Projective-geometric aspects of homogeneous third-order Hamiltonian operators. (English) Zbl 1321.37071
The authors investigate homogeneous third-order Hamiltonian operators \(P\) of differential-geometric type. The operator \(P\) is called Hamiltonian when it is formally skew-adjoint, that is, \(P^{*}=-P\), and its Schouten bracket vanishes, that is, \([P,P]=0\). The presentation makes extensive use of local coordinates.
Three main examples of integrable systems possessing Hamiltonian structure of differential-geometric type are recalled. They are integrable PDEs of Monge-Ampère type that come from the theory of the Witten-Dijkgraaf-Verlinde-Verlinde equation of 2-dimensional topological field theory.
Based on the correspondence with quadratic line complexes, a complete list of homogeneous third-order Hamiltonian operators \(P\) of differential-geometric type with \(n\leq3\) components is obtained. The classification is based on the relation with the Monge metrics of quadratic line complexes. The invariance under the full projective group is proven.
Some of the computations were performed by using the software package CDIFF of the REDUCE computer algebra system.
Three main examples of integrable systems possessing Hamiltonian structure of differential-geometric type are recalled. They are integrable PDEs of Monge-Ampère type that come from the theory of the Witten-Dijkgraaf-Verlinde-Verlinde equation of 2-dimensional topological field theory.
Based on the correspondence with quadratic line complexes, a complete list of homogeneous third-order Hamiltonian operators \(P\) of differential-geometric type with \(n\leq3\) components is obtained. The classification is based on the relation with the Monge metrics of quadratic line complexes. The invariance under the full projective group is proven.
Some of the computations were performed by using the software package CDIFF of the REDUCE computer algebra system.
Reviewer: Antonio De Nicola (Coimbra)
MSC:
| 37K25 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry |
| 37K05 | Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010) |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 53A20 | Projective differential geometry |
Keywords:
Hamiltonian operator; Jacobi identity; projective group; quadratic complex; Monge metric; reciprocal transformationReferences:
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