Abstract
A quantitative description of Hamiltonian chaos, based on a Riemannian geometrization of Newtonian dynamics, is discussed here for a model, introduced by Contopoulos, describing the dynamics of a test star in a galactic potential. A statistical treatment of the geometry of dynamics, effective in the limit of a large number N of degrees of freedom, is here applied to the N = 3 case of the Contopoulos model, discussing how the statistical model has to be modified in order to quantitatively account for the chaoticity of the dynamics.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Binney, J. and Treuraine, S.: 1987, Galactic Dynamics, Princeton University Press, Princeton, N.J., Chapter 3.
Casetti, L.: 1995, Efficient symplectic algorithms for numerical simulations of Hamiltonian flows, Phys. Scr. 51, 29–34.
Casetti, L., Cerruti-Sola, M., Pettini, M. and Cohen, E.G.D.: 1997, The Fermi-Pasta-Ulam problem revisited: Stochasticity thresholds in nonlinear Hamiltonian systems, Phys. Rev. E 55, 6566–6574.
Casetti, L., Clementi, C. and Pettini, M.: 1996, Riemannian theory of Hamiltonian chaos and Lyapunov exponents, Phys. Rev. E 54, 5969–5984.
Casetti, L., Pettini, M. and Cohen, E.G.D.: 2000, Geometric approach to Hamiltonian dynamics and statistical mechanics, Phys. Rep. 337, 237–341.
Cerruti-Sola, M. and Pettini, M.: 1996, Geometric description of chaos in two degrees of freedom Hamiltonian systems, Phys. Rev. E 53, 179–188.
Cerruti-Sola, M., Franzosi, R. and Pettini, M.: 1997, Lyapunov exponents from geodesic spread in configuration space, Phys. Rev. E 56, 4872–4875.
Contopoulos, G. and Barbanis, B.: 1989, Lyapunov characteristic numbers and the structure of phase space, Astron. and Astrophys. 222, 329–343.
Contopoulos, G.: 1986, Qualitative changes in 3-dimensional dynamical systems, Astron. and Astrophys. 161, 244–256.
Contopoulos, G.: 1988, In: A.E. Roy (ed.), Long Term Dynamical Behaviour of Natural and Artificial N-Body Systems, 301.
Contopoulos, G. and Grosbol, P.: 1989, Orbits in barred galaxies, Astron. Astrophys. Rev. 1, 261–289.
Contopoulos, G. and Magnenat, P.: 1985, Simple three dimensional periodic orbits in a galactic-type potential, Cel. Mech. 37, 387–414.
Do Carmo, M.P.: 1992, Riemannian Geometry, Birkhäuser, Boston.
Eisenhart, L.P.: 1929, Dynamical Trajectories and Geodesics, Ann. Math. 30, 591–606.
Guckenheimer, J. and Holmes, P.J.: 1983, Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields, Springer-Verlag, New York.
Hawn, M. and Heiles, C.: 1964, The applicability of the third integral of motion: some numerical experiments, Astron. J. 69, 73–79.
Henrard, J. and Ferraz-Mello, S. (eds): 1999, Cel. Mech. 73, Nos. 1–4, Special Issue on impact of modem dynamics in astronomy.
Kandrup, H.E., Sideris, I.V. and Bohn, C.L.: 2001, Chaos, ergodicity, and the thermodynamics of lower-dimensional Hamiltonian systems, astro-ph/0108038.
Pettini, M.: 1993, Geometrical hints for a non perturbative approach to Hamiltonian dynamics, Phys. Rev. E 47, 828–850.
Pettini, M. and Valdettaro, R.: 1995, On the Riemannian description of chaotic instability in Hamiltonian dynamics, Chaos 5, 646–652.
Poincaré, H.: 1892, Les Méthodes Nouvelles de la Mécanique Celeste, Gauthier-Villars, Paris.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer Science+Business Media Dordrecht
About this paper
Cite this paper
Ciraolo, G., Pettini, M. (2002). Geometry of Chaos in Models of Stellar Dynamics. In: Celletti, A., Ferraz-Mello, S., Henrard, J. (eds) Modern Celestial Mechanics: From Theory to Applications. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2304-6_11
Download citation
DOI: https://doi.org/10.1007/978-94-017-2304-6_11
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-6078-5
Online ISBN: 978-94-017-2304-6
eBook Packages: Springer Book Archive