Gravitational Waves from Inspiralling Compact Binaries in ... - LUTH

Method of variation of constants
Fn v = V(l) obtained by inserting u = U(l) in v = V (u), i.e. V(l) ≡ V (U(l)).
et and eφ are some time and angular eccentricity and ft and gt are
certain functions of c1 and c2, appearing at the 2PN order.
Use the following exact relation for v − u, which is also periodic in u,
given by
βφ = 1−
1−e2 φ
. eφ v − u = 2 tan −1
βφ sin u
,
1 − βφ cos u
Use explicit unperturbed solution, for the construction of the general
solution of the perturbed system. Keep exactly the same functional
form for r, ˙r, φ and ˙ φ, as functions of l and λ, i.e. by writing
r = S(l; c1, c2) ; ˙r = n ∂S
∂l (l; c1, c2) ,
φ = λ + W (l; c1, c2) ; ˙ φ = (1 + k)n + n ∂W
∂l (l; c1, c2),
but allowing temporal variation in c1 = c1(t) and c2 = c2(t)
BRI-IHP06-I – p.106/??