicating a dance started long ago.

  4. Paul Vaughan says:

    “I am really shocked that this unavoidable degeneracy had such a simple explanation, which came unnoticed for 40 years.”

    The conventional mainstream gets exhausted and disoriented from self-indulgent white water rafting on the excessively overextended shallow floods of algebra it concocts to cosmetically define itself, while elsewhere in unexplored natural territory, undiscovered still waters may be simply deep.

  5. Paul Vaughan says:

    “[…] strangely enough, Deprit’s reduction did not have a similar mathematical success. This fact might be partly due to the pessimistic attitude of Deprit himself, who […] writes: […] At least for planetary theories, the answer is likely to be negative […]”.”

    Same obstacle in climate discussion culture — ideological coercion to believe deeper, simpler understanding’s strictly impossible (as if by devil’s decree). There’s a simple solution to that part of the major western fault.

    Elsewhere: After this https://tallbloke.wordpress.com/2017/04/29/ian-wilson-help-needed-to-solve-an-interesting-lunar-puzzle/#comment-125590 I intro’d something pretty simple that ties back to something discussed at the talkshop a long time ago and all of this goes right back to what I noticed during my first month of climate exploration in November 2007. Let’s give Semi a minute to orient…

    I think there were 2 or 3 threads where I featured that 6.4 year graph. In 1 of them I gave some derivations I want to point to to tie in what I intro’d on the Ian Wilson Lunar Puzzle thread. Does anyone remember which thread that was? (I’ll dig for it sometime this week as/when time permits, but if someone points to it we may be able to expedite tardy clue-in.)

  6. oldbrew says:

    PV: Numerous 6.4 refs here…

    Includes this:

  7. tallbloke says:

    Paul V says in the Ian Wilson Lunar puzzle thread he links:
    “Antiresonance is the attractor.
    (The source of perturbations away from antiresonance could be another line of exploration for anyone who choses to pursue it.)”

    This is what I’ve been saying for ages now. The planetary timings are attracted towards the minimum resonance condition (Phi related ratios), but these also lie close to resonant, Fibonacci pair ratios (8:13 EV for example). This is, I think, part of the natural efficiency of the solar system. Energy NEEDS to be passed between planets via resonance in order to reshape their orbits to produce the longer period ‘cog slipping’ which keeps the whole system stable. So the perturbations AWAY from “antiresonance” are in fact an integral part of the long term attraction TOWARDS “antiresonance”.

    Instead of looking at short term perturbations and wondering what their source is, we need to better understand the long term antiresonance attractor and its underlying drivers. We can then see that the shorter term perturbations necessarily arise.

    This is the insight I got when I wrote my main PRP paper when I discovered the simple near-whole-number ratios between the planet-pair conjunction cycle precession periods:

    EV 1199yrs is in a near 2:1 with JS ~2400yr which is in a 3:2 with UN ~3600yr

    Those longer term precessions give rise to the shorter term perturbations which produce the ‘cog slippages’ which in turn produce the long term precessions which keep the system stable.

    In short, the feedbacks are holistically integrated. Why Phi? Because Phi. But we still need to know HOW Phi?

    The observable answer may lie in the microcosmic as well as in the macrocosmic.

    The reason I posted this piece is that I have a hunch (and it’s no more than that at present, but I have a good record on hunches) that those orbital resonance insights might just be able to inform and underpin the modified KAM theory:

    Gabriella Pinzari explained that the source of the degeneracy in the application of the KAM theorem to the Solar system was the necessity to mod out rotations!! Once one mods out rotational symmetry in the Laplace plane, the system becomes nondegenerate

    Could this be analogous to the resultant of the resonantly linked planet pair conjunction cycle precession periods? i.e. could these resonantly linked long-term precessions ‘mod out’ the perturbatory forces arising from the quasi-chaotic orbital/resonant interactions evident in short term solar system motion?

  8. oldbrew says:

    Then there’s the 3:2 length of day ratio between Venus and Mercury. How/why does that work?


  9. oldmanK says:

    The above is all ‘beyond my ken’. But tallbloke’s part comment, quote: “The planetary timings are attracted towards the minimum resonance condition” rings a bell – somewhere?

    I recall from a career tending finicky animals -aka turbogenerators-, that anything which hinted at resonance, at the end of a ‘fatiguing’ life, being short or long depending its level of resonance, tended to “degenerate”, ultimately to destruction. We always aimed for a lifetime which generously exceeded the lifetime of the plant, but that was not always possible. Just the odd quarter century, not millions of years.

    But -I ask- is degenerate same as destruct in the above? If yes, then the ‘minimum resonance condition’ is conditional for survival.

  10. tallbloke says:

    OB: The speculation I tend towards is that Mercury is the escaped Moon of Venus. That was Tom Van Flandern’s idea.

    OldmanK: Yes, I think ‘degenerate’ means ‘become unstable to the point of flying apart’. The Solar System is a whirling integrated system of out of balance forces generating feedbacks, which cause the harmonically linked resonances to reorientate the constituent elements in such a way as to maintain overall stability despite varying internal imbalances. The application of KAM theory to the problem of HOW that can happen is one of many attempts to explain this stability, because standard perturbation theory cannot do that. One of the reasons it can’t is that perturbation theory is incomplete heuristics.

  11. Paul Vaughan says:

    TB suggested: “Instead of looking at short term perturbations and wondering what their source is, we need to better understand the long term antiresonance attractor and its underlying drivers.”

    Both are interesting. Exploratory niches are available.

    The spatiotemporal attractors — and spatiotemporal aggregation criteria more generally — have always attracted my attention.

    Although I’m open-minded about taking a multitude of exploratory perspectives, I don’t normally look at attractors as being “driven”. They’re basins of opportunity. They’re places where things can stably exist if things can get to them.

    There are strong attractors and weak attractors. If the attractor is very strong, oldschool Fourier methods will be enough for exploration, but as we well know from too many years of observation, we’re in for a good snicker and laugh when narrow-minds try to explore weak attractors with tools that are too limited by underlying contextually-false theoretical assumptions.

    When the code for the general solution (cookbook recipe for assessing stability from initial conditions) comes out it will frighten people that something so simple was unknown when we as a civilization felt so smart and advanced.

    TB wrote: “Why Phi? Because Phi. But we still need to know HOW Phi? “

    There’s a general potential frame that’s universally determined by geometry. There’s mystery of neither “why” nor “how” at that level of generalization. It’s just geometry.

    …but I suspect you’re more focused in your speculation on sets of stable matrix convergence possibilities as a function of initial conditions (including physical parameters) in specific cases.

    That’s a pathway problem. Existence of geometric basins alone is not enough to ensure accessibility. There has to be a (physically accessible) path.

    It’s a tractable problem. It’s curious that so many intelligent mainstream explorers blindly run integrations without stepping back to summarize from third eye perspective.

    Downhill is why. By accessible path from initial conditions is how.

    This generalizes beyond the special case of physically limited systems. Those narrowing their focus to physically limited systems will be coding rules for determining physically accessible pathways to stability, but I would advise them to start from the perspective of more general awareness.

  12. Paul Vaughan says:

    Here’s another of the 6.4 year links…
    …but that’s not the page with the derivations I had in mind.
    I’m sidetracked by other pursuits. I’ll resume when possible.

  13. oldbrew says:

    The eccentricity of asteroid 2015BZ509 is 0.617²

    From: A retrograde co-orbital asteroid of Jupiter
    Paul Wiegert, Martin Connors & Christian Veillet

    Wrong-way asteroid plays ‘chicken’ with Jupiter

    It survives the chicken game.

  14. tallbloke says:

    Paul V: Downhill is why. By accessible path from initial conditions is how.
    I suspect you’re more focused in your speculation on sets of stable matrix convergence possibilities as a function of initial conditions (including physical parameters) in specific cases.

    Achieving cosmological generality is satisfying for mathematicians

    Uncovering mechanism specifics is satisfying for engineers. 🙂

    The ‘How Phi’; question is aimed at an elucidation of the interacting forces which create the hills and valleys, and push particles over them and into them.

    A stable solar system is a negentropic entity. There must be an underlying principle which organises the flow of energy through it.

  15. Paul Vaughan says:

    The conventional mainstream has had decades of physically precise numerical integration and they have failed to progress from particular intuition to general awareness.

    It will be orders of magnitude more efficient to begin with general awareness (that applies in any domain whether physical, population-genetic, or whatever) and then easily progress to particular insight.

    Someone missing the bigger picture will be just completely and utterly lost …but it’s just a simple assignment for a technician to move from general awareness to particular solution. They’re not going to understand why this wasn’t known before once they see the dead simple matrix calculation.

  16. Paul Vaughan says:

    Does everyone remember what Cuk said about climate modeling in his seminar video on JEV coupling at eccentricity timescale?

  17. tallbloke says:

    Paul V: The conventional mainstream has had decades of physically precise numerical integration and they have failed to progress from particular intuition to general awareness.

    The problematic word here is ‘physically’. They may have a reasonably precise integration but I don’t believe it based on straight PHYSICS. So far as I know it is based on orbit theory modulated by HEURISTIC equations derived from OBSERVATIONS.

    Happy to be corrected if I’m wrong about that.

    If I’m right, then current perturbation theory is not going to tell us HOW PHI?

    If I’m right, then the big thing to say about the big picture is that we’re still in the dark while we’re trying to look at it.

    The reason I made this post is that I have a hunch (I won’t go so far as to call it an intuition) that KAM theory might, just might, be a better approach that might, possibly, offer a potential bridge between mathematical reasoning and the actual physical mechanism that maintains the solar system’s stability. Currently, the keystone of that bridge hasn’t yet been carved, and we might, just might, be able to correctly shape it using our phi concepts.

  18. tallbloke says:

    Paul V: Does everyone remember what Cuk said about climate modeling in his seminar video on JEV coupling at eccentricity timescale?

    Please, pretty please, save us all some time and just link the video and tell us. 🙂

  19. oldbrew says:

    NASA: Deciphering the Mysterious Math of the Solar Wind

    A constant stream of particles and electromagnetic waves streams from the sun toward Earth
    …Understanding the sun and how the material and energy it sends out affects the solar system is crucial
    …Of course, having an equation doesn’t yet tell us the reason why the waves in the solar wind are shaped in this way.


  20. tallbloke says:

    I found a video of the presentation by Pinzari and Chierchia at the Congress of Mathematicians in Seoul 2014

    I think this is similar to the talk summarized in the headline post. Most of the slides in Gabriela’s talk appear in this publication on her researchgate account

  21. tallbloke says:

    Intro to KAM

    Click to access introkam.pdf

    The problem which the KAM theory was developed to solve first arose in
    celestial mechanics. More than 300 years ago, Newton wrote down the differential
    equations satisfied by a system of massive bodies interacting through
    gravitational forces. If there are only two bodies, these equations can be explicitly
    solved and one finds that the bodies revolve on Keplerian ellipses about
    their center of mass. If one considers a third body (the “three-body-problem”),
    no exact solution exists – even if, as in the solar system, two of the bodies are
    much lighter then the third. In this case, however, one observes that the mutual
    gravitational force between these two “planets” is much weaker than that between
    either planet and the sun. Under these circumstances one can try to solve
    the problem perturbatively, first ignoring the interactions between the planets.
    This gives an integrable system, or one which can be solved explicitly, with
    each planet revolving around the sun oblivious of the other’s existence. One
    can then try to systematically include the interaction between the planets in
    a perturbative fashion. Physicists and astronomers used this method extensively
    throughout the nineteenth century, developing series expansions for the
    solutions of these equations in the small parameter represented by the ratio of
    the mass of the planet to the mass of the sun. However, the convergence of
    these series was never established – not even when the King of Sweden offered
    a very substantial prize to anyone who succeeded in doing so. The difficulty in
    establishing the convergence of these series comes from the fact that the terms
    in the series have small denominators which we shall consider in some detail
    later in these lectures. One can obtain some physical insight into the origin of
    these convergence problems in the following way. As one learns in an elementary
    course in differential equations, a harmonic oscillator has a certain natural
    frequency at which it oscillates. If one subjects such an oscillator to an external
    force of the same frequency as the natural frequency of the oscillator, one has
    resonance effects and the motion of the oscillator becomes unbounded. Indeed,
    if one has a typical nonlinear oscillator, then whenever the perturbing force has
    a frequency that is a rational multiple of the natural frequency of the oscillator,
    one will have resonances, because the nonlinearity will generate oscillations of
    all multiples of the basic driving frequency.
    In a similar way, one planet exerts a periodic force on the motion of a second,
    and if the orbital periods of the two are commensurate, this can lead to resonance
    and instability. Even if the two periods are not exactly commensurate, but only
    approximately so the effects lead to convergence problems in the perturbation
    It was not until 1954 that A. N. Kolmogorov [8] in an address to the ICM
    in Amsterdam suggested a way in which these problems could be overcome.
    His suggestions contained two ideas which are central to all applications of the
    KAM techniques. These two basic ideas are:
    • Linearize the problem about an approximate solution and solve the linearized
    problem – it is at this point that one must deal with the small
    • Inductively improve the approximate solution by using the solution of the
    linearized problem as the basis of a Newton’s method argument.
    These ideas were then fleshed out, extended, and applied in numerous other
    contexts by V. Arnold and J. Moser, ([1], [9]) over the next ten years or so,
    leading to what we now know as the KAM theory

  22. oldbrew says:

    ‘More than 300 years ago, Newton wrote down the differential
    equations satisfied by a system of massive bodies interacting through
    gravitational forces.’

    But what do these ‘gravitational forces’ consist of?
    – – –
    “Understanding the role that magnetic fields play in the evolution of galaxies and their environment is a fundamental question in astronomy that remains to be answered.”

    According to Prof. Bryan Gaensler, Director of the Dunlap Institute for Astronomy & Astrophysics, University of Toronto, and a co-author on the paper, “Not only are entire galaxies magnetic, but the faint delicate threads joining galaxies are magnetic, too. Everywhere we look in the sky, we find magnetism.”

    Read more at: http://phys.org/news/2017-05-magnetic-bridge-nearest-galactic-neighbours.html