Discussion on Perturbation to Weak Noether Symmetry and Adiabatic Invariants for Lagrange Systems

We study a new symmetric perturbation, i.e. weakly Noether symmetric perturbation (WNSP). The criterion and definition of WNSP and Noether symmetric perturbation (NSP) are given. A theorem between WNSP and adiabatic invariants is established. It is concluded that WNSP is different from NSP, the sufficient condition when WNSP is NSP can be presented, and the
former is broader. We apply our results to the planar Kepler problem.