{ital R}-matrix theory, formal Casimirs and the periodic Toda lattice
- Dipartimento di Matematica, Politecnico di Milano, Piazza L. da Vinci 32, I-20133 Milano (Italy)
- Dipartimento di Matematica, Universita di Milano, Via C. Saldini 50, I-20133 Milano (Italy)
The nonunitary {ital r}-matrix theory and the associated bi- and triHamiltonian schemes are considered. The language of Poisson pencils and of their formal Casimirs is applied in this framework to characterize the biHamiltonian chains of integrals of motion, pointing out the role of the Schur polynomials in these constructions. This formalism is subsequently applied to the periodic Toda lattice. Some different algebraic settings and Lax formulations proposed in the literature for this system are analyzed in detail, and their full equivalence is exploited. In particular, the equivalence between the loop algebra approach and the method of differential-difference operators is illustrated; moreover, two alternative Lax formulations are considered, and appropriate reduction algorithms are found in both cases, allowing us to derive the multiHamiltonian formalism from {ital r}-matrix theory. The systems of integrals for the periodic Toda lattice known after Flaschka and H{acute e}non, and their functional relations, are recovered through systematic application of the previously outlined schemes. {copyright} {ital 1996 American Institute of Physics.}
- OSTI ID:
- 286928
- Journal Information:
- Journal of Mathematical Physics, Vol. 37, Issue 9; Other Information: PBD: Sep 1996
- Country of Publication:
- United States
- Language:
- English
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