Second post-Newtonian Lagrangian dynamics of spinning compact binaries

The leading-order spin-orbit coupling is included in a post-Newtonian Lagrangian formulation of spinning compact binaries, which consists of the Newtonian term, first post-Newtonian (1PN) and 2PN non-spin terms and 2PN spin-spin coupling. This makes a 3PN spin-spin coupling occur in the derived Hamiltonian. The spin-spin couplings are mainly responsible for chaos in the Hamiltonians. However, the 3PN spin-spin Hamiltonian is small and has different signs, compared with the 2PN spin-spin Hamiltonian equivalent to the 2PN spin-spin Lagrangian. As a result, the probability of the occurrence of chaos in the Lagrangian formulation without the spin-orbit coupling is larger than that in the Lagrangian formulation with the spin-orbit coupling. Numerical evidences support the claim.