= Frictionless sliding ladder
For a uniform slender ladder of length $2l$ sliding while touching a smooth wall and floor, use its angle $\theta$ from the vertical. Its <centre of mass> is $(l\sin\theta,l\cos\theta)$ and its central <moment of inertia> is $ml^2/3$. Hence the <Lagrangian> is $L=2ml^2\dot\theta^2/3-mgl\cos\theta$, and its <Euler-Lagrange equation> is $\ddot\theta=3g\sin\theta/(4l)$. The contacts are physically possible only while their <normal forces> are nonnegative. If released from rest at $\theta_0$, <energy conservation> gives $\dot\theta^2=3g(\cos\theta_0-\cos\theta)/(2l)$, and the wall force becomes $3mg\sin\theta(3\cos\theta-2\cos\theta_0)/4$. Contact is lost at $\cos\theta=2\cos\theta_0/3$, after which the constrained equation no longer describes the motion.
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