  Hamiltonian systems such as the gravitational N-body problem have
time-reversal symmetry. However, all numerical N-body integration schemes,
including symplectic ones, respect this property only approximately. In this
paper, we present the new N-body integrator JANUS, for which we achieve exact
time-reversal symmetry by combining integer and floating point arithmetic.
JANUS is explicit, formally symplectic and satisfies Liouville's theorem
exactly. Its order is even and can be adjusted between two and ten. We discuss
the implementation ofJANUS and present tests of its accuracy and speed by
performing and analyzing long-term integrations of the Solar System. We show
that JANUS is fast and accurate enough to tackle a broad class of dynamical
problems. We also discuss the practical and philosophical implications of
running exactly time-reversible simulations.
