Geodesic
Note on a ball rolling over a sphere: integrable Chaplygin system with an invariant measure without Chaplygin Hamiltonization
Theoretical and applied mechanics, Tome 46 (2019) no. 1.

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In this note we consider the nonholonomic problem of rolling without slipping and twisting of an $n$-dimensional balanced ball over a fixed sphere. This is a $SO(n)$--Chaplygin system with an invariant measure that reduces to the cotangent bundle $T^*S^{n-1}$. For the rigid body inertia operator $\mathbb I\omega=I\omega+\omega I$, $I=\operatorname{diag}(I_1,\dots,I_n)$ with a symmetry $I_1=I_2=\dots=I_{r} \ne I_{r+1}=I_{r+2}=\dots=I_n$, we prove that the reduced system is integrable, 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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     author = {Bo\v{z}idar Jovanovi\'c},
     title = {Note on a ball rolling over a sphere: integrable {Chaplygin} system with an invariant measure without {Chaplygin} {Hamiltonization}},
     journal = {Theoretical and applied mechanics},
     pages = {97 - 108},
     publisher = {mathdoc},
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     year = {2019},
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Božidar Jovanović. Note on a ball rolling over a sphere: integrable Chaplygin system with an invariant measure without Chaplygin Hamiltonization. Theoretical and applied mechanics, Tome 46 (2019) no. 1. http://geodesic.mathdoc.fr/item/TAM_2019_46_1_a7/