Can we measure stellar accelerations? Many people concern themselves with understanding the Galactic potential. To me, the most obvious way is to measure the acceleration of a star, as this is directly related to the potential. So, one could measure a LOS velocity (or proper motion) at different years and measure its change. I understand that orbital timescales of stars are huge however, so can this be done? If not, how far (order of magnitude) away are we from having the required uncertainties for achieving this?
A commenter points out that accelerations in the galactic gravitational potential are extremely small:


  The Sun's galactic orbital speed is roughly 200 km/s and its distance from the galactic centre is around 8 kpc. Using $a=v^2/r$ that gives an acceleration of roughly $1.6×10^{−10}\rm\,m/s^2$.


A century is only $\pi×10^9$ seconds, so in a hundred years the $\Delta\vec v$ for the Sun and its neighbors is only half a meter per second --- and that's mostly a difference of direction, not magnitude.  We can't measure three-dimensional stellar motions to a precision of a half-meter per second, and the career of a professional astronomer is usually shorter than a century.