The Toda flow on a generic orbit is integrable. (English) Zbl 0606.58020
The Toda flow is generated by the Hamiltonian
\[
H=\frac{1}{2}\sum^{n}_{i=1}y_ i^ 2+\sum^{n-1}_{i=1}\exp (x_ i-x_{i+1})
\]
on \({\mathbb{R}}^ n\). Flaschka [Phys. Rev. B 9 (1974)] showed that this flow is completely integrable by making an change of variables which converts these equations to a Lax pair. In terms of Lie groups, these equations are equivalent to the equations of motion generated on the symplectic manifold \((O_{s_ 0},\omega_ K)\) by the Hamiltonian \((1/2) tr S^ 2\), where \(O_{s_ 0}\) is the ((2n-2)- dimensional, nongeneric) orbit through the real, symmetric, tridiagonal matrix \(S_ 0\) of the co-adjoint action of the lower triangular group on its dual Lie algebra, identified with the symmetric matrices, and \(\omega_ K\) is the Kirillov 2-form.
The purpose of this paper is to describe sufficient integrals in involution to show that the Toda flow on generic orbits (of dimension 2[\(\frac{1}{4}n^ 2])\) is completely integrable. For any \(n\times n\) matrix M and for \(0\leq k\leq [n]\), let \((M)_ k\) denote the \((n-k)\times (n-k)\) matrix obtained by deleting the first k rows and last k columns of M, and consider the generalized eigenvalue problem \[ (*)_ k\quad (M)_ k\omega_{rk}=\lambda_{rk}(I)_ k\omega_{rk},\quad I=n\times n\quad identity\quad matrix. \] Generically, (*) has n-2k roots \(\lambda_{1k},...,\lambda_{n-2k,k}\). Up to a choice of signs, generic orbits are determined by fixing the \([(n+1)]\) sums \(\sum^{n- 2k}_{r=1}\lambda_{rk}\), \(0\leq k\leq [(n-1)]\). The main result of this paper asserts that the [\(\frac{1}{4}n^ 2]\) eigenvalues \(\lambda_{rk}\), \(0\leq k\leq [(n-1)]\), \(2\leq r\leq n-2k\), are in involution with each other and with the Toda Hamiltonian \((1/2)tr S^ 2\). Furthermore, the associated angle variables are (essentially) the last components of the generalized eigenvectors \(\omega_{rk}\), suitably normalized. By Liouville’s theorem, the connected invariant sets are products of lines and circles: it turns out that there are precisely as many circles as there are pairs of complex conjugate roots \(\lambda_{rk}\).
The purpose of this paper is to describe sufficient integrals in involution to show that the Toda flow on generic orbits (of dimension 2[\(\frac{1}{4}n^ 2])\) is completely integrable. For any \(n\times n\) matrix M and for \(0\leq k\leq [n]\), let \((M)_ k\) denote the \((n-k)\times (n-k)\) matrix obtained by deleting the first k rows and last k columns of M, and consider the generalized eigenvalue problem \[ (*)_ k\quad (M)_ k\omega_{rk}=\lambda_{rk}(I)_ k\omega_{rk},\quad I=n\times n\quad identity\quad matrix. \] Generically, (*) has n-2k roots \(\lambda_{1k},...,\lambda_{n-2k,k}\). Up to a choice of signs, generic orbits are determined by fixing the \([(n+1)]\) sums \(\sum^{n- 2k}_{r=1}\lambda_{rk}\), \(0\leq k\leq [(n-1)]\). The main result of this paper asserts that the [\(\frac{1}{4}n^ 2]\) eigenvalues \(\lambda_{rk}\), \(0\leq k\leq [(n-1)]\), \(2\leq r\leq n-2k\), are in involution with each other and with the Toda Hamiltonian \((1/2)tr S^ 2\). Furthermore, the associated angle variables are (essentially) the last components of the generalized eigenvectors \(\omega_{rk}\), suitably normalized. By Liouville’s theorem, the connected invariant sets are products of lines and circles: it turns out that there are precisely as many circles as there are pairs of complex conjugate roots \(\lambda_{rk}\).
Reviewer: R.Devaney
MSC:
| 37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |
| 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.) |
| 37C80 | Symmetries, equivariant dynamical systems (MSC2010) |
| 37K35 | Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems |
Keywords:
Toda flow; Hamiltonian; completely integrable; Lax pair; integrals in involution; generic orbitsReferences:
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