Download Introduction a la theorie des ensembles by Paul Richard Halmos (Jean Gardelle traducteur) PDF

By Paul Richard Halmos (Jean Gardelle traducteur)

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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.

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