  Rabinowitz action functional is the Lagrange multiplier functional of the
negative area functional to a constraint given by the mean value of a
Hamiltonian. In this note we show that on a symplectization there is a
one-to-one correspondence between gradient flow lines of Rabinowitz action
functional and gradient flow lines of the restriction of the negative area
functional to the constraint. In the appendix we explain the motivation behind
this result. Namely that the restricted functional satisfies Chas-Sullivan
additivity for concatenation of loops which the Rabinowitz action functional
does in general not do.
