*(English)*Zbl 0759.58018

The Hamiltonian system under consideration is governed by equations of the form

where $L\left(t\right)$ is a positive definite matrix and further technical conditions, among other things, ensure that the origin is a local maximum of $V$ for all $t$. The authors first reconsider a theorem by Rabinowitz and Tanaka concerning the existence of a homoclinic orbit emanating from 0. Using a new compact imbedding theorem, they are able to show that the Palais- Smale condition is satisfied, which in turn makes it possible to prove the above cited theorem by the more traditional techniques relying on the Mountain Pass Theorem. If, in addition, $W$ is an even function for all $t$, they make use of the symmetric mountain pass theorem to prove the existence of an unbounded sequence of homoclinic orbits.