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Abstract <p> We consider a Hamiltonian system on the symplectic space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(( \mathbb{R} ^{2n}, dy\wedge dx)\)</EquationSource> </InlineEquation> with a real-analytic Hamiltonian <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(H : \mathbb{R} ^{2n}\to \mathbb{R} \)</EquationSource> </InlineEquation>. We assume that the system has a nondegenerate equilibrium position at the origin. Under some nonresonance assumptions, we prove the following. </p> <p> For any positive integer <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(M\)</EquationSource> </InlineEquation>, there exists a real-analytic function <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(F: \mathbb{R} ^{2n}\to \mathbb{R} \)</EquationSource> </InlineEquation> such that </p> <p> (1) <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(F = O\big( (|x|+|y|)^{M+1} \big)\)</EquationSource> </InlineEquation> at the origin, </p> <p> (2) the system with Hamiltonian <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(H+F\)</EquationSource> </InlineEquation> is completely integrable in <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\( \mathbb{R} ^{2n}\)</EquationSource> </InlineEquation>. </p>

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Integrable Perturbations of Polynomial Hamiltonian Systems

  • 期刊论文

  • 发表日期：

    2026年3月27日

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  • D. Treschev

摘要

Abstract

We consider a Hamiltonian system on the symplectic space \(( \mathbb{R} ^{2n}, dy\wedge dx)\) with a real-analytic Hamiltonian \(H : \mathbb{R} ^{2n}\to \mathbb{R} \) . We assume that the system has a nondegenerate equilibrium position at the origin. Under some nonresonance assumptions, we prove the following.

For any positive integer \(M\) , there exists a real-analytic function \(F: \mathbb{R} ^{2n}\to \mathbb{R} \) such that

(1) \(F = O\big( (|x|+|y|)^{M+1} \big)\) at the origin,

(2) the system with Hamiltonian \(H+F\) is completely integrable in \( \mathbb{R} ^{2n}\) .