Integrable classical system
In mathematical physics, the Garnier integrable system, also known as the classical Gaudin model is a classical mechanical system 
discovered by René Garnier in 1919 by taking the 'Painlevé simplification' or 'autonomous limit' of the Schlesinger equations. It is a classical analogue to the quantum Gaudin model due to Michel Gaudin (similarly, the Schlesinger equations are a classical analogue to the Knizhnik–Zamolodchikov equations). The classical Gaudin models are integrable.
They are also a specific case of Hitchin integrable systems, when the algebraic curve that the theory is defined on is the Riemann sphere and the system is tamely ramified.
As a limit of the Schlesinger equations.
The Schlesinger equations are a system of differential equations for formula_1 matrix-valued functions formula_2, given by
formula_3
formula_4
The 'autonomous limit' is given by replacing the formula_5 dependence in the denominator by constants formula_6 with formula_7:
formula_8
formula_9
This is the Garnier system in the form originally derived by Garnier.
As the classical Gaudin model.
There is a formulation of the Garnier system as a classical mechanical system, the classical Gaudin model, which quantizes to the quantum Gaudin model and whose equations of motion are equivalent to the Garnier system. This section describes this formulation.
As for any classical system, the Gaudin model is specified by a Poisson manifold formula_10 referred to as the phase space, and a smooth function on the manifold called the Hamiltonian.
Phase space.
Let formula_11 be a quadratic Lie algebra, that is, a Lie algebra with a non-degenerate invariant bilinear form formula_12. If formula_11 is complex and simple, this can be taken to be the Killing form.
The dual, denoted formula_14, can be made into a linear Poisson structure by the Kirillov–Kostant bracket.
The phase space formula_10 of the classical Gaudin model is then the Cartesian product of formula_16 copies of formula_14 for formula_16 a positive integer.
Sites.
Associated to each of these copies is a point in formula_19, denoted formula_20, and referred to as sites.
Lax matrix.
Fixing a basis of the Lie algebra formula_21 with structure constants formula_22, there are functions formula_23 with formula_24 on the phase space satisfying the Poisson bracket
formula_25
These in turn are used to define formula_11-valued functions
formula_27
with implicit summation.
Next, these are used to define the Lax matrix which is also a formula_11 valued function on the phase space which in addition depends meromorphically on a spectral parameter formula_29,
formula_30
and formula_31 is a constant element in formula_11, in the sense that it Poisson commutes (has vanishing Poisson bracket) with all functions.
(Quadratic) Hamiltonian.
The (quadratic) Hamiltonian is
formula_33
which is indeed a function on the phase space, which is additionally dependent on a spectral parameter formula_29. This can be written as
formula_35
with 
formula_36
and
formula_37
From the Poisson bracket relation
formula_38
by varying formula_29 and formula_40 it must be true that the formula_41's, the formula_42's and formula_43 are all in involution. It can be shown that the formula_42's and formula_43 Poisson commute with all functions on the phase space, but the formula_41's do not in general. These are the conserved charges in involution for the purposes of Arnol'd Liouville integrability.
Lax equation.
One can show
formula_47
so the Lax matrix satisfies the Lax equation when time evolution is given by any of the Hamiltonians formula_41, as well as any linear combination of them.
Higher Hamiltonians.
The quadratic Casimir gives corresponds to a quadratic Weyl invariant polynomial for the Lie algebra formula_11, but in fact many more commuting conserved charges can be generated using formula_11-invariant polynomials. These invariant polynomials can be found using the Harish-Chandra isomorphism in the case formula_11 is complex, simple and finite.
Integrable field theories as classical Gaudin models.
Certain integrable classical field theories can be formulated as classical affine Gaudin models, where formula_11 is an affine Lie algebra. Such classical field theories include the principal chiral model, coset sigma models and affine Toda field theory. As such, affine Gaudin models can be seen as a 'master theory' for integrable systems, but is most naturally formulated in the Hamiltonian formalism, unlike other master theories like four-dimensional Chern–Simons theory or anti-self-dual Yang–Mills.
Quantum Gaudin models.
A great deal is known about the integrable structure of quantum Gaudin models. In particular, Feigin, Frenkel and Reshetikhin studied them using the theory of vertex operator algebras, showing the relation of Gaudin models to topics in mathematics including the Knizhnik–Zamolodchikov equations and the geometric Langlands correspondence.
References.
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