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No abstract

Abstract. When a Hamiltonian system has a "Kinetic + Potential" structure, the resulting flow is locally a geodesic flow. But there may be singularities of the geodesic structure; so the local structure does not always imply that the flow is globally a geodesic flow. In order for a flow to be a geodesic flow, the underlying manifold must have the structure of a unit tangent bundle. We develop homological conditions for a manifold to have such a structure.We apply these criteria to several classical examples: a particle in a potential well, the double spherical pendulum, the Kovalevskaya top, and the N -body problem. We show that the flow of the reduced planar N -body problem and the reduced spatial 3-body are never geodesic flows except when the angular momentum is zero and the energy is positive.

No abstract

We study the isosceles three body problem with fixed symmetry line for arbitrary masses, as a subsystem of the N-body problem. Our goal is to construct minimizing noncollision periodic orbits using a symmetric variational method with a finite order symmetry group. The solution of this variational problem gives existence of noncollision periodic orbits which realize certain symbolic sequences of rotations and oscillations in the isosceles three body problem for any choice of the mass ratio. The Maslov index for these periodic orbits is used to prove the main result, Theorem 4.1, which states that the minimizing curves in the three dimensional reduced energy momentum surface can be extended to periodic curves which are generically hyperbolic. This reminds one of a theorem of Poincaré [8], concerning minimizing periodic geodesics on orientable 2D surfaces. The results in this paper are novel in two directions: in addition to the higher dimensional setting, the minimization in the current problem is over a symmetry class, rather than a loop space.