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Bifurcation diagrams and Fomenko’s surgery on Liouville tori of the Kolossoff potential \(U=\rho+(1/\rho)-k\cos\varphi\). (English) Zbl 0797.34042
The problem of motion of a particle of unit mass on the plane \((x,y)\) in a potential field (1) \(U= a\rho+ b/\rho+ c\cos\varphi+ d\sin\varphi\), where \(a\), \(b\), \(c\), \(d\in \mathbb{R}\) and \(x=\rho\cos \varphi\), \(y=\rho\sin\varphi\) is considered. Without loss of generality one may suppose (after a rotation and \(\mathbb{R}\)-linear change of \(\rho\) and \(\varphi\)) that (2) \(U(x,y)= \pm\rho\pm 1/\rho-k\cos\varphi\), \(k\in\mathbb{R}\), and the corresponding Hamiltonian function \(H\) is of the form (3) \(H={1\over 2}(p^ 2_ x+ p^ 2_ y)+ U(x,y)\). The Hamiltonian system with the Hamiltonian function (3) is integrable and the energy level sets \((H= h)\subset \mathbb{R}^ 4\) are compact if (4) \(U=\rho+(1/\rho)- k\cos\varphi\). A complete description of the topology and bifurcations of the invariant level sets of the Hamiltonian system with the potential (4) is given.

MSC:
34C23 Bifurcation theory for ordinary differential equations
70H05 Hamilton’s equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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