  Classical Hamiltonian systems with a mixed phase space and some asymmetry may
exhibit chaotic ratchet effects. The most significant such effect is a directed
momentum current or acceleration. In known model systems, this effect may arise
only for sufficiently strong chaos. In this paper, a Hamiltonian ratchet
accelerator is introduced, featuring a momentum current for arbitrarily weak
chaos. The system is a realistic, generalized kicked rotor and is exactly
solvable to some extent, leading to analytical expressions for the momentum
current. While this current arises also for relatively strong chaos, the
maximal current is shown to occur, at least in one case, precisely in a limit
of arbitrarily weak chaos.
