Geometric description of chaos in self-gravitating systems.

Abstract

This paper tackles the problem of the origin of dynamic chaos in Hamiltonian systems, with a special emphasis on the self-gravitating N-body systems. A Riemannian approach is adopted. The relationship between dynamic instability and curvature properties of the configuration space manifold is the main concern of this paper. Dynamic instability is studied with the aid of the Jacobi--Levi-Civita (JLC) equation for the geodesic spread. We point out that the approximations introduced so far to make the JLC equation handy, that is, to obtain a scalar equation describing the dynamical instability, are still too severe. In order to assess the validity limits of these approximations, the aid of numerical simulations is essential. For this reason, our analysis is supported by the numerical study of the dynamics of 10 and 100 gravitationally interacting point masses. The self-gravitating N-body systems provide an illuminating example of the relevant difference between geodesic flows of abstract ergodic theory and geodesic flows of physical interest. In fact, even though they correspond to manifolds of almost everywhere negative scalar curvature, this does not determine by itself the degree of chaos of these systems. We show that the quantities determining the instability of nearby trajectories do not simply coincide with scalar or Ricci curvature; rather they involve also other ingredients that are ultimately responsible for the existence of two different mechanisms to make chaos. The quantities mentioned enter a Hill equation and give instability either when they are negative or---when positive---because of parametric resonance. From numerical computations the \ensuremath{\epsilon} (energy density) dependence of the dynamic instability exponents is found to be \ensuremath{\sim}${\mathrm{\ensuremath{\epsilon}}}^{3/2}$. Our paper aims at warning about the possibility of misleading conclusions that might be drawn from the geometric approach if the existence of the problems discussed here is ignored. Finally, we briefly discuss the relationship of the Riemannian geometric description of chaos with Lyapunov exponents in the special case of gravitational N-body systems.