T.i.l that kinematical symmetries are symmetries generated by a Killing vector field $ \pi_*(X_q) $, $$ {\cal L}_{\pi_*(X_q)}g=0,$$ which is given by the pushforward $ \pi_*(X_q) $ of $X_q$, where $$ \pi: M=T^*Q \rightarrow Q $$ is a cotangent bundle on a Riemannian configuration space $(Q,g)$, $q$ is a conserved observable on $M$, and $ X_q $ is the associated Hamiltonian vector field on $M$ generated by $q$. Dynamical symmetries are any other symmetry. See e.g. Lecture 28 by Prof. Frederic Schuller.
What is the importance of this distinction? Are dynamical symmetries generated only by potentials in the action?
Important note: i'm only talking about classical mechanics, i don't know anything about the symplectic/phase space structure of Quantum theories yet.
