Projective geometry of homogeneous second-order Hamiltonian operators

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 ⩽ 9 . As a by-product, we identify such operators as linear line congruences, that are distinguished algebraic varieties in Plücker’s space of lines. Systems of first-order conservation laws that are Hamiltonian with respect to such operators are also explicitly found. The geometry and integrability of the systems is discussed in detail.