  An almost K\"ahler structure on a symplectic manifold $(N, \omega)$ consists
of a Riemannian metric $g$ and an almost complex structure $J$ such that the
symplectic form $\omega$ satisfies $\omega(\cdot, \cdot)=g(J(\cdot), \cdot)$.
Any symplectic manifold admits an almost K\"ahler structure and we refer to
$(N, \omega, g, J)$ as an almost K\"ahler manifold. In this article, we propose
a natural evolution equation to investigate the deformation of Lagrangian
submanifolds in almost K\"ahler manifolds. A metric and complex connection
$\hn$ on $TN$ defines a generalized mean curvature vector field along any
Lagrangian submanifold $M$ of $N$. We study the evolution of $M$ along this
vector field, which turns out to be a Lagrangian deformation, as long as the
connection $\hn$ satisfies an Einstein condition. This can be viewed as a
generalization of the classical Lagrangian mean curvature flow in
K\"ahler-Einstein manifolds where the connection $\hn$ is the Levi-Civita
connection of $g$. Our result applies to the important case of Lagrangian
submanifolds in a cotangent bundle equipped with the canonical almost K\"ahler
structure and to other generalization of Lagrangian mean curvature flows, such
as the flow considered by Behrndt \cite{b} in K\"ahler manifolds that are
almost Einstein.
