Persistence of lower dimensional invariant tori on sub-manifolds in Hamiltonian systems

Abstract

Chow, Li and Yi in [2] proved that the majority of the unperturbed tori on submanifolds will persist for standard Hamiltonian systems. Motivated by their work, in this paper, we study the persistence and tangent frequencies preservation of lower dimensional invariant tori on smooth sub-manifolds for real analytic, nearly integrable Hamiltonian systems. The surviving tori might be elliptic, hyperbolic, or of mixed type.

Partial preservation of frequencies in KAM theory

“…In a subsequent paper [52], Li and Yi carried over their theorem on partial preservation of frequencies to hyperbolic lower dimensional invariant tori (recall that an invariant torus of a Hamiltonian system is said to be lower dimensional if its dimension is less than the number of degrees of freedom). Then Liu [53] generalized the results of [30] on partial preservation of frequencies and frequency ratios to lower dimensional invariant tori of arbitrary type (hyperbolic, elliptic and mixed; see section 4 below for precise definitions of hyperbolicity and ellipticity in this context).…”

“…The proofs given in [30,52,53] are extremely complicated and involve the so-called quasilinear infinite iterative procedure used also in [50] where Li and Yi explored the break-up of resonant tori of integrable Hamiltonian systems under small perturbations.…”

“…with nondegeneracy conditions involving partial derivatives of the frequency map ω of orders 2) miraculously easily. Moreover, we treat the 'classical' case of Lagrangian invariant tori and the case of lower dimensional invariant tori in a unified way and allow lower dimensional tori to be of arbitrary type (as in [53]). For each of the two cases (Lagrangian tori and lower dimensional tori), we consider both types of partial preservation-partial preservation of frequencies and that of frequency ratios at a fixed energy value (as in [30,53]).…”

“…Remark 2.3. Chow and co-workers [30,52,53] considered only principal minors of symmetric matrices, such as the Hesse matrix of E or the matrix (1.2). This is justified by the following well known fact: whenever S is a symmetric matrix of rank d, there is a non-zero principal minor of S of order d (see, e.g.…”

Persistence of hyperbolic tori in Hamiltonian systems

“…Then we give an example to illustrate the persistence of invariant tori on sub-manifolds in generalized Hamiltonian systems. For the persistence of elliptic invariant tori and mixed type of invariant tori on sub-manifolds, the reader can refer to [11] for details.…”