Non-integrability of a Hamiltonian system, based on a problem of nonlinear vibration of an elastic string

Abstract

In this paper we study the problem for non-integrability of a Hamiltonian system, based on the nonlinear vibrations of an elastic string. We have the following hamiltonian: \begin{equation} H(q, p)=\frac{1}{2}\sum_{k=1}^N { p_k}^2 (t)+\frac{c_1}{2}\sum_{k=1}^N k^2 {q_k}^2(t)-\frac{c_2}{2}\sum_{k=1}^N {q_k}^2(t)+ \nonumber \end{equation} \begin{equation} +\frac{h1}{8}\left(\sum_{k=1}^N k^2 {q_k}^2(t)\right)^2-\frac{h_2}{8}\left(\sum_{k=1}^N {q_k}^2(t)\right)^2=const \nonumber \end{equation} The main result is that the responding Hamiltonian system is non-integrable, except in the cases $N > 2$ and $h_1 = 0$ and $N = 2$ and $h_1 = 0$ or $h_2 = 4 h_1 $. In the proof we use the Morales - Ramis theorem based on Differential Galois Theory.