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So, the problem is this: equations of motion of a point mass [tex]m[/tex] on the end of a light rigid rod (length [tex]l[/tex]), the other end of which is attached to a light, frictionless carriage, constrained to move only in the horizontal direction.

So, the problem has two degrees of freedom: [tex]x[/tex] and [tex]\theta[/tex].

By my reckoning, the position vector of the pendulum bob, counting angle clockwise from the upwards vertical, is:

[tex] \mathbf{r}=(x + l \sin\theta, l \cos\theta) [/tex]

This gives a Lagrangian of:

[tex] L = \frac{m \dot{\mathbf{r}}^2}{2} - m g \cos{\theta} = \frac{m \dot{x}^2}{2} + \frac{m l^2 \dot{\theta}^2}{2} + m \dot{x} \dot{\theta} l \cos{\theta} - m g \cos{\theta}[/tex]

Now, the Euler-Lagrange equations end up as

[tex]\ddot{x} + \ddot{\theta} l \cos{\theta} - \dot{\theta}^2 l \sin{\theta} = 0[/tex]

[tex]\ddot{\theta} + \ddot{x} \cos\theta / l - g \sin\theta/l=0[/tex]

which in turn reduce to:

[tex]\ddot{x} = \frac{\dot{\theta}^2 l - g \cos \theta }{\sin\theta}[/tex]

[tex]\ddot{\theta} = \frac{g / l - \dot{\theta}^2 \cos\theta}{\sin\theta}[/tex]

Do you spot the problem yet? There's a nasty singularity when [tex]\theta \to 0[/tex].

Can anyone tell me what's going wrong here?