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

Method of variation of constants
By definition, the oscillatory part ˜ Gα(l) is a periodic function with
zero average over l. Assuming that ˜cα is always small
(˜cα = O(Gα) = O(c −5 )), one can expand the RHS of the exact
evolution system, as
d¯cα
dl
+ d˜cα
dl
= Gα(l; ¯ca + ˜ca) = Gα(l; ¯ca) + O(G 2 α) ,
= ¯ Gα(l; ¯ca) + ˜ Gα(l; ¯ca) + O(G 2 α) .
Solve, modulo O(G 2 α), the evolution equation by defining ¯cα(l) as a
solution of the ‘averaged system’
d¯cα
dl = ¯ Gα(¯ca) ,
and by defining ˜cα(l) as a solution of the ‘oscillatory part’ of the
system
d˜cα
dl = ˜ Gα(l, ¯ca) .
BRI-IHP06-I – p.112/??