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Let's have a disc which moves along a straight curve on a plane in a uniform gravitational field. There need to discover the stability of it's trajectory.

I represented the possible deviation of the center of masses position by the vertical oscillations around the axis, which is parallel to direction of movement and which contains a point of contact between the cylinder and the plane (look to the picture).

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So, with $X$-axis parallel to the direction of movement, $Y$-axis antiparallel to the gravitational field (look to the picture)

enter image description here

I got for the center of masses coordinates $$ y_{c} = R(cos(\varphi ) - 1), \quad z_{c} = Rsin(\varphi ), $$ and $$ L = \frac{m\dot {x}^{2}}{2} + \frac{I_{3}\omega_{z}^{2}}{2} + \frac{(I_{1} + mR^{2})\omega_{x}^{2}}{2} + mgRcos(\varphi ). $$ I need to associate $ \omega_{x, z}$ with $x_{c}, \varphi $. But I don't know how to use Poinsot formula, $$ \mathbf v_{surface} = \mathbf v_{c} + [\mathbf \omega \times \mathbf R_{cs}], $$

where $\mathbf R_{cs}$ is a vector from center of masses to contact point, for the case of rotation near the axes do not pass through one point. Can you help me?

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