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A Spectral Sequence for a Graded Linear Map

Larry Bates, Martin Bendersky, Richard Churchill

Abstract

We apply the method of spectral sequences to study classical problems in analysis. We illustrate the method by finding polynomial vector fields that commute with a given polynomial vector field and finding integrals of polynomial Hamiltonian systems. For the later we describe the integrals for the Henon-Heiles Hamiltonian which arises in celestial mechanics. The unifying feature is that these problems seek elements in the kernel of a linear operator. The spectral sequence approach emphasizes the obstructions constructed from cokernel of the operator to finding elements in the kernel.

A Spectral Sequence for a Graded Linear Map

Abstract

We apply the method of spectral sequences to study classical problems in analysis. We illustrate the method by finding polynomial vector fields that commute with a given polynomial vector field and finding integrals of polynomial Hamiltonian systems. For the later we describe the integrals for the Henon-Heiles Hamiltonian which arises in celestial mechanics. The unifying feature is that these problems seek elements in the kernel of a linear operator. The spectral sequence approach emphasizes the obstructions constructed from cokernel of the operator to finding elements in the kernel.
Paper Structure (5 sections, 25 theorems, 201 equations)

This paper contains 5 sections, 25 theorems, 201 equations.

Table of Contents

  1. Introduction
  2. Vector Fields
  3. Application to Hamiltonian Vector Fields
  4. Spectral Sequences
  5. Hamiltonian Vector Fields

Key Result

Lemma 2.4

Suppose $F:\mathbb{R}^n \to \mathbb{R}^m$ is a differentiable homogeneous function of degree $r$ one has where

Theorems & Definitions (34)

  • Lemma 2.4: Euler's Formula
  • Corollary 2.5
  • Corollary 2.6
  • Proposition 2.7
  • Corollary 2.9
  • Example 2.11
  • Theorem 2.12
  • Proposition 2.13
  • Lemma 2.14
  • Lemma 2.15
  • ...and 24 more