Abstract
In its original form, the n-particle periodic Toda lattice is given by the Hamiltonian on R 2n
where q n+1 = q 1; the symplectic structure is the canonical one, {q i , q j } = {p i , p j } = 0 and {q i , p j } = δ ij , where 1 ≤ i, j ≤ n. For a mechanical interpretation, consider n unit mass particles on a circle that are connected by exponential springs. In [33], Bogoyavlensky proposed a Lie algebraic generalization, where the original Toda lattice corresponds to the root system a n−1. Denoting by l the rank of the root system, the general form of the Hamiltonian is
where qn = l + 1 for the root systems a l , e6, e7, g2 and n = l for the other root systems. Denoting
the potential V • is given for the root systems that correspond to the classical Lie algebras by the following expressions:
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© 2004 Springer-Verlag Berlin Heidelberg
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Adler, M., van Moerbeke, P., Vanhaecke, P. (2004). Periodic Toda Lattices Associated to Cartan Matrices. In: Algebraic Integrability, Painlevé Geometry and Lie Algebras. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol 47. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05650-9_9
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DOI: https://doi.org/10.1007/978-3-662-05650-9_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-06128-8
Online ISBN: 978-3-662-05650-9
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