### Abstract

The Toda lattice defined by the Hamiltonian H = 1/2 ∑^{n}_{i=1} p^{2}_{i} + ∑^{n-1}_{i=1} ν_{i}eq_{i}-q_{i+1} with ν_{i} ∈ {±1}, which exhibits singular (blowing up) solutions if some of the ν_{i} = -1, can be viewed as the reduced system following from a symmetry reduction of a subsystem of the free particle moving on the group G = SL(n, ℝ). The subsystem is T*G_{e}, where G_{e} = N_{+}AN_{-} consists of the determinant one matrices with positive principal minors, and the reduction is based on the maximal nilpotent group N_{+} × N_{-}. Using the Bruhat decomposition we show that the full reduced system obtained from T*G, which is perfectly regular, contains 2^{n-1} Toda lattices. More precisely, if n is odd the reduced system contains all the possible Toda lattices having different signs for the ν_{i}. If n is even, there exist two non-isomorphic reduced systems with different constituent Toda lattices. The Toda lattices occupy non-intersecting open submanifolds in the reduced phase space, wherein they are regularized by being glued together. We find a model of the reduced phase space as a hypersurface in ℝ^{2n-1}. If ν_{i} = 1 for all i, we prove for n = 2, 3, 4 that the Toda phase space associated with T*G_{e} is a connected component of this hypersurface. The generalization of the construction for the other simple Lie groups is also presented.

Original language | English |
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Pages (from-to) | 97-135 |

Number of pages | 39 |

Journal | Journal of Geometry and Physics |

Volume | 21 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jan 1997 |

### Keywords

- Hamiltonian symmetry reduction
- Toda lattice

### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Geometry and Topology

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## Cite this

*Journal of Geometry and Physics*,

*21*(2), 97-135. https://doi.org/10.1016/S0393-0440(96)00010-1