Geometrical Approaches to Differential Equations pp 318-339 | Cite as

# Periodic solutions of continuous self-gravitating systems

## Abstract

A collection of self-gravitating particles can be described by the nonlinear system consisting of the collision-less Boltzmann equation and the appropriate Poisson-equation. Such a system can be studied by associating it with dynamical systems in a finite-dimensional phase-space. The finite-dimensional problems are treated in the frame-work of KAM-theory by Birkhoff normalization and averaging techniques. This leads to a classification of possible two-parameter families of periodic solutions in these dynamical systems.

The asymptotic approximations of the solutions in two degrees of freedom problems with a discrete symmetric potential produce ring-type solutions of the original continuous system.

## Keywords

Periodic Solution Resonance Case Stable Periodic Solution General Hamiltonian System Birkhoff Normal Form## Preview

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## References

- Arnold, V.I., 1974, The mathematical methods of classical mechanics (in Russian), Moskou; French edition 1976 by ed. Mir, Moskou; English edition 1978 by Springer Verlag, Heidelberg.Google Scholar
- Churchill, R.C., Pecelli, G. and Rod, D.L., 1979, A survey of the Hénon-Heiles Hamiltonian with Applications to Related Examples, Volta Memorial Conference, Como 1977, Lecture Notes in Physics, 93, Springer Verlag, Heidelberg.zbMATHGoogle Scholar
- Cushman, R., 1979, Morse Theory and the Method of Averaging, preprint 119, Mathematisch Instituut, Rijksuniversiteit Utrecht.zbMATHGoogle Scholar
- Jackson, E.A., Nonlinearity and Irreversibility in Lattice Dynamics, Rocky Mountain J. Math. 8, 127Google Scholar
- Lynds, R. and Toomre, A., 1976, On the interpretation of Ring Galaxies: the binary ring system II Hz4, Ap.J. 209, 382.CrossRefGoogle Scholar
- Martinet, L. and Mayer, F., 1975, Galactic orbits and integrals of motion for stars of old galactic populations, Astron. and Astroph. 44, 45.Google Scholar
- Moser, J, 1955, Nonexistence of Integrals for Canonical Systems of Differential Equations, Comm. Pure and Appl. Math. 8, 409.MathSciNetCrossRefzbMATHGoogle Scholar
- Moser, J., 1973, Stable and random motions in dynamical systems, Princeton Univ. Press., Ann. Math. Studies 77.Google Scholar
- Ollongren, A., 1962, Three-dimensional galactic stellar orbits, Bull. Astr. Inst. Neth. 16, 241.Google Scholar
- Sanders, J.A., 1978, Are higher order resonances really interesting? Celes. Mech. 16, 421.MathSciNetCrossRefzbMATHGoogle Scholar
- Sanders, J.A. and Verhulst, F., 1979, Approximations of higher order resonances with an application to Contopoulos'model problem, in "Asymptotic Analysis, from theory to application", ed. F. Verhulst, Lecture Notes in Hath. 711, Springer Verlag, Heidelberg.Google Scholar
- Siegel, C.L. and Moser, J.K., 1971, Lectures in Celestial Mechanics, Springer Verlag.Google Scholar
- Toomre, A., 1977, Theories of spiral structure, Annual Rev. Astron. and Astroph. 15, 437.CrossRefGoogle Scholar
- Van der Aa, E. and Sanders, J., 1979, The 1:2:1:-resonance, its periodic orbits and integrals, in "Asymptotic Analysis, from theory to application", ed. F. Verhulst, Lecture Notes in Math. 711, Springer Verlag, Heidelberg.Google Scholar
- Van der Burgh, A.H.P., 1974, Studies in the asymptotic theory of nonlinear resonances, thesis Techn. Univ. Delft.Google Scholar
- Verhulst, F., 1976, On the theory of averaging, in Long-time predictions in dynamics, eds. V.S. Szebehely and B.D. Tapley, Reidel Publ. Co.Google Scholar
- Verhulst, F., 1979, Discrete-symmetric dynamical systems at the main resonances with applications to axi-symmetric galaxies, Phil. Trans. Roy. Soc. London A, 290, 435.CrossRefzbMATHGoogle Scholar