Re: This Week's Finds in Mathematical Physics (Week 222)

Jonathan Thornburg -- remove -animal to reply wrote: 
Robert C. Helling <robert@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:

There are people doing numerical long term stability
analysis of the solar system. From what I know, they are not just
taking F=ma and Newton's law of gravity, replace dt by delta t and
then integrate but use much fancier spectral methods. Could somebody
please point me to an introduction into these methods?



I don't do this sort of work myself, but the buzzwords you want are
"symplectic ODE integrator".  The basic idea is to use an ODE integration
scheme which conserves energy, angular momentum, and maybe other nice
things, up to floating-point roundoff error, rather than just up to
finite differencing error like a standard ODE integrator would do.


T. Fuse,
Planetary Perturbations on the 2: 3 Mean Motion Resonance with Neptune,
http://astronomy.nju.edu.cn/~xswan/reference/Fuse_PASJ54_493.pdf
uses symplectic integration to study 2:3 resonances numerically.

The thesis
    Time-frequency analysis based on wavelets for Hamiltonian systems
by Vela-Arevalo,
    http://www.cds.caltech.edu/~luzvela/th2s.pdf
contains in Chapter 4 interesting numerical information about chaos,
resonances, and stability in the restricted 3-body problem. Other
interesting papers include:

http://users.auth.gr/~hadjidem/Asymmetric1.pdf
Symmetric and asymmetric librations in planetary and satellite
systems at the 2/1 resonance

astro-ph/0501004
Regimes of Stability and Scaling Relations for the Removal Time
in the Asteroid Belt

astro-ph/0203182
The Resonant Dynamical Evolution of Small Body Orbits
Among Giant Planets

http://cns.physics.gatech.edu/~luzvela/VelaArevaloMarsdenCQG_2004.pdf
Time–frequency analysis of the restricted three-body problem:
transport and resonance transitions

http://www.astro.auth.gr/~varvogli/varv5.ps
The “Third” Integral in the Restricted Three-Body Problem Revisited



Arnold Neumaier

.

Relevant Pages

  • Re: Stable Orbit Formulae?
    ... concentric orbits around Sol. ... planets will evolve into orbits where they avoid mutual interaction ... restricted version of the 1:1 resonance problem. ... You're gonna have to gloss it over, because anything else is Pandora's Box, and you've just chaged your careers from game writer to astrophysicist. ...
    (sci.astro)
  • Re: Stable Orbit Formulae?
    ... Thanks to your participation, and the new references, my interest in the ... able to verify that Bode's Law is not a resonance problem. ... Bode's Law, however stated, has AIUI never been considered as absolutely ... might be found relating the periods of successive planets. ...
    (sci.astro)
  • Re: Quantum theory: A resonance effect?
    ... are examples of resonance of one sort or another. ... travelling waves, rather than standing waves. ... electron orbitals in periodic patterns around the atomic nucleus, ... snowballs to asteroids to planets to stars, ...
    (sci.physics)
  • Re: Quantum theory: A resonance effect?
    ... > are examples of resonance of one sort or another. ... > travelling waves, rather than standing waves. ... > electron orbitals in periodic patterns around the atomic nucleus, ... > snowballs to asteroids to planets to stars, ...
    (sci.physics)