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Consider Arnold's example for Arnold diffusion 1964. $$H=I_1^2/2+I_2^2/2+\epsilon(1-\cos\theta_2)(1+\mu(\sin\theta_1+\sin t)) $$

We can first make it a system of three degrees of freedom.

Then we know this system is not integrable in the Liouville Arnold sense. One integral is the Hamiltonian. Can we find one more integral?

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.

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Maybe the following paper (J. Cresson, Hyperbolicity, transversality and analytic first integrals) will be of some help. – Zurab Silagadze Apr 11 '13 at 4:41
I met a similar problem, but I have no idea what a second integral means. – ElixirX Oct 26 '13 at 17:18
integral = integral of the motion = a conserved quantity – Carlo Beenakker Oct 26 '13 at 20:55

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