On the motion of a material point on a moving surface (Q1065575)

The authors study the following problem: a material point is constrained to move on the surface \(z-\lambda(t)(x^ 2+y^ 2)\) in a constant vertical field of gravity, with the friction force \(-\alpha(\dot x,\dot y,\dot z)=(-\alpha \dot x,-\alpha \dot y,-\alpha \dot z)\), where \(\lambda \in C^ 2({\mathbb{R}}_+,{\mathbb{R}}_+)\) is decreasing and convex, \(\alpha =const>0\). Sufficient conditions on \(\lambda\) and \({\ddot \lambda}\) are given for the stability and instability of equilibrium \(x=y=z=0\).