  Let $\OO$ be an orbit of the group of Hamiltonian symplectomorphisms acting
on the space of Lagrangian submanifolds of a symplectic manifold $(X,\omega).$
We define a functional $\CC:\OO \to \R$ for each differential form $\beta$ of
middle degree satisfying $\beta \wedge \omega = 0$ and an exactness condition.
If the exactness condition does not hold, $\CC$ is defined on the universal
cover of $\OO.$ A particular instance of $\CC$ recovers the Calabi
homomorphism. If $\beta$ is the imaginary part of a holomorphic volume form,
the critical points of $\CC$ are special Lagrangian submanifolds. We present
evidence that $\CC$ is related by mirror symmetry to a functional introduced by
Donaldson to study Einstein-Hermitian metrics on holomorphic vector bundles. In
particular, we show that $\CC$ is convex on an open subspace $\OO^+ \subset
\OO.$ As a prerequisite, we define a Riemannian metric on $\OO^+$ and analyze
its geodesics. Finally, we discuss a generalization of the flux homomorphism to
the space of Lagrangian submanifolds, and a Lagrangian analog of the flux
conjecture.
