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: $$H(q,p)=\frac{1}{2}\sum _{k=1}^N{p_k}^2(t)+\frac{c_1}{2}\sum _{k=1}^N{k}^2{q_k}^2(t)-\frac{c_2}{2}\sum _{k=1}^N{q_k}^2(t)+$$ $$+\frac{h1}{8}{(\sum _{k=1}^N{k}^2{q_k}^2(t))}^2-\frac{h_2}{8}{(\sum _{k=1}^N{q_k}^2(t))}^2=const$$ 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 $h_2=4h_1$. In the proof we use the Morales - Ramis theorem based on Differential Galois Theory.