Lagrange equations of the first kind

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SUMMARY

The discussion centers on the application of Lagrange equations of the first kind in the context of motion under gravity. It establishes that the gravitational force acts in the z direction, remaining perpendicular to the motion occurring in the xy plane, thus having no impact on that motion. The focus shifts to the significance of the inflating and shrinking circle boundary of the intersection, which is crucial for understanding the dynamics involved.

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Homework Statement
A material point of mass m is moving in a constant gravitational field along the intersection of a sphere and a moving plane (z=Rsin(omega*t)). Find Lagrange's equations (first kind), the point's movement, and the constraint forces. Ignore friction and assume that the gravitational field is parallel to the "z" axis.
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Lagrange equations of first kind
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If I take the setting correctly, gravity force (z direction ) is always perpendicular to free degree motion of the body ( on xy plane ) so it makes no effect to the motion. Only the inflating-shrinking circle boundary of the intersection would matter.
 
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