The vanishing twist in the restricted three-body problem

• We demonstrate the existence of twistless tori in the planar circular restricted three-body problem.
• The associated reconnection bifurcations and meandering curves are found.
• The Birkhoff normal form at the Lagrangian triangular equilibrium is calculated to eighth order.
• Numerically integrated Poincare sections agree well with predictions made from the truncated integrable normal form.

This paper demonstrates the existence of twistless tori and the associated reconnection bifurcations and meandering curves in the planar circular restricted three-body problem. Near the Lagrangian equilibrium L4L4 a twistless torus is created near the tripling bifurcation of the short period family. Decreasing the mass ratio leads to twistless bifurcations which are particularly prominent for rotation numbers 3/103/10 and 2/72/7. This scenario is studied by numerically integrating the regularised Hamiltonian flow, and finding rotation numbers of invariant curves in a two-dimensional Poincaré map.To corroborate the numerical results the Birkhoff normal form at L4L4 is calculated to eighth order. Truncating at this order gives an integrable system, and the rotation numbers obtained from the Birkhoff normal form agree well with the numerical results. A global overview for the mass ratio μ∈(μ4,μ3)μ∈(μ4,μ3) is presented by showing lines of constant energy and constant rotation number in action space.