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Dimensionality of Stable and Unstable Directions in the Gravitational N—Body Problem

Published online by Cambridge University Press:  12 April 2016

R.H. Miller*
Affiliation:
University of Chicago, e-mail:[email protected]

Abstract

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The gravitational n—body problem is chaotic. Phase trajectories that start very near each other separate rapidly. The rate looks exponential over long times. At any instant, trajectories separated in certain directions move apart rapidly (unstable directions), while those separated in other directions stay about the same (stable directions). Unstable directions lie along eigenvectors that correspond to positive eigenvalues of the matrix of force gradients. The number of positive eigenvalues of that matrix gives the dimensionality of stable regions. This number has been studied numerically in a series of 100—body integrations. It continues to change as long as the integration continues because the matrix changes extremely rapidly. On average, there are about 1.2n unstable directions out of 3n. Issues of dimensionality arise when the tools of ergodic studies are brought to bear on the problem of trajectory separation. A method of estimating the rate of trajectory separation based on matrix descriptions is presented in this note. Severe approximations are required.

Type
Stellar Systems
Copyright
Copyright © Kluwer 1999

References

Arnold, V.I. and Avez, A.: 1968, Ergodic Problems of Classical Mechanics. New York: Benjamin, W.A..Google Scholar
Contopoulos, G. and Voglis, N.: 1996, ‘Spectra of Stretching Numbers and Helicity Angles in Dynamical Systems’. Celest. Mech. & Dyn. Astron 64, 120.CrossRefGoogle Scholar
Goodman, J., Heggie, D. C., and Hut, P.: 1993, ‘On the exponential instability of N —body systems’. ApJ 415, 715733, Fiche 196-F11.CrossRefGoogle Scholar
Gurzadyan, V.G. and Savvidy, G. K.: 1986, ‘Collective Relaxation of Stellar Systems’. A&A 160, 203210.Google Scholar
Kandrap, H.E.: 1990, ‘Divergence of nearby trajectories for the gravitational N — body problem’. ApJ 364, 420425, Fiche 201-B13.CrossRefGoogle Scholar
Kandrup, H.E., Mahon, M. E., and Smith, H. Jr.: 1994, ‘On the sensitivity of the N —body problem toward small changes in initial conditions.IV’. ApJ 428, 458-465, Fiche 135-G12.CrossRefGoogle Scholar
Kandrup, H.E. and Smith, H. Jr.: 1991, ‘On the sensitivity of the N —body problem to small changes in initial conditions’. ApJ 374, 255265, Fiche 101-C9.CrossRefGoogle Scholar
Kandrup, H.E., Smith, H. Jr., and Willmes, D. E.: 1992, ‘On the sensitivity of the JV-body problem to small changes in initial conditions. III’. ApJ 399, 627633, Fiche 205-F9.CrossRefGoogle Scholar
Krylov, N.S.: 1979, Works on the Foundations of Statistical Physics. Princeton, N.J.: Princeton University Press.Google Scholar
Lorenz, E.N.: 1963, ‘Deterministic Nonperiodic Flow’. J. Atmos. Sci. 20, 130.2.0.CO;2>CrossRefGoogle Scholar
Miller, R.H.: 1964, ‘Irreversibility in Small Stellar Dynamical Systems’. ApJ 140, 250256.CrossRefGoogle Scholar
Miller, R.H.: 1966, ‘Polarization of the Stellar Dynamical Medium’. ApJ 146, 831837.CrossRefGoogle Scholar
Miller, R.H.: 1971, ‘Experimental Studies of the Numerical Stability of the Gravitational n-Body Problem’. J. Comp. Phys. 8, 449463.CrossRefGoogle Scholar
Miller, R.H.: 1972, ‘A Matrix Eigenvalue Problem’. Unpublished Report, Section B in ICR Quarterly Report No. 33.Google Scholar
Miller, R.H.: 1974, ‘Numerical Difficulties with the Gravitational n-Body Problem’. In: Bettis, D.G. (ed.): Proceedings of the Conference on the Numerical Solution of Ordinary Differential Equations. Berlin, pp. 260275. Vol. 362 of Lecture Notes in Mathematics.CrossRefGoogle Scholar
Quinlan, G.D. and Tremaine, S.: 1992, ‘On the reliability of gravitational N — body integrations’. MNRAS 259, 505518.CrossRefGoogle Scholar