A Characteristic Number of Hamiltonian Bundles over $S^2$

Each loop $\psi$ in the group $\text{Ham}(M)$ of Hamiltonian diffeomorphisms of a symplectic manifold $M$ determines a fibration $E$ on $S^2$, whose coupling class \cite{G-L-S} is denoted by $c$. If $VTE$ is the vertical tangent bundle of $E$, we relate the characteristic number $\int_E c_1(VTE)c^n$ with the Maslov index of the linearized flow $\psi_{t*}$ and the Chern class $c_1(TM)$. We give the value of this characteristic number for loops of Hamiltonian symplectomorphisms of Hirzebruch surfaces.