  This paper introduces techniques of symplectic topology to the study of
homoclinic orbits in Hamiltonian systems. The main result is a strong
generalization of homoclinic existence results due to Sere and to Coti-Zelati,
Ekeland and Sere, which were obtained by variational methods. Our existence
result uses a modification of a construction due to Mohnke (originally in the
context of Legendrian chords), and an energy--capacity inequality of Chekanov.
In essence, we show the existence of a homoclinic orbit by showing a certain
Lagrangian embedding cannot exist. We consider a (possibly time dependent)
Hamiltonian system on an exact symplectic manifold (W, \omega = d \lambda) with
a hyperbolic rest point. In the case of periodic time dependence, we show the
existence of an orbit homoclinic to the rest point if \lambda(X_H) - H is
positive and proper, H is positive outside a compact set and proper, and (W,
\omega) admits the structure of a Weinstein domain. In the autonomous case, we
establish the existence of an orbit homoclinic to the rest point if the
critical level is of restricted contact-type, and the critical level has a
Hamiltonian displaceable neighbourhood.
