  We present the first determination of $\rho \pi$ scattering, incorporating
dynamically-coupled partial-waves, using lattice QCD, a first-principles
numerical approach to QCD. Considering the case of isospin-2 $\rho \pi$, we
calculate partial-wave amplitudes with $J \le 3$ and determine the degree of
dynamical mixing between the coupled $S$ and $D$-wave channels with $J^P=1^+$.
The analysis makes use of the relationship between scattering amplitudes and
the discrete spectrum of states in the finite volume lattice. Constraints on
the scattering amplitudes are provided by over one hundred energy levels
computed on two lattice volumes at various overall momenta and in several
irreducible representations of the relevant symmetry groups. The spectra follow
from variational analyses of matrices of correlations functions computed with
large bases of meson-meson operators. Calculations are performed with
degenerate light and strange quarks tuned to the physical strange quark mass so
that $m_\pi \sim 700$ MeV, ensuring that the $\rho$ is stable against strong
decay. This work demonstrates the successful application of techniques, opening
the door to calculations of scattering processes that incorporate the effects
of dynamically-coupled partial-waves, including those involving resonances or
bound states.
