# Symplectomorphism taking a Lagrangian to its isotopic copy

Suppose $$\iota_0:L\xrightarrow{}M$$ is a compact Lagrangian submanifold in a symplectic manifold $$(M,\omega)$$. Let $$\iota_t:L\times I\xrightarrow{} M$$ be an isotopy and denote $$\iota_1(L)$$ by $$L’$$. I wonder if there is a symplectomorphism $$\varphi:M\xrightarrow{}M$$ that takes $$L$$ to $$L’$$. Such a diffeomorphism exists by the Isotopy Extension Theorem, but I don’t see a way to turn that into a symplectomorphism. Any negative results are also appreciated.

Take the unit sphere $$(S^2, d\theta \wedge dh)$$. It is a 2-dimensional symplectic manifold, so every 1-dimensiononal submanifold is Lagrangian.
Let $$L \subset S^2$$ be the equator $$\{h=0\}$$. For every $$-1, the equator $$L$$ is isotopic to the circle $$L_s := \{h=s\}$$. Moreover, all of these circles are compact Lagrangian submanifolds.
However, these circles (for $$s \ne 0$$) cannot be images of $$L$$ through a symplectomorphism! $$L$$ divides the sphere into two parts of equal area, and this property must be preserved under a symplectomorphism. On the other hand, for $$s \ne 0$$, the circle $$L_s$$ divides the sphere into two parts of different areas.