I was thinking about the type of symmetry that the Lenz Vector generates and I came across this discussion: What symmetry causes the Runge-Lenz vector to be conserved?
At some point, it is mentioned that the Lenz Vector generates the following transformation under which the action is invariant: $$\vec{\delta r}=\vec{\epsilon}\times \vec{L}$$ yet it seems to me that this infinitesimal transformation is not the same as that generated if we take the Lenz Vector to be the generator of the transformation and calculate $$\delta{q_i}=\epsilon_j\frac{\partial A_j}{\partial p_i}=2p_iq_j−q_ip_j−q⋅p δij=(-\epsilon_{ijk}L_k-q⋅p δij+p_iq_j)\epsilon_j.$$ My claim is that in the second case we take a canonical transformation and use the action as generator of that canonical transformation, whereas in the first case we only take a transformation that leaves the action invariant. I am new to these topics so I would appreciate it if someone could elaborate a bit more on this.
