classical mechanics - What are the assumptions behind the Lagrangian derivation of energy?

What are the assumptions behind the Lagrangian derivation of energy? I understand that we're searching for a function $L$ that describes a set of physics so that solving the energy minimization problem

$$\begin{array}{rcl} \arg\min\limits_{q} && \int_{t_1}^{t_2} L(q,\dot{q},t) dt\\ \textrm{st}&&q(t_1)=q_1\\ &&q(t_2)=q_2 \end{array}$$

determines the path of an a particle $q:[t_1,t_2]\rightarrow\mathbb{R}^3$ from time $t_1$ to time $t_2$. Eventually, we find that $L(q,\dot{q},t)=\frac{1}{2}m \dot{q}^2$. What's not clear to me are the assumptions behind the setup for the energy minimization problem. It looks like to me that there's an assumption about the lack of forces like gravity. Except, I know that this derivation can be used to eventually derive that $F=m\ddot{q}$, so we don't yet have a concept of force. Also, I know there's a corollary that shows that the eventual solution $q$ is such that $\dot{q}=c$ or $\ddot{q}=0$. I'm sure that makes sense in relation to the problem setup, which is what I'm trying to clarify. Finally, I do know that we need to assume

1. Time and space are homogenous


2. Space is isotropic


3. Galilean invariance

Anyway, I'm looking for the other assumptions and I'm pretty sure it has to do with lack of other forces or some kind of related concept.