In this paper, we consider the first-order Hamiltonian system
\[
J
u
˙
(
t
)
+
∇
H
(
t
,
u
(
t
)
)
=
0
,
t
∈
R
.
J\dot {u}(t)+\nabla H(t,u(t))=0,\quad t\in \mathbb {R}.
\]
Here the classical Ambrosetti-Rabinowitz superlinear condition is replaced by a general super-quadratic condition. We will study the homoclinic orbits for the system. The main idea here lies in an application of a variant generalized weak linking theorem for a strongly indefinite problem developed by Schechter and Zou.