Geodesic
Stability of steady rotations in the non-holonomic Routh problem
Russian journal of nonlinear dynamics, Tome 2 (2006) no. 3, pp. 333-345

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We have discovered a new first integral in the problem of motion of a dynamically symmetric ball, subject to gravity, on the surface of a paraboloid. Using this integral, we have obtained conditions for stability (in the Lyapunov sense) of steady rotations of the ball in the upmost, downmost and saddle point.
Keywords: nonholonomic constraint, stationary rotations, stability. 
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     title = {Stability of steady rotations in the non-holonomic {Routh} problem},
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A. V. Borisov; A. A. Kilin; I. S. Mamaev. Stability of steady rotations in the non-holonomic Routh problem. Russian journal of nonlinear dynamics, Tome 2 (2006) no. 3, pp. 333-345. http://geodesic.mathdoc.fr/item/ND_2006_2_3_a5/