Instability of equilibrium solutions of Hamiltonian systems with \(n\)-degrees of freedom under the existence of a single resonance and an invariant ray (Q1797844)

The authors consider a canonical Hamiltonian system: \[ \dot{q} = \partial H/\partial p\;, \qquad \;\dot{p} = - \partial H/\partial q , \] where \(H(q,p)\) is real analytic and \(H(0,0)=0\). It is assumed the Mac Laurin series of \(H(q,p)\) is \[ H = H_2 + H_3 + \cdots + H_j + \cdots \] with \(H_j\) is an homogeneous polynomial of degree \(j\) in \((q,p)\). The quadratic part has the form \[ H_2 = {\frac{1}{2}}\sum_1^n\omega_k(q_k^2 + p_k^2). \] The linear approximation is assumed to be stable and has a single vector \(k\) of resonance, i.e., \(\sum_{i=1}^n k_i\omega_i =0\), with \(k_i\geq 0\). The Hamiltonian is normalized in Lie normal form up to order \(s\) and written in action-angle variables \((r,\varphi)\). Sufficient conditions for the Hamiltonian system to have an invariant ray-type solution, corresponding to nonlinear instability in the Lyapunov sense, are determined.