In this work, we investigate the existence of quasi-periodic solutions for a class of nonlinear second-order dynamical systems of the form \(\begin{aligned} \ddot{u}(t) + A \dot{u}(t) = \nabla _u H(t, u(t)), \end{aligned}\) where \(u(t) = (u_1(t), \ldots , u_N(t)) \in \mathbb {R}^N\) , and A denotes a constant skew-symmetric matrix. Under appropriate structural assumptions on the potential H(t, u), we develop a variational framework and employ a min–max argument to establish the existence of nontrivial quasi-periodic solutions. The functional setting is designed to capture the balance between the conservative part, represented by the Hamiltonian potential, and the dissipative effects induced by the operator A.
