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 \(1\over2\)(Pi Q·i − Qi Pi·) − ∑i=1N (\(1\over2\)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 δik δj,k + exp{Qk − Qk+1} (δik δj,k+1 + δik+1 δj,k) ,
Aij = ∑k=1n exp{Qk − Qk+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].
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 16 jan 2016