Invariant tori for nearly integrable Hamiltonian systems with degeneracy
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 \((a_1,a_2,\dots, a_n)\in \mathbb{R}^n\). The authors prove that the condition \((**)\) is equivalent to Russmann's condition in the analytic case.
scientific article; zbMATH DE number 1121722