A mechanics system consisting of three mass points on sphere S 2 is considered. The configuration space of the system is a fibre bundle over S 2 . It is proved that first Chern class of the bundle is -2 c 1...A mechanics system consisting of three mass points on sphere S 2 is considered. The configuration space of the system is a fibre bundle over S 2 . It is proved that first Chern class of the bundle is -2 c 1(γ) where γ is the canonical line bundle over the complex projective space CP 1=S 2 , which shows the bundle is non trivial. The information about the first Chern class makes the cohomology groups and homotopy groups of the configuration space worked out. In addition the effects of these topolo gical properties of the configuration space on the behavior in large scale of the system, as the number of equilibrium positions, periodic orbits and reduced phase space, are discussed.展开更多
文摘A mechanics system consisting of three mass points on sphere S 2 is considered. The configuration space of the system is a fibre bundle over S 2 . It is proved that first Chern class of the bundle is -2 c 1(γ) where γ is the canonical line bundle over the complex projective space CP 1=S 2 , which shows the bundle is non trivial. The information about the first Chern class makes the cohomology groups and homotopy groups of the configuration space worked out. In addition the effects of these topolo gical properties of the configuration space on the behavior in large scale of the system, as the number of equilibrium positions, periodic orbits and reduced phase space, are discussed.
