Invariant tori for nearly integrable Hamiltonian systems with degeneracy
type: Item
instance of: scholarly article
title: Invariant tori for nearly integrable Hamiltonian systems with degeneracy
zbMATH Open document ID: 0899.34030
DOI: 10.1007/PL00004344
author: Junxiang Xu
author: Jiangong You
author: Qingjiu Qiu
published in: Mathematische Zeitschrift
publication date: 1998-11-15T00:00:00Z
review text: The authors study Hamiltonian systems \[ {dp\over dt}= -H_q(p, q)= -\varepsilon f_q(p,q),\quad {dp\over dt}= H_p(p,q)= h_p(p)+ \varepsilon f_p(p,q)\tag{\(*\)} \] with the Hamiltonian \(H(p,q)= h(p)+ \varepsilon f(p,q)\), \((p,q)\in\Omega\times T^n\), \(n\geq 2\), where \(p= (p_1,p_2,\dots, p_n)\) are action variables varying over some bounded connected domain \(\Omega\in\mathbb{R}^n\), and \(q= (q_1,q_2,\dots, q_n)\) are conjugate angular variables whose domain \(T^n\) is the usual \(n\)-torus obtained by identifying the points whose components differ by integer multiples of \(2\pi\). Suppose \(f(p, q)\) has period \(2\pi\) in every component of \(q\) and \(H(p, q)\) is analytic in \(\overline\Omega\times T^n\), where \(\overline\Omega\) is the closure of \(\Omega\). The main result is the following theorem: If for some \(p\in\overline\Omega\), \[ \text{rank}\Biggl\{\omega, {\partial^\alpha\omega\over \partial p^\alpha}: |\alpha|\leq n-1\Biggr\}= n,\tag{\(**\)} \] with \(\omega(p)\neq h_p(p)\) and \({\partial^\alpha\omega\over \partial p^\alpha}= \left({\partial^\alpha\omega_1\over \partial p^\alpha},{\partial^\alpha\omega_2\over \partial p^\alpha},\dots,{\partial^\alpha\omega_n\over \partial p^\alpha}\right)\), then for every \(\Delta>0\) sufficiently small, there exists \(\varepsilon_0= \varepsilon_0(\Delta)> 0\) such that if \(|\varepsilon|< \varepsilon_0\), there exists a nonempty Cantor subset \(\Omega_\varepsilon\subset\Omega\) such that system \((*)\) admits a family of invariant tori \(\{I_p: p\in\Omega_\varepsilon\}\), whose frequencies \(\omega_*(p)\) satisfy \(|\omega_*(p)- \omega(p)|\leq c\varepsilon\) with \(c\) being a constant independent of \(\varepsilon\). Moreover, \(\text{mes}(\Omega- \Omega_\varepsilon)= o(\Delta)\), where \(o(\Delta)\) is infinitively small as \(\Delta\to 0\). H. Russmann announced 1990 that system \((*)\) possesses any invariant tori if on \(\Omega\), \(a_1h_{p_1}+ a_2h_{p_2}+\cdots+ a_nh_{p_n}\neq 0\) for any \((a_1,a_2,\dots,…
reviewed by: He Chungyou
Mathematics Subject Classification ID: 34C30
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 70H05
zbMATH DE Number: 1121722
zbMATH Keywords: nondegeneracy condition
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: Cantor subset
zbMATH Keywords: invariant tori
MaRDI profile type: Publication
full work available at URL: pl00004344
OpenAlex ID: W2006638341
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description: scientific article; zbMATH DE number 1121722