Projective geometry of homogeneous second order Hamiltonian operators

TL;DR

This paper classifies homogeneous second-order Hamiltonian operators using projective geometry and 3-forms, explores their invariance under reciprocal transformations, and analyzes the integrability of associated conservation law systems.

Contribution

It establishes a novel correspondence between second-order Hamiltonian operators and 3-forms, enabling classification and integrability analysis in higher dimensions.

Findings

Classification of second-order Hamiltonian operators in dimensions up to 9

Explicit construction of Hamiltonian systems of conservation laws

Discussion of integrability conditions for these systems

Abstract

We prove the invariance of homogeneous second-order Hamiltonian operators under the action of projective reciprocal transformations. We establish a correspondence between such operators in dimension $n$ and $3$-forms in dimension $n + 1$. In this way we classify second order Hamiltonian operators using the known classification of $3$-forms in dimensions $\leq$ 9. Systems of first-order conservation laws that are Hamiltonian with respect to such operators are also explicitly found. The integrability of the systems is discussed in detail.