On the sensitivity of the N-body problem to small changes in initial conditions

Kandrup, Henry E.; Smith, Howard G.

doi:10.1086/170114

Cited by 54 publications

(43 citation statements)

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“…This implies that the contributions of the "mean field" and of the nearest pair to the growth are nearly the same, as observed by Kandrup et al (1992).…”

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confidence: 70%

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

Miller¹

1999

International Astronomical Union Colloquium

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Abstract. 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 .In 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.

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“…This implies that the contributions of the "mean field" and of the nearest pair to the growth are nearly the same, as observed by Kandrup et al (1992).…”

supporting

confidence: 70%

“…7. This formulation shows why separations projected onto the configuration space and onto the velocity space each grow at the same rate, as noted by Kandrup and Smith (1991). They're tightly coupled, since the full system is actually described by the 6ra x 6ra-matrix M. 8.…”

mentioning

confidence: 70%

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

Miller¹

1999

International Astronomical Union Colloquium

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“…Bocaletti et al [29] obtained such estimates using the virial theorem assuming that the systems in question are close to stable equilibrium. Numerical simulations by Kandrup and Smith [31,32] and other authors confirmed the theoretical estimates of τ v.r. ∼ t cr obtained by Kandrup [23][24][25] and Bocaletti et al [29].…”

Section: Introductionsupporting

confidence: 56%

Correlations, spectra, and instability of phase-space density fluctuations in open-cluster models

Данилов

Putkov

2013

Astrophys. Bull.

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Abstract-The dynamical evolution of six open star cluster models is analyzed using the correlation and spectral analysis of phase-space density fluctuations. The two-time and mutual correlation functions are computed for the fluctuations of the phase-space density of cluster models. The data for two-time and two-particle correlations are used to determine the correlation time for phase-space density fluctuations ((0.1-1) τ v.r. , where τ v.r. is the violent relaxation time of the model) and the average phase velocities of the propagation of such fluctuations in cluster models. These velocities are 2-20 times smaller than the root mean square velocities of the stars in the cluster core. The power spectra and dispersion curves of phase-space density fluctuations are computed using the Fourier transform of mutual correlation functions. The results confirm the presence of known unstable phase-space density fluctuations due to homologous fluctuations of the cluster cores. The models are found to exhibit a number of new unstable phase-space density fluctuations (up to 32-41 pairs of fluctuations with different complex conjugate frequencies in each model; the e-folding time of the amplitude growth of such fluctuations is (0.4-10) τ v.r. and their phases are distributed rather uniformly). Astrophysical applications of the obtained results (irregular structure of open star clusters, formation and decay of quasi-stationary states in such clusters) are discussed. DOI: 10.1134/S199034131302003X Keywords: stars: kinematics and dynamics-Galaxy: open clusters and associations: general INTRODUCTIONRecently, Chavanis [1-3] derived theoretical estimates for the fluctuations of phase-space density (PSD) and the corresponding correlation functions for spatially uniform and nonuniform systems with long-range interactions (including self-gravitating systems). To this end, the above author used kinetic equations written with a number of simplifying assumptions. The formulas for the correlation functions derived by Chavanis [1-3] have a rather complex form making them difficult to use for analyzing dynamic processes in such systems. In our opinion, it is more productive to compute the correlation functions directly by numerically integrating the equations of motion of gravitating particles in the problems of clustering of galaxies and the evolution of the Universe (see, e.g., [4][5][6]), and also when modeling the dynamics of open star clusters (OCl) [7].In one of our previous papers [7] we computed the two-time correlation functions for r = |r|, v = |v|, and the energy ε per unit star mass, as well as twoparticle correlations of r, v, ε, stellar number density n = n(r, t) and PSD f = f (r, v, t) for open star

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“…In fact, there are indications that the growth rate of Miller's instability remains constant or even increases with N ( Kandrup & Smith 1991;Heggie 1991;Goodman, Heggie, & Hut 1993), although this result is uncertain since published numerical experiments have been limited to Nd10 3 . Here we describe the application of a new, '' systolic '' Nbody algorithm to this problem, which allows us to treat systems with N as large as 10 5 .…”

Section: Introductionmentioning

confidence: 99%