2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/73047We define a nonnegative integer $\la(L,L_0;ϕ)$ for a pair of diffeomorphic closed Lagrangian surfaces $L_0,L$ embedded in a symplectic 4-manifold $(M,\w)$ and a diffeomorphism $ϕ\in\Diff^+(M)$ satisfying $ϕ(L_0)=L$. We prove that if there exists $ϕ\in\Diff^+_o(M)$ with $ϕ(L_0)=L$ and $\la(L,L_0;ϕ)=0$, then $L_0,L$ are symplectomorphic. We also define a second invariant $n(L_1,L_0;[L_t])=n(L_1,L_0,[ϕ_t])$ for a smooth isotopy $L_t=ϕ_t(L_0)$ between two Lagrangian surfaces $L_0$ and $L_1$ with $\la (L_1,L_0;ϕ_1)=0$, which serves as an obstruction of deforming $L_t$ to a Lagrangian isotopy with $L_0,L_1$ preserved.14 pages. Some mistakes corrected. Abstract revisedSymplectic GeometryDifferential Geometry57R17, 57R52, 53D12, 53D35, 53C15text