= Compound Atwood machine
A <compound Atwood machine> replaces one hanging <mass> by another <pulley> carrying two <masses>. Let the upper <mechanical string> carry <mass> $M$ opposite the moving <pulley>, and let the lower <mechanical string> carry <masses> $a,b$ at upward heights $x,y$. The length constraints on the <mechanical strings> give the height of $M$ as $-(x+y)/2$ up to a constant. Thus its <Lagrangian> is
$$
\mathcal L=\frac M8(\dot x+\dot y)^2+\frac a2\dot x^2+\frac b2\dot y^2-g\left[\left(a-\frac M2\right)x+\left(b-\frac M2\right)y\right].
$$
This follows by adding the three <kinetic energies> and subtracting their <gravitational potential energies>. Constant coordinate shifts annihilating the linear <potential energy> give conserved <generalized momentum> components by <Noether's theorem>.
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