Discussion:Mécanique, enseignée via l'Histoire des Sciences/Résonances en astronomie
qq notes
Morbidelli modifier
cf modern Modern Celestial Mechanics ISBN 0-415-27938-0 (pbk) ? 2002 Taylor & Francis
Contents :
- PREFACE INTRODUCTION
- ELEMENTARY CELESTIAL AND HAMILTONIAN MECHANICS
- QUASI-INTEGRABLE HAMILTONIAN SYSTEMS
- KAM TORI
- SINGLE RESONANCE DYNAMICS
- NUMERICAL TOOLS FOR THE DETECTION 0F CHAOS 8!
- INTERACTIONS AMONG RESONANCES
- SECULAR DYNAMICS 0F THE PLANETS
- SECULAR DYNAMICS 0F SMALL BODIES
- MEAN MOTION RESONANCES
- THREE-BODY RESONANCES
- SECULAR DYNAMICS INSIDE MEAN MOTION RESONANCES
- GLOBAL DYNAMICAL STRUCTURE 0F THE BELTS 0F SMALL BODIES
il se trouve que j'ai ce livre jusqu'à Lundi28/08/06 , alors , je vais essayer d'en profiter un max ; Wikialement sylvie--Guerinsylvie 23 août 2006 à 15:48 (CEST)
- PREFACE INTRODUCTION :
In the last 20 years, three spectacular results are:
- chaotic dynamics of the planets,
- asteroids escape from the main belt and reach Earth-threatening orbits,
- origin of the internal heating of the Galilean satellites,
the first six chapters present what one should know of Hamiltonian theory to work at ease in Celestial M
3hapter 2 explains Hamiltonian perturbation theory based on Lie series. Chs
Chapters 3 and 4 illustrate the properties of invariant tori and resonances, respectively.
^hapter 5 to discussin: the numerical tools that are useful for the detection of chaos.
Chapter 6 :1 the interaction of its resonances, < 1 details how these structures can bi identified with numerical explorations.
The second part ol is more technical,
Chapters 9-12, conversely, are devoted to the difficult subject of mean motion resonances.
Chapter 11 discusses the secular dynamics inside mean motion resonances, which in my opinion is one of the most complicated topics thanks to :I. Festou and C. Froeschlé. -. Guzzo, D. Nesvorny and F. Thomas - f< - A. Giorgiliï, V. Gurzadyan, J. Henrard and J. Laskar - to thank A. Cellino, S. Ferraz-Mello, J. Henrard, M. Holman, Z. Knezevic, J. Laskar, E. Lega, A. Lemaitre, C. Murray, N. Murray and D. Nesvorny for p Farinella, Migliorini, Michèle Moons.
Introduction modifier
general overview of the Solar System Newtonannouncement (1687):. Since then, the computation of precise ephemerides has become essentially a mathematical challange. Lagrange and Laplace :a quasi-resonance between the orbital periods ofjupiter and Saturn.
the motion of the planets is in fact chaotic.
Since then, a major goal has been to understand the reasons for this chaotic motion-
even more interesting is the dynamics of the asteroids. called the main belt^>ome puzzling features immediately appear e\ These features are named Kirkwood gaps, from in 1866. They coincides with the location ofthe main mean motion resonances with Jupiter,but concnetration with the location of the 3:2
Only recently has quite a complete solution for the problem of the origin of the Kirkwood gaps been provided.
resonance ofa new type, called secular resonance. Secular resonance occurs when an integer combination of the precession rates ofthe asteroid
close encounters with the ter- restrial planets. " named Apollos, Amors, Atens and Mars-crossers (fig3p5):near-Earth asteroid popu (NEAs). he typical lifetime of NEAs is 10 My.
Beyond the orbit of Neptune, , second belt o as the Kuiper belt. As J As June 9, 2000, 279 trans-Neptunian o The "planet" Pluto is also in the Kuiper belt. In : Pluto should be better regarded as the biggest object 1 r in the Kuiper belt.
Kuiper belt is believed to be responsible for sustaining the population of the so-called Jupiter-family comets. low-inclined, short-period comi
The long-period comets and the so-called Halley-type comets, rather form a quasi-spherical reser- voir, called the Oort cloud. Th( at the frontier of the Solar Sys- semimajor axis ~10,000 AU. At such a distance from the Sun, the gravitational potential of the entire galaxy becomes a strong perturbation ol the dynamical effect of this Galactic tide is crucial to understand the for nation of the Oort cloud, a
The ring systems of Jupiter, Sat For instance, the Cassini division is determined by the the 2:1nance with Mimas, while the outer edge of the A ring is produced by the7:6 resonance with the co-orbital satellites Janus and Epimetheus.
ratios of large integer numbers, as in the case of the 32:31resonance with Prometheus, at the location of the so-called Keeler gap.
The shape of this ring seems to be dictated by the presence of the so-called shepherding satellites, orbiting along each of its sides.
, the systems of satellites of the giant planets c > can be considered as miniature solar systems, ; + " the tides exerted by the planet force the satellitesslow outward migration. Because the relative orbital peri- ods change with the semimajor axes, the satellites must pass through several resonant configurations :: lo orbits twice around Jupiter for each orbit of Europa, which in turn orbits twice for each orbit of Ganymede.
The interplay between tidal forces and orbital dynamics also explainsthe large inclination ofthe Urania.n satellite Miranda resonance with Umbriel,
The tides also tend to slow down the spin frequ ; But sometimes , chaos : Hyperion.
And also to heat : ex : Io
End.Bon , ya rien à glaner que l on ne sache . C est une bonne révision . point.
Hamiltonian mecha modifier
Equations of motion . . . 3rbital elements . Perturbations of the two-body probk amiltonian systems and the two-body problec Perturbations in Hamiltonian form . . . uionical transformations . . . Integrable Hamiltonians Action-angle variables [ntegrable dynamics .
Voyons voir :
cf DANBY1962; Kepler laws; a & e ; theta et E ; E(t) tq E-e sinE )= wt = M(t) , w² = G(m1+m2):a^3 ; Omega, omega, finally {a,e,i,Omega, omega et M)
si i =0 , then pi-surligné = Omega +omega ; si e=0 , omega perd sa signification et on prend lambda = M +pi-
H(E) est l énergie conservée . G le moment cinétique et H= G.cosi
Ok
Perturbations modifier
de lagrange inutilisable à hamilton :Whittaker(1937) donne la solution: pb restreint : Ho + Hperturb, si on se donne le mvt des grosses ,alors H= H(t) = Ho+h1(t)
trans de contact : 3 critères : Crochets de Poisson cf (Whittaker 1937 , Gantmacher1975, 131)
democratic heliocentric(p24): Koseleff(Koseleff, 1993, 1996; Touma and Wisdom, 1994b; Duncan et al., 1998; Chambers, 1999
critère2 : the transformation (v, x) -^(v ) is canonical if there exis1 a function S(vf, x) (called a generating function) such that : ordinary
critère3 : the transformation ) is canonical if there exists a Hamiltonian "-alled a generating Hamiltonian) and a parameter epsilon such that : cf Gantmacher1975,133-134)
DoncGantmacher F. (1975) Lectures in Analytical Mechanics. MIR,
H flow modifier
Th de Liouville : div F = 0 : donc interdiction de voir la dimension diminuer ( un volume reste un volume. cf chap 4.
Th de H(t) : ok ; attention à T1 et à To non nuls !
Th df:dt = {f, H} Alors le développement en série sera via Taylor :
f(t) = f(0) + (t:1!)L1f (0) + (t²:2!)L2f (0) + ...
soit f( v(t), x(t)) = g(v(0),x(0), t ) ; ceci utilisé en particulier via le critère 3
pour avoir un "temps" dit epsilon.
Integrable H modifier
def d'Arnold chap 4. donc difficile.
mais Th2 de Liouville : si on a n cstes PHI1, PHIn en involution ( un ECOC). Mais c'est dur à trouver : Hénon (1964) impossible pour Cub-galactique et Hénon (1974) possible pour Toda ! Poincaré (1892) désespéré. Mais existe critère de non-intégrabilité.
Donc on intègre ceux qu'on sait : Ceux H(I1, ...,In) (actions-angles. Celui H(v,x) 1D , of course Celui H(v1, I2 I3 , ,x1) puisque H et (I2 I3 In)=cstes donc c'est simplement le cas 2.
Action-angles modifier
th2d'Arnold-Liouville : si PHI1, PHIn est compacte alors existent I et phi avec H(I) et les angles phi(t) = wi.t
Alors les H-intégrables seront H(I) et on étudiera les H(I) + perturb
Pour un H(I) , on aura la notion de TORES commutatifs. Les I seront construits sur chacun des cycles. Puis la fonction S sera S(I,x) = int sigma v(I,x) dx et alors phi sera d-rond S(p,x) / d-rond p. On sera donc passé de H(v,x) à H(I). Dnc on a un moyen constructif de construire H(I) .
- Delaunay : (Epstein1916), Sommerfeld1922, Born1927the mechanics of the atom.
C'est incroyable de voir que Born a déjà fait le travail en 1927 ( Pauli date de 1925-1926). c'est la première fois que je lis ce travail : il me reste àl 'apprendre. Cela paraît tellement proche du Cordani !
- Delauney et perturb :pb restreint et pb planétaire
- Règles quantiques de d'Alembert :
on examine le développement en sériede Fourier , et ses symétries : coef réels, invariance par rotation Oz ; invariance par rapport Oz /Oz', non-singularités => Poincaré variables et à nouveau des règles quantiques
Int dynamics modifier
H(I) et phi' = w(I) et I' = 0 Donc mvt sur un tore invariant et l'espace est feuilleté par ces tores. Mvt non résonant ou quasi-périodique. Mvt complètement résonant Mvt à résonance de multiplicité m avec ORDRE de la résonance : = min |k| Comme les w sont des w(I), on se préoccupe de la dégénérescence du système . Si non-dégénéré , c'est facile . Mais Kepler est deux fois dégénéré , donc les perturb seront atroces à classifier .
Nous voilà prévenus !
Laskar modifier
cf Goutelas
Autres modifier
le Gallavotti, bien sûr le Ramis-Morales, bien sûr. les séries de Lie .
il est clair que le cours SICARDY contient déjà beaucoup. Il faudrait qu'un informaticien_graphique prenne en charge les animations.