\(r\)-matrix theory, formal Casimirs and the periodic Toda lattice. (English) Zbl 0863.58038
Summary: The nonunitary \(r\)-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 approximate reduction algorithms are found in both cases, allowing us to derive the multiHamiltonian formalism from \(r\)-matrix theory. The system of integrals for the periodic Toda lattice known after Flaschka and Hénon, and their functional relations, are recovered through systematic application of the previously outlined schemes.
MSC:
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |
References:
| [1] | DOI: 10.1090/conm/132/1188442 · doi:10.1090/conm/132/1188442 |
| [2] | DOI: 10.1007/BF01076717 · Zbl 0535.58031 · doi:10.1007/BF01076717 |
| [3] | DOI: 10.1007/BF01077848 · Zbl 0513.58028 · doi:10.1007/BF01077848 |
| [4] | DOI: 10.1016/0378-4371(89)90398-1 · Zbl 0717.35081 · doi:10.1016/0378-4371(89)90398-1 |
| [5] | DOI: 10.1007/BF01228340 · Zbl 0695.58011 · doi:10.1007/BF01228340 |
| [6] | DOI: 10.1007/BF01410079 · Zbl 0393.35058 · doi:10.1007/BF01410079 |
| [7] | Kostant B., Lond. Math. Soc. Lect. Notes 34 pp 287– (1979) |
| [8] | DOI: 10.1007/BF01390312 · Zbl 0474.58009 · doi:10.1007/BF01390312 |
| [9] | DOI: 10.1063/1.530763 · Zbl 0822.58017 · doi:10.1063/1.530763 |
| [10] | DOI: 10.1063/1.530807 · Zbl 0823.58013 · doi:10.1063/1.530807 |
| [11] | DOI: 10.1016/0001-8708(80)90007-9 · Zbl 0455.58017 · doi:10.1016/0001-8708(80)90007-9 |
| [12] | Kupershmidt B. A., Astérisque 123 (1985) |
| [13] | DOI: 10.1016/0001-8708(79)90057-4 · Zbl 0433.22008 · doi:10.1016/0001-8708(79)90057-4 |
| [14] | DOI: 10.1063/1.528691 · Zbl 0718.58030 · doi:10.1063/1.528691 |
| [15] | DOI: 10.1016/0375-9601(93)90293-9 · doi:10.1016/0375-9601(93)90293-9 |
| [16] | Lichnérowicz A., J. Diff. Geom. 12 pp 253– (1977) |
| [17] | Weinstein A., J. Diff. Geom. 18 pp 523– (1983) |
| [18] | DOI: 10.1063/1.523777 · Zbl 0383.35065 · doi:10.1063/1.523777 |
| [19] | DOI: 10.1016/0167-2789(81)90004-X · Zbl 1194.37114 · doi:10.1016/0167-2789(81)90004-X |
| [20] | DOI: 10.1007/BF01466596 · Zbl 0602.58017 · doi:10.1007/BF01466596 |
| [21] | DOI: 10.1007/BF00398428 · Zbl 0602.58016 · doi:10.1007/BF00398428 |
| [22] | DOI: 10.1103/PhysRevB.9.1924 · Zbl 0942.37504 · doi:10.1103/PhysRevB.9.1924 |
| [23] | DOI: 10.1103/PhysRevB.9.1921 · Zbl 0942.37503 · doi:10.1103/PhysRevB.9.1921 |
| [24] | Leverrier U. J. J., J. Math 5 pp 230– (1840) |
| [25] | DOI: 10.1007/BF02105860 · Zbl 0578.58040 · doi:10.1007/BF02105860 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
