On the antiplectic pair connected with the Adler-Gel'fand-Dikii bracket
Copyright policy )On the antiplectic pair connected with the Adler-Gel'fand-Dikii bracket
After a five-page introduction the antiplectic formalism is reviewed following the author [Q. J. Math., Oxf. II. Ser. 42, 227-256 (1991; Zbl 0755.58030)]. The so-called special 2-forms are introduced and comparisons with the standard Hamiltonian formalism are drawn. Several properties of the bilinear concomitant on a differential field are then examined. A tensor is defined and shown to be a closed special 2-form, which determines a Hamiltonian operator \(\ell\). Two spaces forming an antiplectic pair are then used to recover the Adler-Gel'fand-Dikij (AGD) bracket. The kernel of the AGD operator \(\ell\) is shown to be given by a generalization of the squared eigenfunctions of the Schrödinger case. In the final section the whole construction is regarded from the standpoint of loop groups.
- Imperial College London United Kingdom
composition of ordinary differential operators, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, General models, approaches, and methods in mechanics of particles and systems, Ordinary differential operators, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Adler-Gel'fand-Dikii bracket, loop groups, Loop groups and related constructions, group-theoretic treatment, antiplectic formalism, Hamiltonian and Lagrangian mechanics
composition of ordinary differential operators, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, General models, approaches, and methods in mechanics of particles and systems, Ordinary differential operators, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Adler-Gel'fand-Dikii bracket, loop groups, Loop groups and related constructions, group-theoretic treatment, antiplectic formalism, Hamiltonian and Lagrangian mechanics
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