科研之友
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摘要
In general, there are differences between Lagrangian and Hamiltonian approaches at the same post-Newtonian (PN) order in a coordinate system under a coordinate gauge. They are from truncation of higher-order PN terms. They do not affect qualitative and quantitative results of the two approaches for a weak gravitational system such as the Solar System. Nevertheless, they may make the two approaches have somewhat or completely different dynamical qualitative features of integrability and nonintegrability (or order and chaos) for a strong gravitational field. Even if the two approaches have the same qualitative features, they have different quantitative results when the distances among compact objects are appropriately small. For a relativistic circular restricted three-body problem with the 1PN contribution from the circular motion of the primaries, although the two 1PN Lagrangian and Hamiltonian approaches are nonintegrable, their dynamics are somewhat nonequivalent for a small quantity of separations between the primaries when the initial conditions and other parameters are given. Particularly for comparable mass compact binaries with two arbitrary spins and spin effects restricted to the leading-order spin-orbit interaction, as an important example of extremely strong gravitational fields, the 2PN Arnowitt-Deser-Misner Lagrangian formulation is always nonintegrable and can be chaotic under some appropriate conditions because its equivalent higher-order PN canonical Hamiltonian includes many spin-spin couplings resulting in the absence of a fifth integral in a ten-dimensional phase space and is not integrable. However, the 2PN Arnowitt-Deser-Misner Hamiltonian is integrable and nonchaotic due to the presence of five constants of motion in the ten-dimensional phase space.
