A top is a material body that is symmetrical about an axis and terminates in a sharp point (called the apex, top, or vertex) at one end of the axis (Whittaker 1944, p. 155). The problem of the motion under gravity of a top is not, in general, soluble exactly in terms of quadratures. For many years, (1) the top having its fixed point the same as its center of gravity (so that gravity does not influence the motion) and (2) the top in which the fixed point and center of gravity lie on an axis of symmetry were the only known integrable cases. Kovalevskaya (1888) subsequently found an amazing analytical solution for the case of a top in which two of the principal moments of inertia at the fixed point are equal and double the third and when the center of gravity is in the plane of the equal moments of inertia (Whittaker 1944, p. 164).
Euler (1758) was the first to treat problem (1), with Rueb (1834) subsequently applying elliptic functions and Jacobi
(1849) completing the solution (Whittaker 1944, p. 144), which is most easily expressed in terms of Jacobi
elliptic functions. Eric Weisstein's World of Math Lagrange (1888-1889) solved problem (2) completely (Whittaker 1944, p. 156), and the solution
involves Weierstrass elliptic, Eric Weisstein's World of Math sigma, Eric Weisstein's World of Math and zeta functions Eric Weisstein's World of Math
Gyroscope, Kovalevskaya Top, Tippe Top
References
Audin, M. Spinning Tops: A Course on Integrable Systems. New York: Cambridge University Press, 1996.
Crabtree, H. An Elementary Treatment of the Theory of Spinning Tops and Gyroscopic Motion. London: Longmans, Green and Co., 1909.
Euler, L. "Du mouvement de rotation des corps solides autour d'un axe variable." Mém. de l'Acad. Sci. Berlin 14, 154-193, 1758.
Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, p. 214, 1980.
Golubev, V. V. Lectures on the Integration of the Equations of Motion of a Rigid Body about a Fixed Point. National Science Foundation, 1953.
Jacobi, J. "Sur la rotation d'un corps." Comptes Rendus de l'Acad. Sci. 29, 97-106, 1849.
Jacobi, J. "Sur la rotation d'un corps." J. reine angew. Math. 39, 293-350, 1849.
Kettelkamp, L. Spinning Tops. New York: Morrow, 1966.
Klein, F. The Mathematical Theory of the Top. New York: Scribner's, 1897.
Lagrange, J. L. Mécanique analytique, 4. ed., 2 vols. Paris: Gauthier-Villars, 1888-1889. Reprinted in Oeuvres, vol. 12, p. 251.
Perry, J. Spinning Tops and Gyroscopic Motion. London: Sheldon Press, 1929.
Rueb, A. S. "Specimen inaugurale de motu gyratorio corporis rigidi nulla vi accelatrici sollicitati auct." In Roterodamensi, Trajecti ad Rhenum. Utrecht, Netherlands, 1834.
Sierpinski, W. Congruence of Sets and Other Monographs: On the Congruence of Sets and Their Equivalence by Finite Decomposition: The Mathematical Theory of the Top. New York: Chelsea, 1960. Whittaker, E. T. A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: With an Introduction to the Problem of Three Bodies. New York: Dover, 1944.
