Find a second integral for Arnold's example - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T10:59:20Z http://mathoverflow.net/feeds/question/69826 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69826/find-a-second-integral-for-arnolds-example Find a second integral for Arnold's example Jinxin Xue 2011-07-08T20:22:47Z 2011-07-08T20:22:47Z <p>Consider Arnold's example for Arnold diffusion 1964. $$H=I_1^2/2+I_2^2/2+\epsilon(1-\cos\theta_2)(1+\mu(\sin\theta_1+\sin t))$$</p> <p>We can first make it a system of three degrees of freedom. </p> <p>Then we know this system is not integrable in the Liouville Arnold sense. One integral is the Hamiltonian. Can we find one more integral? </p> <p>I think the answer should be no. But it is not easy to prove. Maybe one more piece of information is useful. In Arnold's 1964 paper, he calculated two different nonvanishing Melnikov functions. </p>