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Rolling and noslip bouncing in cylinders
1.  Department of Mathematics and Statistics, Mount Holyoke College, 50 College St, South Hadley, MA 01075, USA 
2.  Department of Mathematics, Tarleton State University, Box T0470, Stephenville, TX 76401, USA 
3.  Department of Mathematics and Statistics, Washington University, Campus Box 1146, St. Louis, MO 63130, USA 
We compare a classical nonholonomic system—a sphere rolling against the inner surface of a vertical cylinder under gravity—with certain discrete dynamical systems called noslip billiards in similar configurations. A feature of the former is that its height function is bounded and oscillates harmonically up and down. We investigate whether similar bounded behavior is observed in the noslip billiard counterpart. For circular cylinders in dimension $ 3 $, noslip billiards indeed have bounded orbits, and very closely approximate rolling motion, for a class of initial conditions we call transversal rolling impact. When this condition does not hold, trajectories undergo vertical oscillations superimposed to overall downward acceleration. Concerning different crosssections, we show that noslip billiards between two parallel hyperplanes in arbitrary dimensions are always bounded even under a constant force parallel to the plates; for general cylinders, when the orbit of the transverse system (a concept relying on a factorization of the motion into transversal and longitudinal components) has period two, the motion, under no forces, is generically not bounded. This commonly occurs in planar noslip billiards.
References:
[1] 
A. M. Bloch, Nonholonomic Mechanics and Control, vol. 24 of Interdisciplinary Applied Mathematics, 2nd edition, Springer, New York, 2015. doi: 10.1007/9781493930173. 
[2] 
A. V. Borisov, I. S. Mamaev and A. A. Kilin, Rolling of a ball on a surface. New integrals and hierarchy of dynamics, Regul. Chaotic Dyn., 7 (2002), 201219. doi: 10.1070/RD2002v007n02ABEH000205. 
[3] 
D. S. Broomhead and E. Gutkin, The dynamics of billiards with noslip collisions, Phys. D, 67 (1993), 188197. doi: 10.1016/01672789(93)90205F. 
[4] 
C. Cox and R. Feres, Noslip billiards in dimension two, in Dynamical Systems, Ergodic theory, and Probability: In Memory of Kolya Chernov, vol. 698 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2017, 91–110. doi: 10.1090/conm/698/14032. 
[5] 
C. Cox, R. Feres and H.K. Zhang, Stability of periodic orbits in noslip billiards, Nonlinearity, 31 (2018), 44434471. doi: 10.1088/13616544/aacc43. 
[6] 
C. Cox and R. Feres, Differential geometry of rigid bodies collisions and nonstandard billiards, Discrete Contin. Dyn. Syst., 36 (2016), 60656099. doi: 10.3934/dcds.2016065. 
[7] 
R. L. Garwin, Kinematics of an ultraelastic rough ball, American Journal of Physics, 37 (1969), 8892. 
[8] 
M. Gualtieri, T. Tokieda, L. AdvisGaete, B. Carry, E. Reffet and C. Guthmann, Golfer's dilemma, American Journal of Physics, 74 (2006), 497501. 
[9] 
H. Larralde, F. Leyvraz and C. MejíaMonasterio, Transport properties of a modified Lorentz gas, J. Statist. Phys., 113 (2003), 197231. doi: 10.1023/A:1025726905782. 
[10] 
J. M. Lee, Introduction to Smooth Manifolds, vol. 218 of Graduate Texts in Mathematics, SpringerVerlag, New York, 2003. doi: 10.1007/9780387217529. 
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C. MejíaMonasterio, H. Larralde and F. Leyvraz, Coupled normal heat and matter transport in a simple model system, Phys. Rev. Lett., 86 (2001), 54175420. 
[12] 
J. I. Neǐmark and N. A. Fufaev, Dynamics of Nonholonomic Systems, vol. 33 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1972. 
[13] 
M. P. Wojtkowski, The system of two spinning disks in the torus, Phys. D, 71 (1994), 430439. doi: 10.1016/01672789(94)900094. 
show all references
References:
[1] 
A. M. Bloch, Nonholonomic Mechanics and Control, vol. 24 of Interdisciplinary Applied Mathematics, 2nd edition, Springer, New York, 2015. doi: 10.1007/9781493930173. 
[2] 
A. V. Borisov, I. S. Mamaev and A. A. Kilin, Rolling of a ball on a surface. New integrals and hierarchy of dynamics, Regul. Chaotic Dyn., 7 (2002), 201219. doi: 10.1070/RD2002v007n02ABEH000205. 
[3] 
D. S. Broomhead and E. Gutkin, The dynamics of billiards with noslip collisions, Phys. D, 67 (1993), 188197. doi: 10.1016/01672789(93)90205F. 
[4] 
C. Cox and R. Feres, Noslip billiards in dimension two, in Dynamical Systems, Ergodic theory, and Probability: In Memory of Kolya Chernov, vol. 698 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2017, 91–110. doi: 10.1090/conm/698/14032. 
[5] 
C. Cox, R. Feres and H.K. Zhang, Stability of periodic orbits in noslip billiards, Nonlinearity, 31 (2018), 44434471. doi: 10.1088/13616544/aacc43. 
[6] 
C. Cox and R. Feres, Differential geometry of rigid bodies collisions and nonstandard billiards, Discrete Contin. Dyn. Syst., 36 (2016), 60656099. doi: 10.3934/dcds.2016065. 
[7] 
R. L. Garwin, Kinematics of an ultraelastic rough ball, American Journal of Physics, 37 (1969), 8892. 
[8] 
M. Gualtieri, T. Tokieda, L. AdvisGaete, B. Carry, E. Reffet and C. Guthmann, Golfer's dilemma, American Journal of Physics, 74 (2006), 497501. 
[9] 
H. Larralde, F. Leyvraz and C. MejíaMonasterio, Transport properties of a modified Lorentz gas, J. Statist. Phys., 113 (2003), 197231. doi: 10.1023/A:1025726905782. 
[10] 
J. M. Lee, Introduction to Smooth Manifolds, vol. 218 of Graduate Texts in Mathematics, SpringerVerlag, New York, 2003. doi: 10.1007/9780387217529. 
[11] 
C. MejíaMonasterio, H. Larralde and F. Leyvraz, Coupled normal heat and matter transport in a simple model system, Phys. Rev. Lett., 86 (2001), 54175420. 
[12] 
J. I. Neǐmark and N. A. Fufaev, Dynamics of Nonholonomic Systems, vol. 33 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1972. 
[13] 
M. P. Wojtkowski, The system of two spinning disks in the torus, Phys. D, 71 (1994), 430439. doi: 10.1016/01672789(94)900094. 
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