Integrable Systems |

**In General** > s.a. integrable
quantum systems; types of integrable systems.

* __Idea__: (Liouville) A system
with *n* degrees of freedom is integrable if it has *n* conserved
quantities in involution (commuting); Then in principle the Hamilton equations
can be reduced to quadratures.

* __Lax pair / equation__: An
integrable Hamiltonian system has an associated Lax pair of matrices
*S* and *U*, satisfying the equation d*S*/d*t*
= [*S*,*U*], equivalent to the dynamical equations for the
system; As a consequence, the quantities
*I*_{k}:=
*k*^{−1}
tr *S*^{ k}
are a sequence of invariants of the system.

* __Relationships__: The non-linear
equations of motion / field equations are the integrability conditions for
systems of linear differential equations with a spectral parameter.

* __Conserved quantities__:
If the system is in the form *H* = *g*^{ab}
*p*_{a} *p*_{b}
and *k*^{a} is a Killing vector field of
*g _{ab}*,

**Special Concepts and Techniques**

@ __General references__: Haak JMP(94) [symmetries, generalized Bäcklund transformations];
Fokas & Gelfand ed-96 [algebraic];
Mironov ht/96-conf [group theoretic];
Rasin & Schiff JPA(13) [Gardner method to generate symmetries and conservation laws];
> s.a. random matrices.

@ __Symmetries__: Chavchanidze JPA(04)mp/03 [non-Noether];
Rastelli a1001
[on S^{2}, with Platonic symmetries];
> s.a. symmetries.

@ __Approximating a non-integrable one__:
Kaasalainen & Binney PRL(94).

@ __Relationships between wave equations, AKS theorem__:
Symes PhyD(80).

@ __Related topics__: Dullin & Wittek JPA(94) [actions, numerical calculation];
Bogoyavlenskij CMP(96) [tensor invariants];
Labrunie & Conte JMP(96) [finding integrals];
Alvarez et al NPB(98)ht/97 [new approach];
Fock et al JHEP(00)ht/99 [duality];
Leach et al JNMP(00)n.SI [from Yang-Baxter];
Nutku & Pavlov JMP(02)ht/01 [multi-Lagrangians];
Maciejewski & Przybylska PLA(11) [integrable deformations].

> __Related topics__: see harmonic maps;
KAM Theorem and weak chaos [perturbations];
Painlevé Analysis; symplectic structures [KdV].

**References**

@ __General__: Das 89,
& Okubo AP(89),
& Huang JMP(90);
Perelomov 90;
Zakharov ed-91;
Dorfman 93;
Frønsdal FP(93);
Fiorani IJGMP(08)-a0802 [rev, including recent developments];
Zuparic PhD(09)-a0908 [and quantum];
Gómez-Ullate et al ed-JPA(10)#43;
Tudoran JGP(12)-a1106 [unified formulation];
Doikou IJMPA(12)-a1110 [rev];
Torrielli JPA-a1606-ln.

@ __ Lax equations / pairs__:
Cariñena & Martínez IJMPA(94);
Rosquist gq/94,
Rosquist & Goliath GRG(98)gq/97 [tensorial interpretation];
Baleanu & Baskal MPLA(00)gq/01;
Przybylska JGP(01) [generalization];
Sakovich nlin.SI/01 [true and fake];
Cariglia et al PRD(13)-a1210 [on a curved manifold, geometric formulation].

@ __Geometrical aspects__: Pyatov & Solodukhin ed-96;
Strachan JGP(97)ht/96 [deformed differential calculus];
Prykarpatsky & Mykytiuk 98 [and Lie-algebraic approach];
Lesfari JGP(99) [algebraic geometry];
Grant JGP(01) [Grassmannn structures];
Clementi & Pettini ap/01;
Motter & Letelier PRD(02) [coordinate invariance of integrability];
Cieslinski et al ed-JPA(09) [non-linearity and geometry];
Ibort & Marmo TMP(12)-a1203.

@ __Space of integrable theories__: Mironov & Morozov PLB(02)ht/01 [canonical transformations and flows].

**Generalizations**

* __Quasi-integrable systems__: Classical
quasi-integrable systems are ones that can be considered ads integrable systems
with an added non-integrable small perturbation; They have Lyapunov times orders
of magnitude shorter than their ergodic time, the most clear example being the
Solar System.

* __Nekhoroshev theorem__: An
*n*-degree-of-freedom system, with *k* constants of the motion
in involution, *k* ≤ *n*, has persistent *k*-dimensional
invariant tori, and local partial action-angle coordinates, under suitable
non-degeneracy conditions; Thus, it interpolates between the Poincaré-Lyapunov
theorem (*k* = 1) and the Liouville-Arnold theorem (*k* = *n*);
The crucial tool for the proof is a generalization of the Poincaré map.

@ __Partially solvable systems__: Gaeta AP(02) [Nekhoroshev theorem];
Shabat et al ed-04.

@ __Quasi-integrable systems__: Ferreira & Zakrzewski JHEP(11)-a1011 [example];
Goldfriend & Kurchan a1909 [quantum].

@ __Nearly-integrable systems__: Fasso & Sansonetto a1601 [almost-symplectic].

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