Ken Meyer

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 present a procedure for the normalization of perturbed Keplerian problems in $n$ dimensions based onMoser regularization of the Kepler problem and the invariants associated to the reduction process. The approach allows us not only to circumvent the problems introduced by certain classical variables used in the normalization of this kind of problems, but also to do both the normalization and reduction in one step. The technique is introduced for any dimensions and is illustrated for $n = 2, 3$ by relating Moser coordinates with Delaunay-like variables. The theory is applied to the spatial circular restricted three-body problem for the study of the existence of periodic and quasi-periodic solutions of rectilinear type.

We give a fairly simple geometric proof that an equilibrium point of a Hamiltonian system of two degrees of freedom is Liapunov stable in a degenerate case. That is the $1 : −1$ resonance case where the linearized system has double pure imaginary eigenvalues $\pm i \omega$, $\omega \ne 0$ and the Hamiltonian is indefinite. The linear system is weakly unstable, but if a particular coefficient in the normalized Hamiltonian is of the correct sign then Moser’s invariant curve theorem can be applied to show that the equilibrium point is encased in invariant tori and thus it is stable.
This result implies the stability of the Lagrange equilateral triangle libration points, $\mathcal{L}_4$ and $\mathcal{L}_5$, in the planar circular restricted three-body problem when the mass ratio parameter is equal to $\mu_R$, the critical value of Routh.

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.