# Deriving the Lagrangian of a set of interacting particles only from symmetry

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.

1. What do they mean by it is found'? Do they mean that experiment shows this to be the case?

2. Is there any way to justify adding some (potential) function only by appealing to symmetry?

• Right off the bat, I'd wonder how they account for particles in magnetic fields. Perhaps that's thought of as an external agency rather than an interaction between the particles themselves, though. – Michael Seifert Sep 9 '19 at 17:27