9.2 Lagrangian and Hamiltonian formulations
The conservative part of the equations of motion in harmonic coordinates (168) 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 [85], 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 [158] 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. [98]). 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 [103
]
Witness the accelerations occuring at the 2PN and 3PN orders; see also the gauge-dependent logarithms of
and
. We refer to [103
] for the explicit expressions of the ten conserved quantities
corresponding to the integrals of energy (also given in Equation (170)), 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 (174) 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 [95
]. 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 [103
] 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. [103
] 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 (175), to the ADM Hamiltonian of Ref. [95
]. 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 Equation (169), 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
Inserting into this equation all our explicit expressions we find
The notation is the same as in Equation (174), 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 [139
, 140
, 141, 95
, 103
]
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 (174) and
ADM (178, 179), describe rigorously the same physics. On the other hand, the higher complexity of the
harmonic-coordinates Lagrangian (174) 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.