By Babelon O., Bernard D., Talon M.

Introducing the reader to classical integrable platforms and their functions, this booklet synthesizes different ways to the topic, delivering a suite of interconnected equipment for fixing difficulties in mathematical physics. The authors introduce and clarify each one technique, and display the way it could be utilized to specific examples. instead of providing an exhaustive checklist of a number of the integrable platforms, they specialize in classical gadgets that have famous quantum opposite numbers, or are the semi-classical limits of quantum gadgets. They hence allow readers to appreciate the literature on quantum integrable platforms.

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Inserting eq. 16) into eq. 10) one gets Mk = P (k) (L, λ) . More− over, in this formula the Laurent expansions of the coeﬃcients of P (k) can be truncated at some positive power of λ − λk since a high enough power cannot contribute to the singular part, yielding a polynomial with coeﬃcients Laurent polynomials in λ − λk . It is important to realize that the dynamical variables are the matrix elements of the Lax matrix, or the matrix elements of the Lk,r . 2 The Zakharov–Shabat construction the number and the order of the poles of the Lax matrix amounts to specifying a particular model.

The reduced equations of motion are equivalent to the equations of motion for the Neumann model. Indeed, the reduced system is characterized by the conditions t XX = 1 and t XY = 0, but the equations of motion eq. 23) do not preserve the second condition. We need to perform simultaneously a time-dependent gauge transformation Y → Y +λ(t)X to keep the motion on the gauge surface. Writing: 0= d ˙ − λJX) (X, Y + λX) = (−JX, Y + λX) + (X, −JY − L0 X + λX dt = −(X, L0 X) + λ˙ since J is antisymmetric, gives λ˙ = (X, L0 X).

Note that this is a two-poles r-matrix with poles at λ = ±µ. Example 2. We consider next the Neumann model. Recall that the Lax matrix reads L(λ) = L0 + λ1 J − λ12 K. As in the Euler top it satisﬁes t L(−λ) = L(λ). Hence we are dealing with the graded orthogonal group t g −1 (−λ) = g(λ). Let us check that the matrix L (λ) is an orbit under − the coadjoint action. As a matter of fact: (g(λ)L− (λ)g −1 (λ))− = − 1 1 g0 K t g 0 + (g0 J t g 0 − g1 K t g 0 + g0 K t g 1 ) 2 λ λ with g(λ) = g0 +λg1 +. ..