An infinitesimal Liouville-Arnold theorem as a criterion of reducibility for variational Hamiltonian equations
✍ Scribed by Sergej B. Kuksin
- Book ID
- 103980679
- Publisher
- Elsevier Science
- Year
- 1992
- Tongue
- English
- Weight
- 843 KB
- Volume
- 2
- Category
- Article
- ISSN
- 0960-0779
No coin nor oath required. For personal study only.
✦ Synopsis
W-We prove a criterion for the variational equation about a quasiperiodic solution of a
Hamiltonian equation being reducible to a constant coefficient equation. We discuss applications of this criterion to the stability problem for lower dimensional invariant tori.
1. INTBODWXlON
The subject of investigation is a Hamiltonian vector-field Hr on a ZN-dimensional symplectic manifold (Mu. CC)). which is integrable on some invariant symplectic submanifold -7 C M. dim 7 = 2n < 2A". So 7 is foliated into invariant tori TE depending on an n-dimensional parameter p E P CC R". and the flow on every torus TF is of the form 4 = V,f,(p) cfo is a restriction of the Hamiltonian f to ?). Let (_T 2)' C T ;A# = U mE .7Tm N be the (skew-)normal bundle of 7. If S, is a flow of H,, then the normal bundle (T 7)" is invariant for the tangent flow S,,. We call the restriction of S,, on (T -7)-'the flow of the normal variational equation (NVE) of Hf along 7. and study the question: under what conditions is this flow reducible to the flow of a linear equation with coefficients independent of the point 4 E TF (so-called reducibility problem; see e.g. [l]
). If such reducibility occurs then in the 'nondegenerate case' 7 is 'KAM-stable". That is most of the tori T:? p E P. survive after a small Hamiltonian perturbation of the system (this results from a perturbation theorem for lower-dimensional invariant toti of a linear system.