Duality theory and homoclinic orbits of gyroscopic systems (Q1381010)

Homoclinic orbits in Hamiltonian systems on \(\mathbb{R}^{4n}\) are studied using variational methods. Here, the Hamiltonian is of the particular form \[ H(t,x,y)= \frac 1 2 y^2 + \frac{\alpha^2}{2} x^2 + (\alpha J y, x) - R(t,x) - \frac 1 2 (Ax,x), \] where \(x,y\in\mathbb{R}^{2n}\) and \(J\) is the standard symplectic matrix on \(\mathbb{R}^{2n}\). The Hamiltonian is assumed to be periodic in time \(t\). Moreover \(R\) is convex with superquadratic growth and \(A\) is symmetric and positive definite. The author proves the existence of two homoclinic solutions, both converging to zero for \(| t|\to \infty \). The methods follow the lines of \textit{V. Coti Zelati, I. Ekeland} and \textit{E. Séré} [Math. Ann. 288, No. 1, 133-160 (1990; Zbl 0731.34050)]. The variational problem is transformed via Clarke duality and compactness is restored using the concentration-compactness lemma. The proofs are exposed with many details. The motivation and the relation to gyroscopic systems is rather poorly described.