Integrable Systems| 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 dS/dt
= [S,U], equivalent to the dynamical equations for the
system; As a consequence, the quantities
Ik:=
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 = gab
pa pb
and ka is a Killing vector field of
gab, I1
= ka pa;
If kab is a Killing tensor, ...
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 S2, 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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