Topology and Integrability in Lagrangian Mechanics

Open problems, questions and challenges in finite- dimensional integrable systems

“…We also would like to draw attention to two recent papers containing open problems on integrable natural Hamiltonian systems on two-dimensional manifolds: Burns & Matveev [59] (see §10.2 there), and Butler [60] (see § §3.3-3.5 there).…”

Invariant tori for a class of singly thermostated hamiltonians

“…B H is the union of the interiors of the edges of Γ H , and L = ψ −1 (B H ), then ψ|L is a proper submersion whose fibres are circles. Classical constructions yield the existence of angle-action variables (θ, I) : L −→ T 1 × R where the disjoint union is taken over the edge set of Γ H [3]. In these variables, H = H(I) and H I > 0 since Ĩ is monotone increasing in H. If σ is a saddle vertex, then as γ −→ σ (from above or below), H I ( Ĩ(γ)) −→ 0 since the period goes to ∞; if σ is a local minimum vertex, then as γ σ, H I ( Ĩ(γ)) −→ ω σ > 0 where ω σ is the frequency of the linearized oscillations at σ.…”