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 \(1\over2\)(Pi Q·iQi Pi·) – ∑i=1N (\(1\over2\)Pi2 + fi exp{QiQi+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–1Qi} – fi exp{QiQi+1} .

* Lax pair: The pair (Lij, Aij) satisfying dL/dt = [L, A], given by

Lij = ∑k=1n Pk δik δj,k + exp{QkQk+1} (δik δj,k+1 + δik+1 δj,k) ,

Aij = ∑k=1n exp{QkQk+1} (δik δj,k+1 – δik+1 δj,k) .

* Approximation: Note that the Hénon-Heiles approximation is chaotic!
> Online resources: see Gerald Teschl page; MathWorld page; Wikipedia page.

References > s.a. self-dual solutions of general relativity; Lieb-Robinson Bounds.
@ General: Toda PRP(74); Das & Okubo AP(89); Toda 89; in Perelomov 90; Corrigan PW(92)dec; Krichever & Vaninsky ht/00-in; Tomei a1508 [rev].
@ Hamiltonian: Gekhtman LMP(98) [non-abelian]; Carlet LMP(05)mp/04 [2D, and R-matrices]; Tsiganov JPA(07) [bi-Hamiltonian structures]; Fehér PLA(13) [action-angle map and duality]; Evripidou a1504 [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 PhyA(06)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]; Vereschagin PLA(10) [integrable boundary problems].
@ 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]; Iwao JPA(10) [generalized periodic discrete Toda equation, theta-function solution].
@ Nearby systems: Christiansen et al LMP(93) [integrable].


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