Projective geometry of homogeneous second order Hamiltonian operators

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 $\le$ 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.

References in corpus (3)

• Applications of Nijenhuis geometry III: Frobenius pencils and compatible non-homogeneous Poisson structures
• Flat Pencils of Symplectic Connections and Hamiltonian Operators of Degree 2
• Geometry of inhomogeneous Poisson brackets, multicomponent Harry Dym hierarchies and multicomponent Hunter-Saxton equations

Cited by in corpus (3)

• Hamiltonian formulation and aspects of integrability of generalised hydrodynamics
• Miura-reciprocal transformations and localizable Poisson pencils
• Bi-Hamiltonian structures of KdV type, cyclic Frobenius algebrae and Monge metrics