New Phenomena in the Spatial Isosceles Three-Body Problem with Unequal Masses

This work studies periodic orbits as action minimizers in the spatial isosceles three-body problem with mass M = [1,m, 1]. In each period, the body with mass m moves up and down on a vertical line, while the other two bodies have the same mass 1, and rotate about this vertical line symmetrically. For given m > 0, such periodic orbits form a one-parameter set with a rotation angle θ as the parameter. Two new phenomena are found for this set. First, for each m > 0, this set of periodic orbits bifurcate from a circular Euler (central configuration) orbit to a Broucke (collision) orbit as θ increases from 0 to π. There exists a critical rotation angle θ0(m), where the orbit is a circular Euler orbit if 0 < θ ≤ θ0(m); a spatial orbit if θ0(m) < θ < π; and a Broucke (collision) orbit if θ = π. The exact formula of θ0(m) is numerically proved to be θ0(m) = π 4 4m+1 m+2. Second, oscillating behaviors occur at rotation angle θ = π/2 for all m ∈ [0.1, 3]. Actually, the orbit with θ = π/2 runs on its initial periodi...