Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On variational bivectors

I. S. Krasil'shchik

TL;DR

The work examines variational bivectors for PDEs and revisits the known result that nondegenerate symbols force bivectors to be Hamiltonian, showing explicit counterexamples in degenerate-symbol cases where bivectors have nonzero Schouten brackets and generate nontrivial $3$-vectors. Through detailed constructions for the wave equation $u_{xy}=0$, the third-order equation $u_{xyz}=0$, and the Laplace equation $u_{xx}+u_{yy}=0$, the authors produce explicit bivectors and analyze their Poisson properties and compatibility, revealing both Poisson and non-Poisson instances. The main contribution is showing that degeneracy of the symbol permits non-Hamiltonian variational bivectors, highlighting the necessity of nondegeneracy for the classical Hamiltonian framework in this setting, and proposing a conjecture about higher-dimensional behavior. These results deepen the understanding of variational structures in PDEs and their implications for Hamiltonian representations.

Abstract

We construct examples of variational bivectors that are not Poissonian.

On variational bivectors

TL;DR

The work examines variational bivectors for PDEs and revisits the known result that nondegenerate symbols force bivectors to be Hamiltonian, showing explicit counterexamples in degenerate-symbol cases where bivectors have nonzero Schouten brackets and generate nontrivial -vectors. Through detailed constructions for the wave equation , the third-order equation , and the Laplace equation , the authors produce explicit bivectors and analyze their Poisson properties and compatibility, revealing both Poisson and non-Poisson instances. The main contribution is showing that degeneracy of the symbol permits non-Hamiltonian variational bivectors, highlighting the necessity of nondegeneracy for the classical Hamiltonian framework in this setting, and proposing a conjecture about higher-dimensional behavior. These results deepen the understanding of variational structures in PDEs and their implications for Hamiltonian representations.

Abstract

We construct examples of variational bivectors that are not Poissonian.
Paper Structure (10 sections, 6 theorems, 40 equations)

This paper contains 10 sections, 6 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Variational bivectors
  3. Examples
  4. The wave equation
  5. A third-order equation
  6. The Laplace equation
  7. Concluding remarks
  8. A fact
  9. The 3D Laplace equation
  10. The Poincaré equation

Key Result

Theorem 1

Let $H$ be a bivector on $\mathscr{E}$. Then there exists an element $H_p$ of parity $2$ such that the evolutionary derivation $\mathbf{E}_{\varphi(H)}$ of parity $1$ with the generating section $\varphi(H) = (H_u, H_p)$ is a symmetry of $\mathscr{T}^*\mathscr{E}$.

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Remark 1
  • Theorem 1: cf. GeomJetSpKVV-Springer
  • proof
  • Remark 2
  • Definition 3
  • Remark 3
  • Theorem 2
  • Remark 4
  • ...and 6 more