Lifted transformations on the tangent bundle, and symmetries of particle motion

We define affine transport lifts on the tangent bundle by associating a transport rule for tangent vectors with a vector field on the base manifold. Our aim is to develop tools for the study of kinetic/dynamic symmetries in particle motion. The new lift unifies and generalizes all the various existing lifted vector fields, with clear geometric interpretations. In particular, this includes the important but little-known "matter symmetries" of relativistic kinetic theory. We find the affine dynamical symmetries of general relativistic charged particle motion, and we compare this to previous results and to the alternative concept of "matter symmetry."