  We prove that (under appropriate orientation conditions, depending on $R$) a
Hamiltonian isotopy $\psi^1$ of a symplectic manifold $(M, \omega)$ fixing a
relatively exact Lagrangian $L$ setwise must act trivially on $R_*(L)$, where
$R_*$ is some generalised homology theory. We use a strategy inspired by that
of Hu, Lalonde and Leclercq (\cite{Hu-Lalonde-Leclercq}), who proved an
analogous result over $\mathbb{Z}/2$ and over $\mathbb{Z}$ under stronger
orientation assumptions. However the differences in our approaches let us
deduce that if $L$ is a homotopy sphere, $\psi^1|_L$ is homotopic to the
identity. Our technical set-up differs from both theirs and that of Cohen,
Jones and Segal (\cite{Cohen-Jones-Segal, Cohen}). We also prove (under similar
conditions) that $\psi^1|_L$ acts trivially on $R_*(\mathcal{L} L)$, where
$\mathcal{L} L$ is the free loop space of $L$. From this we deduce that when
$L$ is a surface or a $K(\pi, 1)$, $\psi^1|_L$ is homotopic to the identity.
Using methods of \cite{Lalonde-McDuff}, we also show that given a family of
Lagrangians all of which are Hamiltonian isotopic to $L$ over a sphere or a
torus, the associated fibre bundle cohomologically splits over $\mathbb{Z}/2$.
