The characteristic system for the Euler-Poisson equations (Q2735891)

The author considers the Euler-Poisson equations of moving a solid NEWLINE\[NEWLINEA\dot p= Ap\times p+\gamma\times r, \qquad \dot\gamma= \gamma\times p, \tag{1}NEWLINE\]NEWLINE with \(p,\gamma\in \mathbb{C}^3\), \(A= \text{diag} (A_1, A_2, A_3)\), \(A_i> 0\), \(r\in \mathbb{R}^3\). The system (1) is transformed into: NEWLINE\[NEWLINEA\dot{\widetilde p}= A\widetilde p\times \widetilde p+\widetilde\gamma\times r+ A\widetilde p, \qquad \dot{\widetilde\gamma}= \widetilde\gamma\times \widetilde p+2\widetilde\gamma, \tag{2}NEWLINE\]NEWLINE where the solutions to (1) and (2) are connected by NEWLINE\[NEWLINEp(t)= \frac{1}{t-t_*} \widetilde p(\log (t-t_*)), \qquad \gamma= \frac{1}{(t-t_*)^2} \widetilde \gamma(\log(t-t_*)). \tag{3}NEWLINE\]NEWLINE The solutions \(p(t)\), \(\gamma(t)\) to (1) do not have a singularity at the point \(t_*\) if and only if the corresponding solution to (2) has the asymptotic behaviour \(\widetilde p(\tau)\sim \widetilde p_0\exp \tau\), \(\widetilde\gamma(\tau)\sim \widetilde \gamma_0\exp 2\tau\), when \(\operatorname {Re}\tau\to -\infty\). NEWLINENEWLINENEWLINETo find the singular points of (2), one has to solve the characteristic system \(A\dot{\widetilde p}_0= 0\) and \(\dot{\widetilde\gamma}_0=0\). That is the subject of this paper. The main result is the following: if NEWLINE\[NEWLINE(A_1- A_2)(A_2- A_3)(A_3 -A_1) r_1 r_2 r_3 = 0,NEWLINE\]NEWLINE then the solution to the characteristic system is found effectively; in another case, the solutions may be found, if the roots of some polynomial of 8th order are known. It is interesting that the conditions of the Euler, Lagrange, Kovalevskaya and Grioli cases appear in the solving of characteristic systems in partial cases.