In section 5 of Landau and Lifshitz's Mechanics book, they show that the Lagrangian of a free particle must be proportional to its velocity squared, $\mathcal{L} = \alpha\mathbf{v}^2$ using only symmetry arguments, cf. e.g. this Phys.SE post. As the Lagrangian of two non-interacting systems must be independent, this means that the for a set of non-interacting particles, \begin{equation} \mathcal{L} = \frac{1}{2}\sum_\alpha m_\alpha\mathbf{v}_\alpha^2. \end{equation} To generalise this to a set of interacting particles, they state that `it is found' that the interaction between the particles can be described by adding a function $U(\mathbf{r}_1, \mathbf{r}_2,\ldots)$ to the above equation.
What do they mean by `it is found'? Do they mean that experiment shows this to be the case?
Is there any way to justify adding some (potential) function only by appealing to symmetry?
