Topological classification of integrable Hamiltonian systems with two degrees of freedom. List of systems of small complexity

Foliations of isonergy surfaces and singularities of curves

“…Since we assumed there is no point of the set Σ ess c between the lines H = h 1 and H = h 2 , the Fomenko graphs, together with all their corresponding invariants, will change continuously between the values h 1 and h 2 . Since the graph with all joined numerical invariants is given by a finite set of discrete parameters [14], it follows that all isoenergy surfaces H = h, where h is between h 1 and h 2 , have the same Fomenko invariants. Thus, the isoenergy surfaces H = h 1 and H = h 2 are Liouville equivalent, which concludes the proof.…”

“…with h c ∈ [h 1 , h 2 ]. Assumptions 1-4 imply that with the exception of isolated values of H, the Liouville foliation of isoenergy surfaces may be completely described with the corresponding Fomenko invariants [14]. Vertices of the graph joined to the isoenergy surface H = h correspond exactly to intersections of the line H = h with Σ.…”

“…We say that isoenergy surfaces are Liouville equivalent if they are topologically conjugate and the homeomorphism preserves their Liouville foliations [12,13,14]. with h c ∈ [h 1 , h 2 ].…”

“…Later on, Fomenko, Bolsinov, Oshemkov, Matveev, Zieschang and others studied the topological classification of the isoenergy surfaces of 2 degrees of freedom integrable systems and their description by certain invariants, see [10,12,14,15,25,26,46,52,27] and references therein. They established a beautiful and simple way of representing the Liouville foliation of an isoenergy surface by a certain graph with edges and some subgraphs marked with rational and natural numbers.…”

“…Thus, each smooth family of Liouville tori creates an edge, and edges connect together at vertices that correspond to the singular leaves (see Figure 10 for a simple example). In Appendix 1, we give a more precise description of the work of Fomenko and his school, while all the details can be found in [10,15,12,14,25,26,46] and references therein. In [42] higher dimensional analogs to branched surfaces were developed and constructed for several three degrees of freedom systems.…”