On the Number of Isolating Integrals in Systems with Three Degrees of Freedom

  • Claude Froeschle
Conference paper
Part of the Astrophysics and Space Science Library book series (ASSL, volume 31)


Dynamical systems with three degrees of freedom can be reduced to the study of a four-dimensional mapping. We consider here, as a model problem, the mapping given by the following equations:
$$\left\{ \begin{gathered}{x_1} = {x_0} + {a_1}\sin ({x_0} + {y_0}) + b\sin ({x_0} + {y_0} + {z_0} + {t_0}) \hfill \\ {y_1} = {x_0} + {y_0} \hfill \\ {z_1} = {z_0} + {a_2}\sin ({z_0} + {t_0}) + b\sin ({x_0} + {y_0} + {z_0} + {t_0})(\bmod 2\pi ) \hfill \\ {t_1} = {z_0} + {t_0} \hfill \\ \end{gathered} \right.$$

We have found that as soon as b ≠ 0, i.e. even for a very weak coupling, a dynamical system with three degrees of freedom has in general either two or zero isolating integrals (besides the usual energy integral).


Model Problem Usual Energy Spiral Galaxy Invariant Curve Restricted Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Arnold, V. I.: private communication to Dr M. Hénon.Google Scholar
  2. Contopoulos, G.: 1970, Cont. from the Ast. Dept. Univ. of Thessaloniki, No. 53.Google Scholar
  3. Froeschie, C.: 1970, Astron. Astrophys. 4, 115.ADSGoogle Scholar
  4. Froeschie, C.: 1970, Astron. Astrophys. 5, 177.ADSGoogle Scholar
  5. Hénon, M.: 1969, Quart. Appl. Math. 27, 291.MathSciNetzbMATHGoogle Scholar
  6. Poincaré, H.: 1892, Les Méthodes Nouvelles de la Mécanique Céleste, Gauthier-Villars, Paris.Google Scholar
  7. Taylor, J. B.: 1969, private communication; see Culham Laboratory Progress Report CLM-PR-12.Google Scholar

Copyright information

© D. Reidel Publishing Company, Dordrecht, Holland 1972

Authors and Affiliations

  • Claude Froeschle
    • 1
  1. 1.Observatoire de Nice, Le Mont-Gros06 NiceFrance

Personalised recommendations