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Journal of Mathematical Physics / Volume 52 / Issue 11 / ARTICLES / Methods of Mathematical Physics

J. Math. Phys. 52, 113502 (2011); http://dx.doi.org/10.1063/1.3658279 (20 pages)

The stochastisation hypothesis and the spacing of planetary systems

Jacky Cresson

 Laboratoire de Mathématiques Appliquées de Pau, Université de Pau et des Pays de l'Adour, avenue de l'Université, BP 1155, 64013 Pau Cedex, France and Institut de Mécanique Céleste et de Calcul des Éphémérides, Observatoire de Paris, 77 avenue Denfert-Rochereau, 75014 Paris, France

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(Received 16 June 2011; accepted 6 October 2011; published online 8 November 2011)

We introduce the stochastisation hypothesis which aims to provide a framework to deal with physical systems in random environment. We apply the stochastisation hypothesis in two different cases: in the study of the dynamics of a protoplanetary nebula and in the chaotic long-term behaviour of a generic planetary system using previous works of Albeverio et al. [“A stochastic model for the orbits of planets and satellites: an interpretation of Titius-Bode law,” Expo. Math. 4, 363–373 (1983)], Nottale [“The quantization of the solar system,” Astron. Astrophys. 315, L9 (1996)], and Laskar [“On the spacing of planetary systems,” Phys. Rev. Lett. 84(15), 3240–3243 (2000)10.1103/PhysRevLett.84.3240]. These results give both a particular law for the distribution of planetary orbits.

© 2011 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. REMINDER ABOUT STOCHASTIC EMBEDDING
    1. Notations
    2. Stochastic derivatives
    3. Stochastic embedding
      1. Stochastic differential embedding
      2. Stochastic analogue of dynamical quantities: speed and acceleration
      3. Stochastic variational embedding
    4. Coherence of stochastic embedding
    5. Lifting of symmetries and a stochastic Noether theorem
      1. First integrals of stochastic dynamical systems
      2. Stochastic lifting of symmetries
      3. Stochastic invariance
      4. Stochastic Noether's theorem
      5. An example: rotations and stochastisation
  3. THE STOCHASTISATION HYPOTHESIS
    1. The stochastisation hypothesis
    2. Discussion of the stochastisation hypothesis
    3. Stochastic processes and stochastisation: diffusion processes ?
  4. THE STOCHASTISATION HYPOTHESIS AND THE NEWTON EQUATION
    1. The stochastic Newton equation
    2. The reversible stochastic Newton's equation
  5. APPLICATION: ORGANISATION OF PLANETARY SYSTEMS
    1. Dynamics of a protoplanetary nebula
      1. Model for the dynamics in a protoplanetary nebula
      2. On the nature of collisions
      3. Mechanisms for confinements
      4. Law of repartition for planetary orbits
    2. Long-term behaviour of the solar system
      1. Chaos and Stochastic processes
      2. Hierarchical structures
      3. Is the solar system stable ?
  6. CONCLUSION
    1. Summary
    2. Perspectives
      1. Coalescent processes
      2. Statistics and gravitational structuration constant
      3. Stability of the solar system

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KEYWORDS and PACS

Keywords

Brownian motion, chaos, planetary nebulae, stochastic processes

PACS

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3658279

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

For access to fully linked references, you need to log in.
    Cresson, J. and Darses, S., “Stochastic embedding of dynamical systems,” J. Math. Phys. 48(7), 072703 (2007)JMAPAQ000048000007072703000001.

    Cresson, J. and Greff, I., “A non-differentiable Noether's theorem,” J. Math. Phys. 52, 023513 (2011)JMAPAQ000052000002023513000001.

    Laskar, J., “On the spacing of planetary systems,” Phys. Rev. Lett. 84(15), 3240–3243 (2000).

    Misawa, T. and Yasue, K., “Canonical dynamical systems,” J. Math. Phys. 28(11), 2569–2573 (1987)JMAPAQ000028000011002569000001.


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