I have recently been introduced to Lagrangian mechanics. My previous exposure to Lagrangian math has been in the form of optimizing constrained functions using Lagrange multipliers.
I get the math behind the Euler-Lagrange equations. I understand the proof behind conservation of energy using these equations assuming time-translation invariance. I also believe I understand that symmetries will always result in some conserved quantity. No challenges there.
But to my untrained eye, it appears there are a few assumptions we make in the process, and I'm having some trouble understanding why these assumptions are true. Or, maybe my assumptions are wrong (in which case, I don't get the math after all)?
We appear to know that the principle of stationary action is true for the universe. E.g. there is an excellent answer here about why the Principle of Stationary action is true. I am convinced.
We define Kinetic energy of the system to be to be $T = \sum f(\mathcal{P}_n(\dot{q}))$ where $\mathcal{P}_n$ is a polynomial of some degree.
We define $V(q)$ to be the potential energy of the system.
We assume the system is time-translation invariant.
We define Lagrangian to be $L(\dot{q}, q) = T(\dot{q})-V(q)$.
Questions:
Why is T only a function of $\dot {q}$? How do we know for sure?
Why is V only a function of $q$? How do we know for sure?
I've been trying to understand why these assumptions are true for a few days now, and I find myself going in circles. Can someone give me an intuition (or references) for why these assumptions are true?