Abstract
Let $\mathfrak{S}$ be a compact, connected surface and $H \in C^2(T^* \mathfrak{S})$ a Tonelli Hamiltonian. This note extends V. V. Kozlov's result on the Eulercharacteristic of $\mathfrak{S}$ when $H$ is real-analytically integrable, using a definition of topologically-tame integrability called semisimplicity. Theorem: If $H$ is $2$-semisimple, then $\mathfrak{S}$ has non-negative Eulercharacteristic; if $H$ is $1$-semisimple, then $\mathfrak{S}$ has positive Euler characteristic.
Highlights
Say that a natural mechanical system is a Hamiltonian that is a sum of kinetic and potential energies
Kozlov proved if H enjoys a second, independent analytic integral F, the Euler characteristic of S is non-negative and so it is homeomorphic to S2, T2 or a non-orientable quotient thereof [10]
Let Σ be a compact smooth manifold and H : T ∗Σ → R be a C2 Tonelli Hamiltonian that is geometrically semisimple with respect to (f, L, B)
Summary
It extends Kozlov’s theorem to non-commutatively integrable Tonelli Hamiltonians. We say that H (or A) is non-commutatively integrable if k + l = 2n and the set of regular points is dense. Nehorosev [13], who generalized the Liouville-Arnol’d theorem [1], proved that if H is non-commutatively integrable and p is a regular point, there is a neighbourhood U with coordinate chart (θ, I, x, y) : U → Tk × Rk × R2(n−k), where the Poisson bracket is canonical and H = H(I).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
