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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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##### References:
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