By Babelon O., Bernard D., Talon M.
Introducing the reader to classical integrable structures and their functions, this booklet synthesizes the various techniques to the topic, offering a collection of interconnected tools for fixing difficulties in mathematical physics. The authors introduce and clarify each one procedure, and exhibit the way it might be utilized to specific examples. instead of featuring an exhaustive record of a number of the integrable structures, they concentrate on classical gadgets that have famous quantum opposite numbers, or are the semi-classical limits of quantum items. They therefore let readers to appreciate the literature on quantum integrable platforms.
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Remark. 29) i,j often occurs when calculating r-matrices. It is called the tensor Casimir of gl(N ). 30) This proposition shows that the generic Zakharov–Shabat system, equipped with this symplectic structure, is an integrable Hamiltonian system (the precise counting of independent conserved quantities will be done in Chapter 5). 3 Coadjoint orbits and Hamiltonian formalism The Jacobi identity is satisfied because this r-matrix verifies the classical Yang–Baxter equation (see eq. 12) in Chapter 2): [r12 , r13 ] + [r12 , r23 ] + [r13 , r23 ] = 0 where rij stands for rij (λi , λj ).
A solution is y = 1/2 j g jj pj vj which easily gives: Jkl = − 1 2 (ak − al ) vjk vjl g jj pj j With this, we can compute the conserved quantities Fk , eq. 28) for appropriate bn , n = 1, . . , N − 1, and we have used Fk = 1 to normalize the leading coefficient in the numerator. 29) Following the general strategy of the Liouville theorem, we express the momenta pj in terms of the conserved quantities Fk and the ζj . We have: g jj p2j = lim λ→ζj λ − ζj 1 H(λ) =⇒ p2j = − H(ζj ) u(λ) 4 where we have taken into account the value of the metric tensor and eq.
We end this section by illustrating theses constructions on the examples of the Euler and Lagrange tops and of the Neumann model. We verify that the matrices L(λ) and M (λ) are indeed related as in eq. 15). Example 1. Let us consider the Euler top. We see that L(λ), eq. 1), has a pole at 0 and M (λ) has a pole at ∞. Let us apply the above procedure to remove this pole. There exists a polynomial P (x) = αx2 + βx + γ such that P (I 2 ) = I. We will need the coefficient α = −1/I1 I2 I3 . Redefining M (λ) to M (λ) − λP (L(λ)) one gets M = M0 − (α/λ)J 2 with M0 = Ω − α(I 2 J + JI 2 ) − βJ.