What is the relationship between velocity-dependent potentials and non-Abelian gauge fields?

Question

My (limited) understanding of non-Abelian gauge fields is that they arise from the construction of a theory using a non-Abelian Lie group (as a generalization of the Abelian group underlying E&M) upon writing down symmetry-preserving Lagrangians and promoting the generators to field operators. In this way of thinking, I don't see any reason for the resulting interaction/potential terms to have any specific form, after all there can be many types of non-Abelian Lie groups.

However, I recently noticed an interesting footnote in some notes detailing the transition from Schrodinger-type Quantum Mechanics to the path integral formulation: "The generalization of velocity-dependent potentials to field theory involves the quantization of non-Abelian gauge fields."

I don't immediately see how this connection exists, does this only apply to a specific example of non-Abelian gauge fields, or am I missing something much simpler and more fundamental here? Even the Abelian gauge field theory of electromagnetism has a velocity dependent potential of sorts (current-gauge field coupling term), what does the author of these notes actually mean?

Edit: The sentence I am referring to above is footnote 3 on p.7. in "Path integrals in Quantum Field Theory by Sanjeev S. Seahra dated on May 11, 2000 from the University of Waterloo.

Answer 1

The Lagrangian for Abelian and non-Abelian gauge theories are merely field-theoretic examples of Lagrangians $L=\frac{1}{2}\dot{q}^2-V(q,\color{red}{\dot{q}})$ with velocity-dependent potentials rather than just $L=\frac{1}{2}\dot{q}^2-V(q)$.

It is unclear why the lecture notes (in footnote 3 in Section 2 about non-relativistic QM) chooses to specifically mention non-Abelian gauge theories, as there are many other examples of velocity-dependent potentials. Perhaps the lecture notes later wants to focus on non-Abelian gauge theories?

The lecture notes mentions footnote 3 in connection with time derivatives inside the path-integral (23), which are subtle because of the time-slicing prescription, cf. e.g. this, this & this Phys.SE posts.