Euler's disk is a toy described at http://www.eulersdisk.com. Aspects of its motion are modelled as an ideal disk rolling on a horizontal plane. In the final stages of Euler disk motions, the disk is nearly flat to the plane. Asymptotic approximations to the frequency of finite amplitude oscillations on steady (non-dissipative) rolling motions of the Euler disk are described. There are two different approximations which are appropriate in different limits. When the parameters are such that both apply, the formulae for the frequency agree: this appears to be new and simple. The material has been used in teaching; the teaching, and related, materials are available via the web [Keady, Math2200 Lecture Handouts, UWA]. References M. Batista. The nearly horizontally rolling of a thick disk on a rough plane. of a thick rigid disk Regular and Chaotic Dynamics 13 (4) (2008), 344--354. doi:10.1134/S1560354708040084 H. Caps, S. Dorbolo, S. Ponte, H. Croisier and N. Vandewalle. Rolling and slipping motion of Euler's disk. Physical Review E 69 (6) (2004), 056610. doi:10.1103/PhysRevE.69.056610 R. H. Cushman and J. J. Duistermaat. Nearly flat falling motions of the rolling disk. Regular and Chaotic Dynamics 11 (1) (2006) 31--60. doi:10.1070/RD2006v011n01ABEH000333 G. Keady. MATH2200 Lecture Handouts. The rolling of a disk on a horizontal plane. Part I, General; and Part II, Euler disk motions. http://school.maths.uwa.edu.au/ keady/papers.html H. K. Moffatt. Euler's disk and its finite-time singularity. Nature 404 (2000) 833. doi:10.1038/35009017 A. H. Nayfeh and D. T. Mook. Nonlinear Oscillations (Wiley: 1979). O. M. O'Reilly. The dynamics of rolling disks and sliding disks. Nonlinear Dynamics 10 (1996) 287--305. M. Srinivasan and A. Ruina. Rocking and rolling: a can that appears to rock might actually roll. Phys. Rev. E 78 (6) (2008) 066609. doi:10.1103/PhysRevE.78.066609 A. A. Stanislavsky and K. Weron. Nonlinear oscillations in the rolling motion of Euler's disk. Physica D 156 (2001) 247--259. doi:10.1016/S0167-2789(01)00281-0 J. L. Synge and B. A. Griffith. Principles of Mechanics. (McGraw-Hill, 3rd ed, 1959). W. T. Thomson. Introduction to Space Dynamics. (Dover: 1986 reprinting of Wiley 1963). R. Villanueva and M. Epstein. Vibrations of Euler's disk. Phys. Rev. E 71 (7) (2005) 066609. doi:10.1103/PhysRevE.71.066609 E. T. Whittaker. A treatise on the analytical dynamics of particles and rigid bodies. (CUP, 2nd ed, 1917).
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