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A global Nekhoroshev theorem for particles on the torus with time dependent Hamiltonian

Dario Bambusi

Abstract

We prove a $C^\infty$ global version of Nekhoroshev theorem for time dependent Hamiltonians in $R^d\times T^d$. Precisely, we prove a result showing that for all times the actions of the unperturbed systems are bounded by a constant times $ \langle t\rangle^ε$. We apply the result to the dynamics of a charged particle in $T^d$ subject to a time dependent electromagnetic field.

A global Nekhoroshev theorem for particles on the torus with time dependent Hamiltonian

Abstract

We prove a global version of Nekhoroshev theorem for time dependent Hamiltonians in . Precisely, we prove a result showing that for all times the actions of the unperturbed systems are bounded by a constant times . We apply the result to the dynamics of a charged particle in subject to a time dependent electromagnetic field.
Paper Structure (9 sections, 16 theorems, 80 equations)

This paper contains 9 sections, 16 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Main Result
  3. Analytic Part
  4. Time dependent Lie transform
  5. Solution of the Cohomological equation
  6. End of the proof of Theorem \ref{['norm.form']}
  7. Geometric Part
  8. The partition
  9. Properties of the partition

Key Result

Theorem 2.3

Assume that $P\in C^{\infty}_b(\mathbb R;S^{\tt b}_1)$ with ${\tt b}<2$, then $\forall \epsilon>0$$\exists R_\epsilon$, s.t., if the initial datum fulfills $\left\|p_0\right\|\geq R_\epsilon$ then along the solutions of the Cauchy problem for the Hamilton equation of hami one has

Theorems & Definitions (38)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.4
  • Definition 3.5
  • Remark 3.6
  • ...and 28 more