Improving the accuracy of simulated chaotic N-body orbits using smoothness

ABSTRACT Symplectic integrators are a foundation to the study of dynamical N-body phenomena, at scales ranging from planetary to cosmological. These integrators preserve the Poincaré invariants of Hamiltonian dynamics. The N-body Hamiltonian has another, perhaps overlooked, symmetry: it is smooth, or, in other words, it has infinite differentiability class order (DCO) for particle separations greater than 0. Popular symplectic integrators, such as hybrid methods or block adaptive stepping methods do not come from smooth Hamiltonians and it is perhaps unclear whether they should. We investigate the importance of this symmetry by considering hybrid integrators, whose DCO can be tuned easily. Hybrid methods are smooth, except at a finite number of phase space points. We study chaotic planetary orbits in a test considered by Wisdom. We find that increasing smoothness, at negligible extra computational cost in particular tests, improves the Jacobi constant error of the orbits by about 5 orders of magnitude in long-term simulations. The results from this work suggest that smoothness of the N-body Hamiltonian is a property worth preserving in simulations.

Transversality theorem in o-minimal structures

In this paper we present Thom’s transversality theorem in o-minimal structures (a generalization of semialgebraic and subanalytic geometry). There are no restrictions on the differentiability class and the dimensions of manifolds involved in comparison withthe general case.

Note on Almost Product Manifolds and their Tangent Bundles

Let Mn be an n-dimensional manifold of differentiability class C∞ with an almost product structure . Let have eigenvalue +1 of multiplicity p and eigenvalue -1 of multiplicity q where p+q = n and p≧1, q≧1. Let T(Mn) be the tangent bundle of M. T(Mn) is a 2n dimensional manifold of class C∞. Let xi be the local coordinates of a point P of Mn. The local coordinates of T(Mn) can be expressed by 2n variables (xi, yi) where xi are coordinates of the point P and yi are components of a tangent vector at P with respect to the natural frame constituted by the vectior ∂/∂xi at P.