Nb. This is a problem taken from a past exam paper.
Problem Statement Consider the 3-particle Calogero-Moser system defined by the Lagrangian $$\mathcal{L}=\frac{1}{2}m\sum_{i=1}^3\dot x_i^2-\frac{1}{2}g\sum_{i=1,j=1,\\ \ \ \ \ j\neq i}^3\frac{1}{(x_i-x_j)^2}$$ Derive the equations of motion of the system. Now consider the following continuous transformation $$x_i(t)\to\tilde x_i(t)=\frac{1}{s}x_i(s^2t)$$ Take $s=1+\epsilon$ and find the corresponding infinitesimal transformation $$\zeta_i(x,\dot x,t)$$ and show that under this infinitesimal transformation the Lagrangian transforms as $$\mathcal{L}\to\tilde{\mathcal{L}}=\mathcal{L}+\epsilon\frac{d\Lambda}{dt}$$ Solution Attempt
Using the Euler-Lagrange equation I found the equations of motion to be: $$\left(\frac{2}{(x_1-x_2)^3}+\frac{2}{(x_1-x_3)^3}\right)+m\ddot x_1=0 \ \ \ \ (1)$$ $$\left(\frac{2}{(x_1-x_2)^3}-\frac{2}{(x_2-x_3)^3}\right)-m\ddot x_2=0 \ \ \ \ (2)$$ $$\left(\frac{2}{(x_1-x_3)^3}+\frac{2}{(x_2-x_3)^3}\right)-m\ddot x_3=0 \ \ \ \ (3)$$ I am not entirely sure what the question is asking for after this. I have considered changing all of the $x_i,\dot x_i$ terms to $\tilde x_i,\dot{\tilde{x_i}}$ and proceeding with a substitution of $\tilde x_i(t)=\frac{1}{s}x_i(s^2t)$ but I feel as though this is not the correct method to proceed with. If someone could provide an outline of what the question is asking for and how to proceed that would be great.