On Jacobi's decomposition of the motion of a heavy symmetrical top into the motions of two torque-free triaxial tops

According to Jacobi's decomposition theorem, the motion of a heavy symmetrical top can be expressed as a composite motion of two torque-free triaxial tops. It is natural to seek the connection between the dynamical constants of the three top motions (one heavy symmetrical top and two free triaxial tops). From the investigation, it is found that the formulas connecting these constants are simply projective transformations (fractional linear transformations). However, from the geometrical analysis on the decomposition theorem which will be stated below, it is found that a common constant term must be added to each of the above mentioned connecting formulas for one of the free triaxial tops.

The geometrical analysis is employed to obtain the clear physical contents of the theorem. It is found that this geometrical method gives not only the fundamental requirements for the theorem but also a clear physical picture showing how the composite motion, that is, the motions of two free triaxial tops, is related to the given motion of the heavy symmetrical top. Since an elaborate manipulation of the elliptic functions and the theta functions is involved in Jacobi's analysis, it is difficult to extract a clear picture of the theorem from Jacobi's analysis alone. Instantaneous positions of the three tops are specified by the three sets of Euler angles which can be computed using the connecting formulas mentioned above. However, the geometrical method gives some particular angular relations between the three sets of Euler angles and these relations greatly simplify the computation of the Euler angles and clarify the relative positions of the three tops.