Bifurcation diagram for the Kovalevskaya case on the Lie algebra \(\mathrm{so}(4)\) (Q1945193)

This paper concerns the construction of a bifurcation diagram for a one-parameter family of integrable systems on the pencil of Lie algebras \(\mathrm{so}(4)\), \(\mathrm{e}(3)\) and \(\mathrm{so}(3,1)\), with basis coordinates \((J_1,J_2,J_3,x_1,x_2,x_3)\) and for which the Lie-Poisson bracket on the Lie algebra family is \[ \{J_i,J_j\}=\epsilon_{ijk}J_k, \] \[ \{J_i,x_j\}=\epsilon_{ijk}x_k, \quad \{x_i,x_j\}=\chi\epsilon_{ijk}J_k. \] Here the parameter cases \(\chi>0\), \(\chi=0\), \(\chi<0\) correspond to so(4), e(3) and so(3,1), respectively. The Kovalevskaya Hamiltonian is defined by \( H=J_1^2+J_2^2+2J_3^2+2c_1x_1\), where \(c_1\) is an arbitrary constant; it admits a constant of the motion \[ K=(J_1^2+J_2^2-2c_1x_1+\chi c_1^2)^2+(2J_1J_2+2c_1x_2)^2, \] so the system is integrable on the 4-dimensional phase space \(M_{a,b}\) given by restricting the Casimirs \[ S_1=\chi(J_1^2+J_2^2+J_3^2)+(x_1^2+x_2^2+x_3^2), \quad S_2=x_1J_1+x_2J_2+x_3J_3, \] (necessarily constants of the motion) to \(S_1=a\), \(S_2=b\). The authors concern themselves with the case \(\chi>0\), so(4), where this space is compact. The ``momentum map'' for this system is the function \((J_1,J_2,J_3,x_1,x_2,x_3)\to (H,K)\) taking each point in the phase space \(M_{a,b}\) to the corresponding values of its constants of the motion \((h,k)\). Of interest are the ``critical points'' of this map, the points \((J_1,J_2,J_3,x_1,x_2,x_3)\) where the rank of the map is less than 2. A value \((h,k)\) at a critical point is called a ``critical value''. The set of all critical values \((h,k)\) is called the ``bifurcation diagram'' of the momentum map. It is typically a set of curves in the \((h,k)\)-plane. The authors compute explicitly the bifurcation diagram for the case \(\chi>0\), \(b=0\) and arbitrary \(a>0\). They quote Kharlamov's result for the more complicated \(\mathrm{e}(3)\) case, \(\chi=0\), and show that this result can be obtained as a limit of theirs as \(\chi\to 0\). Only the results are presented; the details of the proofs are omitted.