9.2 Lagrangian and Hamiltonian formulations

The conservative part of the equations of motion in harmonic coordinates (131) is derivable from a generalized Lagrangian, depending not only on the positions and velocities of the bodies, but also on their accelerations: and . As shown by Damour and Deruelle [55], the accelerations in the harmonic-coordinates Lagrangian occur already from the 2PN order. This fact is in accordance with a general result of Martin and Sanz [100] that -body equations of motion cannot be derived from an ordinary Lagrangian beyond the 1PN level, provided that the gauge conditions preserve the Lorentz invariance. Note that we can always arrange for the dependence of the Lagrangian upon the accelerations to be linear, at the price of adding some so-called “multi-zero” terms to the Lagrangian, which do not modify the equations of motion (see, e.g., Ref. [63]). At the 3PN level, we find that the Lagrangian also depends on accelerations. It is notable that these accelerations are sufficient - there is no need to include derivatives of accelerations. Note also that the Lagrangian is not unique because we can always add to it a total time derivative , where depends on the positions and velocities, without changing the dynamics. We find [66]

Witness the accelerations occuring at the 2PN and 3PN orders; see also the gauge-dependent logarithms of and , and the single term containing the regularization ambiguity . We refer to [66] for the explicit expressions of the ten conserved quantities corresponding to the integrals of energy (also given in Eq. (133)), linear and angular momenta, and center-of-mass position. Notice that while it is strictly forbidden to replace the accelerations by the equations of motion in the Lagrangian, this can and should be done in the final expressions of the conserved integrals derived from that Lagrangian.

Now we want to exhibit a transformation of the particles dynamical variables - or contact transformation, as it is called in the jargon - which transforms the 3PN harmonic-coordinates Lagrangian (137) into a new Lagrangian, valid in some ADM or ADM-like coordinate system, and such that the associated Hamiltonian coincides with the 3PN Hamiltonian that has been obtained by Damour, Jaranowski and Schäfer [60]. In ADM coordinates the Lagrangian will be “ordinary”, depending only on the positions and velocities of the bodies. Let this contact transformation be and , where and denote the trajectories in ADM and harmonic coordinates, respectively. For this transformation to be able to remove all the accelerations in the initial Lagrangian up to the 3PN order, we determine [66] it to be necessarily of the form

(and idem ), where is a freely adjustable function of the positions and velocities, made of 2PN and 3PN terms, and where represents a special correction term, that is purely of order 3PN. The point is that once the function is specified there is a unique determination of the correction term for the contact transformation to work (see Ref. [66] for the details). Thus, the freedom we have is entirely coded into the function , and the work then consists in showing that there exists a unique choice of for which our Lagrangian is physically equivalent, via the contact transformation (138), to the ADM Hamiltonian of Ref. [60]. An interesting point is that not only the transformation must remove all the accelerations in , but it should also cancel out all the logarithms and , because there are no logarithms in ADM coordinates. The result we find, which can be checked to be in full agreement with the expression of the gauge vector in Eq. (132), is that involves the logarithmic terms
together with many other non-logarithmic terms (indicated by dots) that are entirely specified by the isometry of the harmonic and ADM descriptions of the motion. For this particular choice of the ADM Lagrangian reads as
Inserting into this equation all our explicit expressions we find

The notation is the same as in Eq. (137), except that we use upper-case letters to denote the ADM-coordinates positions and velocities; thus, for instance and . The Hamiltonian is simply deduced from the latter Lagrangian by applying the usual Legendre transformation. Posing and , we get [8788896066]

Arguably, the results given by the ADM-Hamiltonian formalism (for the problem at hand) look simpler than their harmonic-coordinate counterparts. Indeed, the ADM Lagrangian is ordinary - no accelerations - and there are no logarithms nor associated gauge constants and . But of course, one is free to describe the binary motion in whatever coordinates one likes, and the two formalisms, harmonic (137) and ADM (141, 142), describe rigorously the same physics. On the other hand, the higher complexity of the harmonic-coordinates Lagrangian (137) enables one to perform more tests of the computations, notably by inquiring about the future of the constants and , that we know must disappear from physical quantities such as the center-of-mass energy and the total gravitational-wave flux.