The spring-pendulum system and the Riemann equation (Q2741140)

This paper is concerned with planar spring-pendulum systems governed by Hamiltonian NEWLINE\[NEWLINEH_{\lambda}(r,\theta,p_r,p_\theta)=(1/2) pr^2 + p_{\theta}^2/r^2 + r [(\lambda-1)- \lambda \cos \theta] + (1/2) r^2.NEWLINE\]NEWLINE It is shown that for almost all \(\lambda \in (0,1)\) the system is not integrable when regarded as a complex analytic system. In particular, the authors show that the real version can have no rational integral for almost all \(\lambda \in (0,1)\).NEWLINENEWLINEFor the entire collection see [Zbl 0963.00030].