For example, in order to be sure that the solutions exist he uses the fact that a zero of the derivative at a maximum point of a holomorphic function is necessarily of odd order, relying for that on a theorem proved at the beginning of his thesis which is exactly the Weirerstrass preparation theorem. A un certain In the same way, ergodic KAM tori of dimension 3 in the reduced phase space of the planar Three-Body Problem the three frequencies being the two mean motions and the frequency of the difference of perihelia give rise in the non reduced phase space to 4-dimensional invariant tori which could be resonant.
In other terms, the periodic solutions disappear by couples as do the real roots of algebraic equations. Figure The birth of a pair of subharmonics of period 3T. Moreover, he proves that We have still the periodic solutions of the second kind; if we vary continuously one of the parameters of which depends the problem, for example one of the masses, we see a periodic solution of the first kind deform in a continuous way, the period remaining equal to T. At some point, this solution splits into two, or rather three solutions, I mean that at some point we have three periodic solutions which differ only slightly; one of them still has period T , the period of the other two is a multiple of T.
Those are the periodic solutions of the second kind Analysis of the works. After a chapter XXIX, see He analyses the way the subharmonics wrap around the T -periodic solution and identifies subharmonics with extrema of the action. The rotation number is not far. Mais alors que sont devenus les satellites A stables? Dar- win. Darwin appelle un satellite oscillant.
In a both lucid and moving introduction, he explains that he struggled during two years but could not reach a complete proof. Nevertheless, having proved the theorem in all the special cases he had examined and due to its paramount importance, he decided to submit the problem to geometers. Comparing this question with the one completely understood by J. The application to the Restricted Three-Body Problem with a large Jacobi constant, comes from the fact that, the return map in the Birkhoff annulus of section is a monotone twist map, which allows application of the theorem to its iterates in order to get periodic orbits.
But in fact, the full force of the theorem is not needed in this case or in the companion case of an area preserving diffeomorphism of the plane in the neighborhood of a generic elliptic fixed point. The main features which are understood are represented on Figure 21 see [Ze, C9, Le] : happened with the stable satellites A?
About this, I can only make hypotheses and, in order to get something else one shoud continue the mechanical quadratures of M. But, if one examines the shape of the curves, it seems that at some moment the orbit of the satellite must have passed near Jupiter and that after that he became what M. Darwin calls an oscillating satellite. Hence I had to content myself with some partial results, mainly concerning closed geodesics, which play here the role of periodic solutions of the Three-Body Problem. Generically, the union of these invariant curves is a set whose transversal struture is Cantor-like.
The annuli in between two invari- ant curves and not containing in their interior any invariant curve homotopic to the boundary are called Birkhoff zones of instability. The dynamics in such a zone is quite complicated and certainly not completely understood. These sets, whose existence was initially obtained by S.
Aubry and J. Mather, in each case by minimizing the action functional, were shown by A. Katok to be limits of periodic orbits, thanks to the Birkhoff a priori Lipschitz estimates resulting from the monotone twist condition; which,in the case of a return map, are the trace of the KAM tori on a surface of section. But we still do not know whether periodic points are dense or if hyperbolic behaviour that is non-zero Lyapunov exponents exist on a set of positive Lebesgue measure.
Describing this would lead us too far astray, hence I refer to [Au2] for this part of the story and the references.
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Particularly interesting for mission design are the Halo orbits in the spatial Re- stricted Problem, which bifurcate from a planar Lyapunov family originating from a collinear relative equilibrium. Other much studied special cases are the collinear problem with the remarkable periodic regularized solution discovered by Schubart see an animation in [C4] and the isosceles problem, where one body moves on a line, while the two others, with the same mass, move symmetrically on the orthogonal line resp.
Today, the power of computers allows us to get a pretty good understanding of au- tonomous 2 degrees of freedom systems like the Restricted Problem or its simplified version called the Hill problem. Finally, results of C. See for instance [Si, SST]. But what does it do, how does it find out? Does it smell the nearby paths, and check them against each other? The answer is, yes, it does in a way. Then, developing the note [P10] which was based on a simple geometric reason- ing, he proves that a periodic solution which locally minimizes the action must be dynamically unstable: The very statement of the principle of least action has something shocking to the mind.
To go from one point to another one, a material molecule, taken away from the action of any force, but constrained to move on a surface, will follow the geodesic line, i. It seems that this molecule knows the point where one wants it to go, that it anticipates the time needed to reach it along such or such path, and then chooses the most convenient path.
In a sense, the statement presents this molecule as a free animated being. It is clear that it would be better to replace it by a less shocking statement where, as philosophers would say, the final causes would not appear to replace the efficient ones.
For example, on a torus of revolution around the z axis, the intersections of the torus with the horizontal plane of symmetry are respectively unstable for the short one, along which the curvature is negative, and stable for the long one, along which the curvature is positive: along the first one there are no conjugate points while they exist on the second one. Such assertions hold only for mechanical systems with two degrees of freedom. In- deed, examples studied by Marie-Claude Arnaud in [Ar] show that in higher dimen- sions, a locally action minimizing periodic solution may possess only two directions of instability transverse to the flow in its energy level.
The note of which was quoted in section The methods I used to prove their existence are very simple and can be reduced to the calculus of limits. But one can arrive to this proof by a completely different path, which it will often be useful to follow, but which I did not fully exploit. Suppose, for instance, that one is looking for the geodesics of an indefinite surface whose general shape is the one of a one-sheeted hyperboloid. One can be sure that there exists a closed geodesic corresponding to a periodic solution because, among all the closed curves going around the surface which one can draw, one must be shorter than the others.
Given a relative loop of configurations of the three bodies in the plane i. In fact this was fixing the homology class of the loop of relative configurations. Choosing k and l different from zero implies coercivity i. I have only sketched this method from which there is still probably much to get. Indeed, constraining the homology of the loop of configurations be it relative or absolute most of the time leads to minimizers with collisions there are exceptions found by A.
Venturelli, K. But, if one replaces the homology constraint by a symmetry constraint, the method becomes remarkably efficient, leading in particular to new classes of periodic or quasi-periodic solutions, the choreographies and the Hip-Hops see the references in [C2, Mont1, Mont2],. In The New Methods, there is only one chapter, which indeed displays a quite remarkable intuition, where he copes with solutions approaching collisions, in fact only two-body collisions.
The idea is simple and beautiful: in the limit of zero masses, the planets follow Keplerian orbits till they have a close encounter i. If one replaces almost double collision by almost triple collision, the situation be- comes tremendously richer. The way to analyse it is to compactify by adding the so-called McGehee collision manifold see [McG, C7]. A description of some of the complexity it implies in the phase portrait of the planar Three-Body Problem can be found in the beautiful survey paper by Rick Moeckel [Moe].
As is well-known today, the situation is much more complex in the non restricted case where, the number of degrees of freedom being strictly greater than 2, Lagrangian invariant tori do not separate an energy manifold. This leaves room for a very slow diffusion of the trajectories not belonging to the invariant tori: remaining a long time around the KAM tori which are quite sticky, they could a priori escape very far. The first highly non generic example of this phenomenon was given by Arnold in [A4].
Lower bounds on diffusion time for close to integrable Hamiltonian systems were first given by Nekhoroshev [Ne]. An example of application to the Three-Body Problem is [Nie]. For the long-term evolution of the full solar system, see [Las1].
Following G. Kolmogorov [K] , we ask: Question. In particular, this would imply that the bounded orbits are nowhere dense and no topological stability occurs. Indeed, as he told me in front of a chinese tea, for realistic masses of the planets, instability seems likely, even in the sense of measure! Lagrange first proved it, then Poisson, other proofs followed, other are still to come.
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Were the ancient proofs no sufficient, or is it that the new ones are superfluous? The astonishment of these persons would certainly increase, if they were told that it may happen that some day a mathematician will show, by a rigourous reasoning, that the planetary system is unstable.
Nevertheless, this might well happen; and yet there will be no contradiction, and the ancient proofs will keep their value. The items [S1, S2, S3] were published at that time. We all considered him as the strongest of us all.