Poincare H.: Les Methodes Nouvelles de la Mechanique Celeste, Gauthier Villars, Paris (1892)  Home/Search   Document Not in Database   Summary   Related Articles   Check

....k 1 (mod p) 0 ; where k is the number of the hyperbolic point, according to the Tables 1a,b,c; p is their period. 4. Construction of separatrices of the Poincar e map Let z k , 1 6 k 6 p be hyperbolic points of period p, belonging to a periodic hyperbolic trajectory fl hyp . As it is shown in [10], fl hyp has two invariant curves: the stable W s and the unstable W u separatrices, defined as: W u (fl hyp ) fz 2 S : F n (z) fl hyp as n 1g ; W s (fl hyp ) fz 2 S : F n (z) fl hyp as n Gamma1g ; REGULAR AND CHAOTIC DYNAMICS V. 4, # 1, 1999 109 Gamma A. V. IVANOV It ....

H. Poincar'e. Les Methodes Nouvelles de la Mechanique Celeste. V. 1--3. Gauthier--Villars. Paris. 1892; (Dover, New York, 1957). Reprint.

....some kind of chaos. Apparent regions filled by chaotic trajectories are called stochastic layers. To the best of our knowledge, it was H enon who discovered such regions (see [HH] and also [H] for historical comments) Of course, Poincar e knew the extremely intricated character of the motion [Po]. To illustrate the above description of the coexistence of different kinds of motion, we present here a picture of the phase portrait of the map (1.1) x y 7 0:2 cos 2x Gamma y x defined on the 2 torus T 2 = R 2 =Z 2 : Only one quarter of the torus, namely the domain jxj ....

H. POINCAR ' E, Les M'ethodes Nouvelles de la M'echanique C'eleste,Vols 1--3, Gauthier-- Villars, Paris, 1892, reprinted by Dover, New York (1957).

....in unpublished paper [CS] Nonnullity of implies transversality of the intersection of the separatrices at the point ( y Gamma ) The existence of such a point generates a very complicated picture of intersections of the separatrices W s and W u ; which was described by H. Poincar e [13]. The separatrix W u is an injective immersion of R into the phase space (R=2Z) Theta R: The image of this injection is a noncompact curve passing through the point (0; 0) oscillating with the amplitude growing as the distance from (0; 0) grows, and winding onto itself. The stable separatrix ....

]H. POINCAR ' E, H. , Les M'ethodes Nouvelles de la M'echanique C'eleste,Vols 1--3, Gauthier-- Villars, Paris, 1892, reprinted by Dover, New York (1957).

....= z k 1 (mod p) 0 ; where k is the number of the hyperbolic point, according to the tables 1a,b,c, p is their period. 4 Construction of separatrices of the Poincar e map Let z k , 1 k p be hyperbolic points of period p, belonging to a periodic hyperbolic trajectory fl hyp . As it is shown in [10], fl hyp has two invariant curves: the stable W s and the unstable W u separatrices, defined as: W u (fl hyp ) fz 2 S : F n (z) fl hyp as n 1g; W s (fl hyp ) fz 2 S : F n (z) fl hyp as n Gamma1g; It is not difficult to prove that there are injective maps Phi u;s k : R ....

H.Poincar'e, Les Methodes Nouvelles de la Mechanique Celeste, vols. 1-3, GauthierVillars, Paris, 1892; (Dover, New York, 1957), reprint.

....we could not find any proof of it, nor a proof of even the fact that they are smooth coordinates. Poincar e proved indeed only that the Hamiltonian of the three body problem, written as a function of the Poincar e elements, is an analytic function of x; y in a disk centered in the origin ([8], chapter 1) However, by itself, this fact does not imply that the Poincar e elements are analytic (or just smooth) coordinates in a neighborhood of the circular orbit. This fact does not appear to be completely obvious because of the very definition of the angle , which is the sum of two ....

H. Poincar'e, Les m'ethodes nouvelles de la m'echanique c'eleste, Vol. 1 (Gauthier--Villars, Paris, 1892).

....systems, regions of high vorticity in fully developed turbulence and fractal growth processes. This was already fully appreciated by Poincar e, who, describing his discovery of homoclinic tangles, mused that the complexity of this figure will be striking, and I shall not even try to draw it [1] . Today such drawings are cheap and plentiful; but Poincar e went a step further and noting that hidden in this apparent chaos is a rigid skeleton, a tree of cycles (periodic orbits) of increasing lengths and self similar structure suggested that the cycles should be the key to chaotic dynamics: ....

H. Poincar'e, Les m'ethodes nouvelles de la m'echanique c'eleste (Guthier-Villars, Paris 1892-99)

....equation and can be used to label the curves. For families of maps that contain an integrable i.e. explicitly solvable in closed form system, one can study the functional equation for invariant curves perturbatively and one is led to the Lindstedt expansions of classical mechanics (see [Po] and section 3) Even if these expansions have been in use for over a century, their analytic properties have been very hard to study. For example, due to the presence of small divisors for diophantine frequencies (see section 3) the fact that they have a positive radius of convergence was ....

H. Poincar'e: "Les methodes nouvelles de la m'echanique c'eleste", Gauthier Villars, Paris (1891--1899).

....structure which is not at all apparent in the integrable approximations. However, hidden in this apparent chaos is a rigid skeleton, a tree of cycles (periodic orbits) of increasing lengths and self similar structure. The important role played by periodic orbits was already noted by H. Poincar e [1] , and has been at the core of much of the mathematical work on the theory of the dynamical systems [2] ever since. The insight of the modern dynamical systems theory [3] is that the zeroth order approximations to the harshly chaotic dynamics should be very different from those for the nearly ....

H. Poincar'e, Les m'ethodes nouvelles de la m'echanique c'eleste (Guthier-Villars, Paris 1892-99).

....This approach is basically a non perturbative one, and it proved successful in explaining many qualitative features of the transition. On the other hand, it is still unclear how this picture affects the perturbative methods traditionally used since long time (like e.g. the Lindstedt series [3] [4]) In particular, many questions pertaining to the analytic structure of the invariant tori find their natural setting in the language and formalism of series expansions and perturbation theory. We observe that a similar duality occurs in quantum field theory, where two approaches to ....

Poincar'e. H., Les m'ethodes nouvelles de la m'echanique c'eleste, tome II, Paris, Gauthier--Villars (1893)

....theory for trace formulas for weakly stochastic chaotic dynamics in the standard field theoretic language of Feynman diagrams. Here we approach the same problem from an althogether different direction; the key idea of flattening the neighborhood of a saddlepoint can be traced back to Poincar e [2], and is perhaps not something that a field theorist would instinctively hark to as a method of computing perturbative corrections. In the Feynman diagram approach [1] we observed that the sums of diagrams simplify for saddlepoints corresponding to repeats of shorter periodic orbits, and were ....

H. Poincar'e. Les m'ethodes nouvelles de la m'echanique c'eleste. Guthier-Villars, Paris, 1892.

....than on the sky falling on our heads ) Nevertheless the stability problem for many body systems interacting only through gravitation still stems out as one of the more intriguing and rich problems in mathematics. In modern times outstanding contributions came, above all, from H. Poincar e [13], V. I. Arnold [2] and J. Moser [12] In particular the so called KAM (Kolmogorov, Arnold, Moser) theory (see [1] and references therein) gave a positive answer to the above stability problem in the sense that it proved ( 2] the possibility of the existence of many body systems ( planetary ....

....14 More precisely we omit all the terms such that jR n (L 0 ; G 0 )j G Sa . whose Hessian matrix is not invertible. There are a few well known methods to overcome this minor problem 15 and it turns out that for our purposes the most convenient one is to follow Poincar e s trick [13], which consists in replacing the Hamiltonian H 2 by its square 16 . Therefore we let H 3 j (H 2 ) 2 : H 3 ( g; L; G; 1 2L 2 Gamma G) 2 2 ( 1 2L 2 Gamma G) R( g; L; G) 2 [R( g; L; G) 2 : 3.4) The Hessian of the unperturbed Hamiltonian (H 3 j =0 ) is equal to ....

Poincar`e H.: Les Methodes Nouvelles de la Mechanique Celeste, Gauthier Villars, Paris (1892)

....developed, with the N disk pinball model used as a concrete example and a series of physically interesting cases worked out in detail. 1 permanent address 1 Introduction The periodic orbit theory of classical chaotic dynamical systems has a long and distinguished history; initiated by Poincar e[1], and developed as a mathematical theory of hyperbolic dynamical systems by Smale, Sinai, Bowen, Ruelle and others[2, 3, 4, 5] it has in recent years been applied to many systems of physical interest[6, 7, 8, 9] The periodic orbit theory of quantum mechanical systems largely parallels this ....

H. Poincar'e, Les m'ethodes nouvelles de la m'echanique c'eleste (Guthier-Villars, Paris 1892-99)

....10 Methods for Evolving the Solar System. 10.1 The Legacy of Laplace. When Laplace expanded the mutual perturbations of the planets to first order in their masses, inclinations and eccentricities, he found that the orbits could be expressed as a sum of periodic terms implying stability. Poincar e[33] showed that these expansions don t converge owing to resonances. Using the KAM theorem, Arnold[4] derived 1 In Ancient Greek, chaos was the great abyss out of which Gaia flowed . 6 Figure 2: A comparison of the Earth s eccentricity as calculated by evolving the secular system and explicitly ....

H. Poincar'e, Les methodes nouvelles de la mechanique celeste. Gauthiers-Villars, Paris, (1892).

....necessary and sufficient conditions for uniform integrability of families of Hamiltonian systems. Rafael de la Llave Department of Mathematics University of Texas at Austin Austin TX 78712 1082 To the memory of Ricardo Ma n e Abstract. In [Po] Ch. V, specially x81, H. Poincar e discussed an obstruction to uniform integrability of families of Hamiltonians. That is, the existence of changes of variables analytic in the parameter ffl and in the variables that make the family of Hamiltonians a function of only action variables) We ....

....one could find a family of canonical transformations g ffl in such a way that (1:1) H ffl ffi g ffl = I ffl where I ffl is only a function of the actions. The fact that these hopes can not be fulfilled even in the very weak sense of formal power series was demonstrated very eloquently in [Po]. The whole of chapter V is devoted to finding obstructions for uniform integrability. In particular, in x81 one can find obstructions for the existence of analytic solutions of (1.1) that depend analytically on ffl (uniform integrability) and in x84, the conditions are extended slightly and ....

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H. Poincar'e: "Les methodes nouvelles de la m'echanique c'eleste", Gauthier Villars, Paris (1891--1899).

....Max Planck Institut fur Stromungsforschung, Bunsenstrae 10, D 37073 Gottingen, Germany z e mail: wolfram chaos.gwdg.de 2 1. Introduction One century ago Poincar e introduced the concept of time discrete dynamical systems in his study of two dimensional autonomous differential equations [1]. Such construction of Poincar e maps has become one of the very basic tools in nonlinear dynamics and is nowadays contained in every textbook on nonlinear dynamics (c.f. e.g. 2] The method is to a good deal at the heart of e.g. data analysis [3] control in nonlinear systems [4] and numerical ....

H. Poincar'e, Les Methodes Nouvelles de la M'echanique C'eleste, Gauthier--Villars, Paris 1899.

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Poincare H.: Les Methodes Nouvelles de la Mechanique Celeste, Gauthier Villars, Paris (1892)

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H. Poincare, Les Methodes Nouvelles de la Mechanique Celeste, Gauthier Villars, Paris, 1892.

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H.Poincar'e, Les Methodes Nouvelles de la Mechanique Celeste, vols. 1-3, GauthierVillars, Paris, 1892; (Dover, New York, 1957), reprint. 44

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H. Poincar'e, Les m'ethodes nouvelles de la m'echanique c'eleste (Guthier-Villars, Paris 1892-99)

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H. Poincar'e, Les m'ethodes nouvelles de la m'echanique c'eleste, (Paris 1892-99)

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H. Poincar'e, H. 1892. Les methodes nouvelles de la mechanique celeste. Gauthiers-Villars. Paris. Quinn, T.; N. Katz; J. Stadel and G. Lake. 1997. Time stepping N-body simulations. In preparation.

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