Revisiting the Spinning Top

(Abstract) This paper revisits the problem of a spinning top in a uniform gravitational field when one point on the symmetry axis is fixed in space. It is an instructive and synthetic work of which the theoretical part includes all necessary issues to formulate the full differential equations governing the general motion of the spinning top under arbitrary initial conditions. Both Euler and Lagrange formulations are discussed. Moreover, closed form analytical solutions are derived for the regular precession and the nutation. The numerical integration of the equations was achieved using the standard Runge-Kutta scheme ODE45 available in MATLAB®, which was initially applied to the totality of Euler's equations and then to Lagrange's equations. Also, in house RK2 and RK4 Runge-Kutta as well as Crank-Nicolson schemes were applied in conjunction with the constraint for energy conservation. The quality of the numerical solution was evaluated by testing the conservation of total energy as well as angular momenta in the form of residuals in the corresponding Euler's dynamic equations.

[1]  Sophie Kowalevski,et al.  Sur le probleme de la rotation d'un corps solide autour d'un point fixe , 1889 .

[2]  Sophie Kowalevski.  Sur une propriété du système d'équations différentielles qui définit la rotation d'un corps solide autour d'un point fixe , 1890 .

[3]  Harold Crabtree.  An elementary treatment of the theory of spinning tops and gyroscopic motion , 1909 .

[4]  L. Brand,et al.  Vector and tensor analysis , 1947 .

[5]  Saul Gorn.  The Automatic Analysis and Control of Computing Errors , 1954 .

[6]  John Perry,et al.  Spinning Tops And Gyroscopic Motions , 1957 .

[7]  V. Golubev.  Lectures on integration of the equations of motion of a rigid body about a fixed point , 1960 .

[8]  An Elementary Introduction to Elliptic Functions Based on the Theory of Nutation , 1961 .

[9]  Theoretical mechanics : a short course , 1967 .

[10]  D. Mcgill,et al.  Stability regions for an unsymmetrical rigid body spinning about an axis through a fixed point , 1975 .

[11]  T. Ratiu,et al.  The Lagrange rigid body motion , 1982 .

[12]  Roger Cooke,et al.  The mathematics of Sonya Kovalevskaya , 1984 .

[13]  J. C. Simo,et al.  Unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum , 1991 .

[14]  J. C. Simo,et al.  Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics , 1992 .

[15]  A. C. Or,et al.  The Dynamics of a Tippe Top , 1994, SIAM J. Appl. Math..

[16]  Michèle Audin,et al.  Spinning Tops: A Course on Integrable Systems , 1996 .

[17]  Arne Marthinsen,et al.  Runge-Kutta Methods Adapted to Manifolds and Based on Rigid Frames , 1999 .

[18]  Alain Goriely,et al.  Kovalevskaya rods and Kovalevskaya waves , 2000 .

[19]  Zdzislaw Jackiewicz,et al.  Construction of Runge–Kutta methods of Crouch–Grossman type of high order , 2000, Adv. Comput. Math..

[20]  K. Engø,et al.  A Note on the Numerical Solution of the Heavy Top Equations , 2001 .

[21]  Richard Cushman,et al.  Global Aspects of Classical Integrable Systems , 2004 .

[22]  Jerrold E. Marsden,et al.  Tippe Top Inversion as a Dissipation-Induced Instability , 2004, SIAM J. Appl. Dyn. Syst..

[23]  M. Romano.  Exact analytic solutions for the rotation of an axially symmetric rigid body subjected to a constant torque , 2008, 1204.3419.

[24]  Roman Kozlov.  High-order conservative discretizations for some cases of the rigid body motion , 2008 .

[25]  Pragya Shukla,et al.  Slow manifold and Hannay angle in the spinning top , 2010 .

[26]  E. Whittaker.  A Treatise on the Analytical Dynamics of Particles and Rigid Bodies; With an Introduction to the Problem of Three Bodies , 2012 .