Fairmont Queen Elizabeth (Montreal), December 7 - 10, 2012
In this year, Fukuda, Ozaki, Taniguchi and the present author proved this conjecture for the planar equal-mass three-body problem under the Newton potential($\alpha=1$) and the strong force potential($\alpha=2$). In this talk, I will review our work.
This is a joint work with Hiroshi Fukuda,
Hiroshi Ozaki and Tetsuya Taniguchi.
Using the discrete and rotational symmetries, we reduce the problem to a three degrees of freedom Hamiltonian system. In this setting, we show that the central configurations mentioned above are in fact relative equilibria and that $m_0$ marks a pitchfork/steady-state bifurcation. The value $m_c$ marks a Hamiltonian-Hopf bifurcation (i.e., a 1:-1 resonance).