After regular symplectic reduction (here described following the approaches in [W. J. Satzer jun., Indiana Univ. Math. J. 26, 951–976 (1977; Zbl 0358.70016); M. Kummer, Indiana Univ. Math. J. 30, 281–291 (1981; Zbl 0425.70019)]), the dynamics of a symmetric top can be described by a family of natural Hamiltonians on \(T^*\mathbb S^2\) with the symplectic form \(\omega+r \omega_{\mathbb S^2}\), where \(\omega\) is the canonical form, \(\omega_{\mathbb S^2}\) is the standard volume form of \(\mathbb S^2\) and \(r\in \mathbb R\) is the rate of spin of the top parametrizing the family.
In this article, a further symplectic reduction is introduced, defined by a momentum map which is singular, as well as the corresponding reduced spaces, in correspondence of the straight up configuration of the top. It is proved that a neighborhood of the straight up configuration in the reduced space is “isomorphic” to \(T^*(-1,1)/\mathbb Z^2\). The proof is made via Sikorski’s theory of differential spaces, or in general \(C^\infty\)-ringed spaces. The opportunity of this approach to singular reduction is discussed in Remark 3.28. The study of the dynamics is therefore reduced to the analysis of a regular potential \(W(u^2)\), with \(u\in [0,1)\). Lyapunov stability and bifurcations are then discussed. The approach is illustrated for the cases of Lagrange and Kirchhoff tops.