Dieter Schmidt

We study Hamiltonian systems with two degrees of freedom near an equilibrium point, when the linearized system is not semisimple. The invariants of the adjoint linear system determine the normal form of the full Hamiltonian system. For work on stability or bifurcation the problem is typically reduced to a semisimple (diagonalizable) case. Here we study the nilpotent cases directly by looking at the Poisson algebra generated by the polynomials of the linear system and its adjoint.

We started our studies with a planar Eulerian restricted four-body problem (ERFBP) where three masses move in circular orbits such that their configuration is always collinear. The fourth mass is small and does not influence the motion of the three primaries. In our model we assume that one of the primaries has mass 1 and is located at the origin and two masses of size $\mu$ rotate around it uniformly. The problem was studied in [3], where it was shown that there exist noncollinear equilibria, which are Lyapunov stable for small values of $\mu$. KAM theory is used to establish the stability of the equilibria. Our computations do not agree with those given in [3], although our conclusions are similar. The ERFBP is a special case of the $1+N$ restricted body problem with $N=2$. We are able to do the computations for any $N$ and find that the stability results are very similar to those for $N=2$. Since the $1+N$ body configuration can be stable when $N>6$, these results could be of more significance than for the case $N=2$.

In 1882 J. J. Thomson had claimed in his Adams prize essay "The motion of vortex rings" that a ring of seven vortices would be unstable. It was shown later that linear analysis can not decide stability in this case. In 1999 Cabral and Schmidt proved stability by calculating the higher order terms in the normal form of the Hamiltonian with the help of POLYPACK, a personal algebraic processor. The work is repeated here with the help of the more readily available computer algebra system MATHEMATICA.

The equations of motion for $N$ vortices on a sphere were derived by V.A.Bogomolov in 1977. References to related work can be found in the book by P.K.Newton. We use the equations of motion found there to discuss the stability of a ring of $N$ vortices of unit strength at the latitude $z$ together with a vortex of strength $\kappa$ at the north pole. The regions of stability are bounded by curves $\kappa = \kappa(z)$. These curves are computed explicitly for all values of $N$.
When the stability of a configuration changes, for example by varying the strength of the vortex at the north pole, bifurcations to new configurations are possible. We compute the bifurcation equations explicitly for $N = 2, 3$ and $4$. For larger values of $N$ the complexity of the formal computations becomes too great and we use a numerical value for the latitude instead. We thus derive the bifurcation equations in a semi-numerical form. As expected the new configurations look very similar to those which had been found previously for the planar case.