Geodesic
Two Integrable Cases of a Ball Rolling over a Sphere in $\mathbb{R}^n$
Russian journal of nonlinear dynamics, Tome 15 (2019) no. 4, pp. 457-475.

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We consider the nonholonomic problem of rolling without slipping and twisting of a balanced ball over a fixed sphere in $\mathbb{R}^n$. By relating the system to a modified LR system, we prove that the problem always has an invariant measure. Moreover, this is a $SO(n)$-Chaplygin system that reduces to the cotangent bundle $T^*S^{n-1}$. We present two integrable cases. The first one is obtained for a special inertia operator that allows the Chaplygin Hamiltonization of the reduced system. In the second case, we consider the rigid body inertia operator $\mathbb I\omega=I\omega+\omega I$, ${I=diag(I_1,\ldots,I_n)}$ with a symmetry $I_1=I_2=\ldots=I_{r} \ne I_{r+1}=I_{r+2}=\ldots=I_n$. It is shown that general trajectories are quasi-periodic, while for $r\ne 1$, $n-1$ the Chaplygin reducing multiplier method does not apply.
Keywords: nonholonomic Chaplygin systems, invariant measure, integrability. 
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B. Gajić; B. Jovanović. Two Integrable Cases of a Ball Rolling over a Sphere in $\mathbb{R}^n$. Russian journal of nonlinear dynamics, Tome 15 (2019) no. 4, pp. 457-475. http://geodesic.mathdoc.fr/item/ND_2019_15_4_a5/

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