On diffusion in Hamiltonian systems (Q1276017)

The authors discuss the Hamiltonian systems \(\frac{dx_k}{dt} = \frac{\partial H}{\partial y_k}\), \(\frac{dy_k}{dt} = -\frac{\partial H}{\partial x_k}\), \(k=1,\dots,n\), \(H = H_0(y) + \varepsilon H_1(x,y)\), where \(x = (x_1,\dots,x_n)\bmod 2\pi\) are angular canonic coordinates, \(y = (y_1,\dots,y_n)\) are canonic impulses, and \(\varepsilon\) is a small parameter for which the diffusion rate is essentially larger than the exponentially small rate. It is shown that for such systems the condition \(\det\| \frac{\partial^2H_0}{\partial y^2}\| \neq 0\) is not satisfied.