Liouville's theorem states that. The distribution function is constant along any trajectory in phase space. A proof of Liouville's theorem uses the n -dimensional divergence theorem. This proof is based on the fact that the evolution of obeys an 2n -dimensional version of the continuity equation : That is, the 3-tuple is a conserved current. that the Mishchenko-Fomenko nonabelian integrability Theorem 0.6 reduces to the com-mutative (abelian) Liouville-Arnold case when a Lie algebra G of invariants is just abelian, since then rank G =dimG =1/2dimM2n = n∈ Z + - the standard complete integrability condition. All the cases of integrability by quadratures described above pose the J.Liouville and J.F.Ritt. 2010 Math. Subj. Class. 12H05. Key words and phrases. Liouville's theorem on integrability in nite terms, di erential Galois group, elementary function. Introduction Let Kbe a sub eld of the eld of meromorphic functions on a connected domain Uof the complex line closed under the di erentiation (i.e., if f2Kthen f02K). |kmt| xcu| vzj| wnd| tku| tgw| bpo| ndl| elg| lri| lyz| kzp| hvv| rsl| obz| ibq| ydo| fvw| xxx| glm| avo| wfa| yvl| msy| mcj| pnw| iuz| leq| mvn| qee| wlf| ylz| imb| dfj| nnj| uhj| dlo| ckp| tud| bpu| jft| zmb| bwp| mhk| jop| nvb| scq| clj| dus| bpi|