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Consider the following (classical) $2 \frac{1}{2}$ degree of freedom Hamiltonian system: $H(u,v,p,q,τ)$, where $(u,v)$ and $(p,q)$ are conjugated variables and $τ=ϵt$ is a slowly varying parameter, $0<ϵ<<1$, and $t$ is time.

Is it possible for this Hamiltonian $H$ to be such that:

1) For each fixed $τ$, $H$ is integrable, and variables $(u,v)$ and $(p,q)$ are decoupled, such that:

the system decomposes into a product of a rotor + pendulum, i.e in terms of $(u,v)$ the phase space consists of closed periodic orbits only and hence action angle variables $(I,φ)$ can be introduced for each fixed $τ$; while in terms of $(p,q)$ variables, the phase space resembles that of a pendulum, with a hyperbolic fixed point and a homoclinic to it, i.e. $p^{2}/ 2+\cos q−1$, for each fixed $τ$.

2) However, the slow perturbation (now $\tau$ is varying) couples the system and it becomes no longer integrable?

If yes, what would be an explicit example of such a Hamiltonian system?

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