A note on the superintegrability of the Toda lattice (original) (raw)

The finite non-periodic toda lattice: A geometric and topological viewpoint

2008

In 1967, Japanese physicist Morikazu Toda published the seminal papers [78] and [79], exhibiting soliton solutions to a chain of particles with nonlinear interactions between nearest neighbors. In the decades that followed, Toda's system of particles has been generalized in different directions, each with its own analytic, geometric, and topological characteristics that sets it apart from the others. These are known collectively as the Toda lattice. This survey describes and compares several versions of the finite non-periodic Toda lattice from the perspective of their geometry and topology. Contents 1. Outline of the paper 2 2. Finite non-periodic real Toda lattice 3 2.1. Symmetric form 4 2.2. Hessenberg form 6 2.3. Extended real tridiagonal symmetric form 7 2.4. Extended real tridiagonal Hessenberg form 9 2.5. Full Symmetric real Toda lattice 3. Complex Toda lattices 3.1. The moment map 3.2. Complex tridiagonal Hessenberg form 3.3. The full Kostant-Toda lattice 3.4. Nongeneric flows in the full Kostant-Toda lattice 4. Other Extensions of the Toda Lattice 4.1. Isospectral deformation of a general matrix 4.2. Gradient formulation of Toda flows 5. Connections with the KP equation 5.1. The τ-functions for the symmetric Toda lattice hierarchy 5.2. The KP equation and the τ-function 5.3. Grassmannian Gr(k, n) 6. The Toda lattice and integral cohomology of real flag manifolds 6.1. The moment polytope and Weyl group action 6.2. Integral cohomology of G/B 6.3. Blow-ups of the indefinite Toda lattice on G and the cohomology of G/B References 1 Partially supported by NSF grant DMS0404931.

Fifty years of the finite nonperiodic Toda lattice: a geometric and topological viewpoint

Journal of Physics A: Mathematical and Theoretical, 2018

In 1967, Japanese physicist Morikazu Toda published a pair of seminal papers in the Journal of the Physical Society of Japan that exhibited soliton solutions to a chain of particles with nonlinear interactions between nearest neighbors. In the fifty years that followed, Toda's system of particles has been generalized in different directions, each with its own analytic, geometric, and topological characteristics. These are known collectively as the Toda lattice. This survey recounts and compares the various versions of the finite nonperiodic Toda lattice from the perspective of their geometry and topology. In particular, we highlight the polytope structure of the solution spaces as viewed through the moment map, and we explain the connection between the real indefinite Toda flows and the integral cohomology of real flag varieties.

Change of the Time for the Toda Lattice

Journal of Nonlinear Mathematical Physics, 2001

For the Toda lattice we consider properties of the canonical transformations of the extended phase space, which preserve integrability. At the special values of integrals of motion the integral trajectories, separated variables and the action variables are invariant under change of the time. On the other hand, mapping of the time induces shift of the generating function of the Bäcklund transformation.

Canonical transformations of the extended phase space, Toda lattices and the Stäckel family of integrable systems

Journal of Physics A: Mathematical and General, 2000

We consider compositions of the transformations of the time variable and canonical transformations of the other coordinates, which map completely integrable system into other completely integrable system. Change of the time gives rise to transformations of the integrals of motion and the Lax pairs, transformations of the corresponding spectral curves and R-matrices. As an example, we consider canonical transformations of the extended phase space for the Toda lattices and the Stäckel systems.

On the periodic Toda lattice hierarchy with an integral source

Communications in Nonlinear Science and Numerical Simulation, 2017

This work is devoted to the application of inverse spectral problem for integration of the periodic Toda lattice hierarchy with an integral type source. The effective method is presented of constructing the periodic Toda lattice hierarchy with an integral source.

Canonical transformations of the time for the Toda lattice and the Holt system

1999

For the Toda lattice and the Holt system we consider properties of canonical transformations of the extended phase space, which preserve integrability. The separated variables are invariant under change of the time. On the other hand, mapping of the time induces transformations of the action-angles variables and a shift of the generating function of the Bäcklund transformation.

Properties of the canonical transformations of the time for the Toda lattice and the Henon-Heiles system

Journal of Physics A: Mathematical and General, 2000

For the Toda lattice and the Holt system we consider properties of canonical transformations of the extended phase space, which preserve integrability. The separated variables are invariant under change of the time. On the other hand, mapping of the time induces transformations of the action-angles variables and a shift of the generating function of the Bäcklund transformation.

Algebraic Complete Integrability of the (a_4^{\left(2\right)}) Toda lattice

arXiv (Cornell University), 2024

The aim of this work is focused on the investigation of the algebraic complete integrability (a.c.i) of the Toda lattice associated with the twisted affine Lie algebra a (2) 4. First, we prove that the generic fiber of the momentum map for this system is an affine part of an abelian surface. Second, we show that the flows of integrable vector fields on this surface are linear. Finally, using the formal Laurent solutions of the system, we provide a detailed geometric description of these abelian surfaces and the divisor at infinity.

Action-angle variables and novel superintegrable systems

Physics of Particles and Nuclei, 2012

In this paper we demonstrate the effectiveness of the action angle variables in the study of super integrable systems. As an example, we construct the spherical and pseu dospherical generalizations of the two dimensional superintegrable models introduced by Tremblay, Turbiner and Winternitz and by Post and Winternitz.