Georgia Institute of Technology College of Sciences

The iteration theory for Lagrangian Maslov index is a very useful tool in studying the multiplicity of brake orbits of Hamiltonian systems. In this talk, we show how to use this theory to prove that there exist at least $n$ geometrically distinct brake orbits on every $C^2$ compact convex symmetric hypersurface in $\R^{2n}$ satisfying the reversible condition. As a consequence, we show that if the Hamiltonian function is convex and even, then Seifert conjecture of 1948 on the multiplicity of brake orbits holds for any positive integer $n$.