Toda
Lattice| Toda
Lattice |
In General
* Idea: One of the main examples of integrable system.
$ Def: A lattice of points
on a line (infinite or finite) with equal masses, longitudinally oscillating
and subject to nearest neighbor couplings,
decreasing
exponentially with the separation (exponentially repulsive forces).
* Lagrangian (first-order):
L = 
i=1N 
(Pi Q·i – Qi Pi·) – i=1N
( Pi2 + fi
exp{Qi – Qi+1})
,
where N is the number of masses on the line, and fi a
set of real coupling constants; There is also an inequivalent Lagrangian (bi-Hamiltonian
structure, > see types of symplectic structures).
* Equations of motion:
Q·i = Pi , Pi· = fi–1 exp{Qi–1 – Qi} – fi exp{Qi – Qi+1} .
* Lax pair: The pair (Lij, Aij) satisfying dL/dt = [L, A], given by
Lij = k=1n Pk 
ik j,k
+ exp{Qk – Qk+1}
(ik j,k+1
+ ik+1 j,k)
,
Aij =
k=1n
exp{Qk – Qk+1}(ik j,k+1 – ik+1 j,k)
.
* Approximation: Note that the Hénon-Heiles approximation is chaotic!
References > s.a. self-dual solutions
of general relativity.
@ General: Toda PRP(74);
Das & Okubo AP(89);
Toda 89; in Perelomov 90; Krichever & Vaninsky
ht/00-in.
@ Hamiltonian: Gekhtman LMP(98)
[non-abelian]; Carlet mp/04 [2D,
and R-matrices]; Tsiganov JPA(07) [bi-Hamiltonian structures].
@ Properties: Anderson
JMP(96)ht/95 [open N-body,
solution]; Kasman JMP(97) [orthogonal polynomials]; Nimmo & Willox PRS(97)
[Darboux transformations]; Calderbank JGP(00) [geometry];
Vaninsky
JGP(03)mp/02 [open,
and
Atiyah-Hitchin bracket]; Agrotis et al mp/05 [open,
super-integrability]; Likhachev et al PLA(06) [thermodynamics].
@ Related topics: Torrence JPA(87), JPA(88)
[and linear wave equations],
NPPS(88)
[Kac-van Moerbeke lattice];
Rosquist
& Goliath GRG(98)gq/97 [Lax
pair, geometrized]; Gueuvoghlanian a0901 [submanifolds, Lie algebras].
@ Quantum: Ikeda JPA(94);
Matsuyama AP(92), AP(01);
An LMP(09) [complete set of eigenfunctions]; > s.a. deformation
quantization [Moyal].
Related Systems and Generalizations
@ Generalizations: Alber LMP(89)
[relativistic]; Saveliev ht/95 [integrable];
Gervais & Saveliev
NPB(95)ht [higher
grading];
Adler
JPA(01)
[arbitrary planar graph]; Santini et al PRE(04)nl.SI [2D
square lattice].
@ Nearby systems: Christiansen et al LMP(93) [integrable].
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send feedback and suggestions to bombelli at olemiss.edu – modified 4 jul
2009