  We review the implementation of individual particle time-stepping for N-body
dynamics. We present a class of integrators derived from second order
Hamiltonian splitting. In contrast to the usual implementation of individual
time-stepping, these integrators are momentum conserving and show excellent
energy conservation in conjunction with a symmetrized time step criterion. We
use an explicit but approximate formula for the time symmetrization that is
compatible with the use of individual time steps. No iterative scheme is
necessary. We implement these ideas in the HUAYNO (available online at
www.amusecode.org) code and present tests of the integrators and show that the
presented integration schemes shows good energy conservation, with little or no
systematic drift, while conserving momentum and angular momentum to machine
precision for long term integrations.
