  We prove that for a weakly exact magnetic system on a closed connected
Riemannian manifold, almost all energy levels contain a closed orbit. More
precisely, we prove the following stronger statements. Let $(M,g)$ denote a
closed connected Riemannian manifold and $\sigma$ a weakly exact 2-form. Let
$\phi_{t}$ denote the magnetic flow determined by $\sigma$, and let $c$ denote
the Mane critical value of the pair $(g,\sigma)$. We prove that if $k>c$, then
for every non-trivial free homotopy class of loops on $M$ there exists a closed
orbit with energy $k$ whose projection to $M$ belongs to that free homotopy
class. We also prove that for almost all $k<c$ there exists a closed orbit with
energy $k$ whose projection to $M$ is contractible. In particular, when
$c=\infty$ this implies that almost every energy level has a contractible
closed orbit. As a corollary we deduce that if $\sigma$ is not exact and $M$
has an amenable fundamental group (which implies $c=\infty$) then there exist
contractible closed orbits on almost every energy level.
