Poisson Quasi-Nijenhuis Manifolds and the Toda System

Abstract

The notion of Poisson quasi-Nijenhuis manifold generalizes that of Poisson-Nijenhuis manifold. The relevance of the latter in the theory of completely integrable systems is well established since the birth of the bi-Hamiltonian approach to integrability. In this note, we discuss the relevance of the notion of Poisson quasi-Nijenhuis manifold in the context of finite-dimensional integrable systems. Generically (as we show by a class of examples with 3 degrees of freedom) the Poisson quasi-Nijenhuis structure is largely too general to ensure Liouville integrability of a system. However, we present a general scheme connecting Poisson quasi-Nijenhuis and Poisson-Nijenhuis manifolds, and we give sufficient conditions such that the spectral invariants of the “quasi-Nijenhuis recursion operator” of a Poisson quasi-Nijenhuis manifold (obtained by deforming a Poisson-Nijenhuis structure) are in involution. Then we prove that the closed (or periodic) n-particle Toda lattice, along with its relation with the open (or non periodic) Toda system, can be framed in such a geometrical structure.

Appendix: The 4-particle closed Toda case

In this appendix we give more explicit formulas concerning the closed Toda lattice and we justify some assertions done in Section 2, before the beginning of Subsection 2.1.

In the canonical variables (q₁, q₂, q₃, q₄, p₁, p₂, p₃, p₄), we have that

$$ N=\left[ \begin {array}{cccccccc} p_{{1}}&0&0&0&0&1&1&1 \\ 0&p_{{2}}&0&0&-1&0&1&1 \\ 0&0&p_{{3}}&0&-1&-1&0&1 \\ 0&0&0&p_{{4}}&-1&-1&-1&0 \\ 0&-{\mathrm{e}^{q_{{1}}-q_{{2}}}}&0&0&p_{{1}}&0&0&0 \\ {\mathrm{e}^{q_{{1}}-q_{{2}}}}&0&-{\mathrm{e}^{q_{{2}}-q_{{3}}}}&0&0&p_{{2}}&0&0 \\ 0&{\mathrm{e}^{q_{{2}}-q_{{3}}}}&0&-{\mathrm{e}^{q_{{3}}-q_{{4}}}}&0&0&p_{{3}}&0 \\ 0&0&{\mathrm{e}^{q_{{3}}-q_{{4}}}}&0&0&0&0&p_{{4}}\end {array} \right] $$

(63)

and ${\Omega }=\mathrm {e}^{q_{4}-q_{1}}dq_{4}\wedge dq_{1}$, so that $\pi {\Omega }^{\flat }=\mathrm {e}^{q_{4}-q_{1}}\left (\partial _{p_{4}}\otimes dq_{1}-\partial _{p_{1}}\otimes dq_{4}\right )$ is a rank-2 tensor. It can be checked that its torsion vanishes, while that of

$$ \hat N=N-\pi {\Omega}^{\flat}=\left[ \begin {array}{cccccccc} p_{{1}}&0&0&0&0&1&1&1 \\ 0&p_{{2}}&0&0&-1&0&1&1\\ 0&0&p_ {{3}}&0&-1&-1&0&1\\ 0&0&0&p_{{4}}&-1&-1&-1&0 \\ 0&-{\mathrm{e}^{q_{{1}}-q_{{2}}}}&0&-{\mathrm{e}^{q_{{4} }-q_{{1}}}}&p_{{1}}&0&0&0\\ {\mathrm{e}^{q_{{1}}-q_{{2} }}}&0&-{\mathrm{e}^{q_{{2}}-q_{{3}}}}&0&0&p_{{2}}&0&0 \\ 0&{\mathrm{e}^{q_{{2}}-q_{{3}}}}&0&-{\mathrm{e}^{q_{{3} }-q_{{4}}}}&0&0&p_{{3}}&0\\ {\mathrm{e}^{q_{{4}}-q_{{1 }}}}&0&{\mathrm{e}^{q_{{3}}-q_{{4}}}}&0&0&0&0&p_{{4}}\end {array} \right] $$

(64)

turns out to be

$$ T_{\hat N}=\mathrm{e}^{q_{4}-q_{1}}\left( \partial_{p_{1}}\otimes dq_{4}\wedge dI_{1}-\partial_{p_{4}}\otimes dq_{1}\wedge dI_{1}- X_{1}\otimes dq_{1}\wedge dq_{4}\right) , $$

(65)

where X₁ = πdI₁. This is consistent with formula (41). Moreover, one can check that $T_{N}(X,Y)=\pi \left (i_{X\wedge Y}\phi \right )$ is satisfied with

$$ \phi=dI_{1}\wedge{\Omega}=\mathrm{e}^{q_{4}-q_{1}}\left( dI_{1}\wedge dq_{1}\wedge dq_{4}\right) = dI_{1}\wedge d \mathrm{e}^{q_{4}}\wedge d\mathrm{e}^{-q_{1}}=d \left( I_{1} d \mathrm{e}^{q_{4}}\wedge d\mathrm{e}^{-q_{1}}\right) . $$

(66)

If we put $H_{k}=\frac 12 I_{k}=\frac 1{2k}\operatorname {Tr}({\hat N}^{k})$, with k = 1, 2, 3, 4, then we obtain the constants of the motion of the 4-particle closed Toda chain. Here, by “constants of the motion of the 4-particle closed Toda chain” we mean those obtained by taking traces of the powers of the well known Lax matrix (see, e.g., [15])

$$ L=\left[ \begin{array}{cccc} p_{1}&{\mathrm{e}^{\frac12(q_{{1}}-q_{{2}) }}}&0&{\mathrm{e}^{\frac12(q_{{4}}-q_{{1}) }}}\\ {\mathrm{e}^{\frac12(q_{{1}}-q_{{2}) }}}&p_{2}&{\mathrm{e}^{\frac12(q_{{2}}-q_{{3}) }}}&0\\ 0&{\mathrm{e}^{\frac12(q_{{2}}-q_{{3}) }}}&p_{3}&{\mathrm{e}^{\frac12(q_{{3}}-q_{{4}) }}}\\ {\mathrm{e}^{\frac12(q_{{4}}-q_{{1}) }}}&0&{\mathrm{e}^{\frac12(q_{{3}}-q_{{4}) }}}&p_{4} \end{array}\right]\>. $$

(67)

We also have that

$$ \hat\pi^{\prime}=\hat N\pi=\left[ \begin {array}{cccccccc} 0&-1&-1&-1&p_{{1}}&0&0&0\\ 1&0&-1&-1&0&p_{{2}}&0&0\\ 1&1&0&-1&0&0&p_{{3}}&0\\ 1&1&1&0&0&0&0&p_{{4}}\\ -p_{{1}}&0&0&0&0&-{\mathrm{e}^{q_{{1}}-q_{{2}}}}&0&-{\mathrm{e}^{q_{{4}}-q_{{1}}}}\\ 0&-p_{{2}}&0&0&{\mathrm{e}^{q_{{1}}-q_{{2}}}}&0&-{\mathrm{e}^{q_{{2}}-q_{{3}}}}&0\\ 0&0&-p_{{3}}&0&0&{\mathrm{e}^{q_{{2}}-q_{{3}}}}&0&-{\mathrm{e}^{q_{{3}}-q_{{4}}}}\\ 0&0&0&-p_{{4}}&{ \mathrm{e}^{q_{{4}}-q_{{1}}}}&0&{\mathrm{e}^{q_{{3}}-q_{{4}}}}&0\end {array} \right] , $$

(68)

while the corresponding Poisson tensor for the open Toda lattice is

$$ \pi^{\prime}=N\pi= \left[ \begin {array}{cccccccc} 0&-1&-1&-1&p_{{1}}&0&0&0\\ 1&0&-1&-1&0&p_{{2}}&0&0\\ 1&1&0&-1&0&0&p_{{3}}&0\\ 1&1&1&0&0&0&0&p_{{4}}\\ -p_{{1}}&0&0&0&0&-{\mathrm{e}^{q_{{1}}-q_{{2}}}}&0&0\\ 0&-p_{{2}}&0&0&{\mathrm{e}^{q_{{1}}-q_{{2}}}}&0&-{\mathrm{e}^{q_{{2}}-q_{{3}}}}&0\\ 0&0&-p_{{3}}&0&0&{\mathrm{e}^{q_{{2}}-q_{{3}}}}&0&-{\mathrm{e}^{q_{{3}}-q_{{4}}}}\\ 0&0&0&-p_{{4}}&0&0&{\mathrm{e}^{q_{{3}}-q_{{4}}}}&0 \end {array} \right] . $$

(69)

It holds

$$ \hat\pi^{\prime}=\pi^{\prime} +\mathrm{e}^{q_{4}-q_{1}} \partial_{p_{4}}\wedge\partial_{p_{1}} , $$

(70)

and the Schouten bracket of $\hat \pi ^{\prime }$ with itself is

$$ [\hat\pi^{\prime},\hat\pi^{\prime}] =2\mathrm{e}^{q_{4}-q_{1}}\left( X_{1}\wedge \partial_{p_{4}}\wedge \partial_{p_{1}}\right) . $$

(71)

Then we can verify that the second of (6) is satisfied if ϕ is given by (66).

Finally, we explicitly show that the functions I₂, I₃, I₄ are in involution, as stated in Theorem 6. Taking (15) and (46) into account, we obtain

$$ \{I_{2},I_{3}\}={{\langle \phi_{1},X_{2}\rangle}}=2{\Omega}(X_{2},\hat N X_{1}) ,\qquad \{I_{3},I_{4}\}={{\langle \phi_{2},X_{3}\rangle}}=2{\Omega}(X_{3},{\hat N}^{2} X_{1}) . $$

(72)

Since Ω vanishes on the vector fields $Y_{k}={\hat N}^{k-1}X_{1}-X_{k}$, it holds

$$ \{I_{2},I_{3}\}=2{\Omega}(X_{2},X_{2})=0 ,\qquad \{I_{3},I_{4}\}=2{\Omega}(X_{3},X_{3})=0 . $$

(73)

As far as {I₂I₄} is concerned, thanks to (18) and (46) it can be written as

$$ \{I_{2},I_{4}\}={{\langle \phi_{1},X_{3}\rangle}}+{{\langle \phi_{2},X_{2}\rangle}}=2{\Omega}(X_{3},{\hat N} X_{1})+2{\Omega}(X_{2},{\hat N}^{2} X_{1}). $$

(74)

Hence

$$ \{I_{2},I_{4}\}=2{\Omega}(X_{3},X_{2})+2{\Omega}(X_{2},X_{3}) , $$

(75)

which clearly vanishes. Notice however that, e.g.,

$$ {{\langle \phi_{1},X_{3}\rangle}}=2{\Omega}(X_{3},{\hat N} X_{1})=2{\Omega}(X_{3},X_{2}) $$

is not vanishing by itself, as anticipated in Section 2.