I all, I am struggling to grasp the notion of gauge invariant when talking about an object like the canonical momenta $\frac{\partial L}{\partial \dot{q}_i}$ or kinetic momenta $m\dot{q}_i$.
I am very comfortable with gauge theory in the field theory context, starting with a Lagrangian, requiring its invariance under a local symmetry, partial $\rightarrow$ covariant derivatives and the corresponding transformation of the gauge field connections,etc. But showing an object like a momentum is gauge invariant is new for me.
I am looking into the difference between the canonical and kinetic momenta in the case of a charged particle in an EM field, described by the standard Lagrangian
\begin{equation} L = \frac{1}{2} m\dot{r}^2 - q \phi + q \dot{r}\cdot A \end{equation}
The canonical momenta are $\vec{p}_c=m\dot{\vec{r}} +q\vec{A}$ and the kinetic are just $\vec{p}_k=m\dot{\vec{r}}$.
I am trying to figure out how to explicit show that $\vec{p}_c$ are not gauge-invariant (presumably under the U(1) symmetry of EM?) whereas $\vec{p}_k$ are. I know this to be the case by ACuriousMind's answer here, Emilio Pisanty's answer here, and the following section of Wikipedia's minimal coupling article.
Any tips are appreciated! :)