## 15. ÖMG-Kongress

Jahrestagung der Deutschen Mathematikervereinigung

#### 16. bis 22. September 2001 in Wien

**Hauptvortrag**

Mittwoch, 19. September 2001, 11.30-12.30, Audimax der Universität Wien

**Singular reduction and Hamiltonian dynamics**
**Tudor Ratiu**, **Ecole polytechnique fédérale de Lausanne**

This talk will present generalizations of the Weinstein-Moser
theorem about the existence of periodic orbits around stable equilibria
of Hamiltonian systems. The results discussed deal with symmetric
Hamiltonian systems and address the existence and the lower estimates on
the number of bifurcating relative equlibria and relative periodic
orbits emanating from equlibria and relative equlibria. The case of
formally unstable critical elements will also be discussed. Techniques
of singular reduction are used to obtain some of these results. Special
emphasis will be put on the so called ``reconstruction equations'' which
mirror the equations of motion in the singularly reduced spaces but
expressed on the original smooth manifold. The results are obtained by
combining techniques of singular reduction, normal forms, and
topological estimates. If time permits, the symmetric Hamiltonian Hopf
bifurcation will be shown to be obtainable by similar methods which
brings it closer in formulation to the classical dissipative Hopf
bifurcation theorem.

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