  We formulate Non-Relativistic Quantum Chromodynamics (NRQCD) on a lattice
which is boosted relative to the usual discretization frame. Moving NRQCD
(mNRQCD) allows us to treat the momentum for the heavy quark arising from the
frame choice exactly. We derive mNRQCD through O(1/m^2,v^4), as accurate as the
NRQCD action in present use, both in the continuum and on the lattice with
O(a^4) improvements. We have carried out extensive tests of the formalism
through calculations of two-point correlators for both heavy-heavy
(bottomonium) and heavy-light (B_s) mesons in 2+1 flavor lattice QCD and
obtained nonperturbative determinations of energy shift and external momentum
renormalization. Comparison to perturbation theory at O(\alpha_s) is also made.
The results demonstrate the effectiveness of mNRQCD. In particular we show that
the decay constants of heavy-light and heavy-heavy mesons can be calculated
with small systematic errors up to much larger momenta than with standard
NRQCD.
