  The conservation of energy, linear momentum and angular momentum are
important drivers for our physical understanding of the evolution of the
Universe. These quantities are also conserved in Newton's laws of motion under
gravity \citep{Newton:1687}. Numerical integration of the associated equations
of motion is extremely challenging, in particular due to the steady growth of
numerical errors (by round-off and discrete time-stepping,
\cite{1981PAZh....7..752B,1993ApJ...415..715G,1993ApJ...402L..85H,1994LNP...430..131M})
and the exponential divergence \citep{1964ApJ...140..250M,2009MNRAS.392.1051U}
between two nearby solution. As a result, numerical solutions to the general
N-body problem are intrinsically questionable
\citep{2003gmbp.book.....H,1994JAM....61..226L}. Using brute force integrations
to arbitrary numerical precision we demonstrate empirically that ensembles of
different realizations of resonant 3-body interactions produce statistically
indistinguishable results. Although individual solutions using common
integration methods are notoriously unreliable, we conjecture that an ensemble
of approximate 3-body solutions accurately represents an ensemble of true
solutions, so long as the energy during integration is conserved to better than
1/10. We therefore provide an independent confirmation that previous work on
self-gravitating systems can actually be trusted, irrespective of the intrinsic
chaotic nature of the N-body problem.
