  We study the small-mass (overdamped) limit of Langevin equations for a
particle in a potential and/or magnetic field with matrix-valued and
state-dependent drift and diffusion. We utilize a bootstrapping argument to
derive a hierarchy of approximate equations for the position degrees of freedom
that are able to achieve accuracy of order $m^{\ell/2}$ over compact time
intervals for any $\ell\in\mathbb{Z}^+$. This generalizes prior derivations of
the homogenized equation for the position degrees of freedom in the $m\to 0$
limit, which result in order $m^{1/2}$ approximations. Our results cover
bounded forces, for which we prove convergence in $L^p$ norms, and unbounded
forces, in which case we prove convergence in probability.
