  Consider any symplectic ruled surface $(M^g_{\lambda},\omega_{\lambda})$
given by $(\Sigma_g \times S^2, \lambda \sigma_{\Sigma_g} \oplus
\sigma_{S^2})$. We compute all natural equivariant Gromov-Witten invariants
$EGW_{g,0}(M^g_{\lambda};H_k, A-kF)$ for all hamiltonian circle actions $H_k$
on $M^g_{\lambda}$, where $A=[\Sigma_g \times pt]$ and $F= [pt \times S^2]$. We
use these invariants to show the nontriviality of certain higher order
Whitehead products that live in the homotopy groups of the symplectomorphism
groups $G_{\lambda}^g$, $g \geq 0$. Our results are sharper when $g=0,1$ and
enable us to answer a question posed by D.McDuff in the case $g=1$ and provide
a new interpretation of the multiplicative structure in the ring
$H^*(BG^0_{\lambda} ;\Q)$ found by Abreu-McDuff.
