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- #1

The Lagrangian, $\mathcal{L} = T - U$, is the kinetic minus the potential energy.

\begin{alignat*}{3}

\mathcal{L} & = & T - U\\

& = & \frac{1}{2}m\left(\dot{x}_1^2 + \dot{x}_2^2\right) - \frac{1}{2}k(x_1 - x_2 - \ell)^2

\end{alignat*}

Rewrite $\mathcal{L}$ in terms of the new variables $X = \frac{1}{2}(x_1 + x_2)$ (the CM position) and $x$ (the extension), and write down the two Lagrange equations for $X$ and $x$.

The solution has:

Let $x = x_1 - x_2 - \ell$

Where did this piece come from (below). I see that adding them together produces $X$.

\begin{alignat}{3}

x_1 & = & X + \frac{x}{2} + \frac{\ell}{2}\\

x_2 & = & X - \frac{x}{2} - \frac{\ell}{2}

\end{alignat}