Dieter Schmidt
Cincinnati, Ohio 452210025, USA
Department of Mathematical Sciences,University of Cincinnati
Publications:
Schmidt D. S., Vidal C.
Stability of the Planar Equilibrium Solutions of a Restricted $1+N$ Body Problem
2014, vol. 19, no. 5, pp. 533547
Abstract
We started our studies with a planar Eulerian restricted fourbody 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 $N=2$.

Cushman R., Dullin H. R., Hanßmann H., Schmidt D. S.
The $1:\pm2$ Resonance
2007, vol. 12, no. 6, pp. 642663
Abstract
On the linear level elliptic equilibria of Hamiltonian systems are mere superpositions of harmonic oscillators. Nonlinear terms can produce instability, if the ratio of frequencies is rational and the Hamiltonian is indefinite. In this paper we study the frequency ratio $\pm 1/2$ and its unfolding. In particular we show that for the indefinite case ($1:2$) the frequency ratio map in a neighborhood of the origin has a critical point, i.e. the twist condition is violated for one torus on every energy surface near the energy of the equilibrium. In contrast, we show that the frequency map itself is nondegenerate (i.e. the Kolmogorov nondegeneracy condition holds) for every torus in a neighborhood of the equilibrium point. As a by product of our analysis of the frequency map we obtain another proof of fractional monodromy in the $1:2$ resonance.

Schmidt D. S.
The stability of the Thomson heptagon
2004, vol. 9, no. 4, pp. 519528
Abstract
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.

Cabral H. E., Meyer K. R., Schmidt D. S.
Stability and bifurcations for the $N + 1$ vortex problem on the sphere
2003, vol. 8, no. 3, pp. 259282
Abstract
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 seminumerical form. As expected the new configurations look very similar to those which had been found previously for the planar case. 