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 Plucker'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.

Projective geometry of homogeneous second-order Hamiltonian operators

Pierandrea Vergallo;Raffaele Vitolo
2023-01-01

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 <= 9. As a by-product, we identify such operators as linear line congruences, that are distinguished algebraic varieties in Plucker'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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11587/508306
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