  In this study, we extend the HAL QCD method to a case where a total momentum
of a two-particle system is non-zero and apply it to the $I=2$ S-wave $\pi\pi$
scattering in order to confirm its validity. We derive a fundamental relation
of an energy-independent non-local potential defined in the center of mass
frame with NBS wave functions in a laboratory frame. Based on the relation, we
propose the time-dependent method to extract potentials, often used in practice
for the HALQCD method in the center of mass frame. For numerical simulations in
the $I=2$ $\pi\pi$ system, we employ (2+1)-flavor gauge configurations on a
$32^3 \times 64$ lattice at the lattice spacing $a \approx 0.0907$ fm and
$m_{\pi} \approx 700$ MeV. Both effective leading order (LO) potentials and
corresponding phase shifts obtained in laboratory frames agree with those
obtained in the center-of-mass frame by the conventional HAL QCD method within
somewhat larger statistical errors. In addition, we observe a consistency in
scattering phase shifts between ours and results by the finite-volume method as
well. The HAL QCD method with non-zero total momenta, established in this
study, brings more flexibility to the HAL QCD method, which enables us to
handle systems having the same quantum numbers with a vacuum or to access
energy regions prohibited in the center of mass frame.
