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A KAM theorem of symplectic algorithms for nearly integrabel Hamiltonian systems

Zaijiu Shang, Yang Xu

Abstract

In this paper we prove a KAM-like theorem of symplectic algorithms for nearly integrable Hamiltonian systems which generalises the result of \cite{r1} and \cite{r6} for the case of integrable systems.

A KAM theorem of symplectic algorithms for nearly integrabel Hamiltonian systems

Abstract

In this paper we prove a KAM-like theorem of symplectic algorithms for nearly integrable Hamiltonian systems which generalises the result of \cite{r1} and \cite{r6} for the case of integrable systems.
Paper Structure (8 sections, 8 theorems, 90 equations)

This paper contains 8 sections, 8 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Main theorem
  3. Proof of Theorem \ref{['t1']}
  4. One-step Iterative Analysis
  5. Proof of the Iteration Lemma
  6. Proof of convergence
  7. Measure Estimation
  8. Kolmogorov's non-degeneracy condition

Key Result

Theorem 2.1

For the nearly integrable Hamiltonian system 1 , $H$ is real analytic on $D_{r,s} \times V_\kappa$, and $H^0$is analytic on $V_\kappa$. If time step $t$ and disturbance parameter $\epsilon$ are small enough, $\omega$ satisfies the Rüssmann's non-degeneracy condition, then the generating function rep

Theorems & Definitions (13)

  • Remark 1.1
  • Theorem 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Corollary 3.1
  • Corollary 3.2
  • Corollary 3.3
  • Lemma 3.3
  • Lemma 3.4
  • Remark 3.1
  • ...and 3 more