Consider a general curved space-time with torsion. In the standard Einstein-Cartan-Kibble-Sciama theory (ECKS or ECSK), torsion is non-dynamical and doesn't propagates in free space. But a more general theory could allow torsion to be dynamical and propagates. In general, geodesics (the shortest or extremal length curves) and auto-parallels (the straightest curves) are different curves in space-time.
In classical general relativity (without torsion), test-particles without spin should follow geodesics. This is a statement of inertia motion.
But with torsion, what curve should a test-particle follow ? A geodesic or a auto-parallel curve ?
It can be shown that if torsion is totally antisymetric (which is just a special case), geodesics and auto-parallel curves are the same. But in general they aren't.
I feel that Newton's inertia principle is really about the straightest curves, and not the shortest (or extremal) curves.
Is there any indication, clue or argument, that the spineless test-particles should follow auto-parallel curves in space-time, instead of geodesics ? Would it be more natural in some way ?
If the auto-parallel curves are more fundamental (from an inertia point of view), then would it imply that the lagrangian method for fields is falling apart as a general principle, since it's (i.e. was) motivated by the inertia principle ?