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

ISBN-10: 052182267X

ISBN-13: 9780521822671

Introducing the reader to classical integrable platforms and their purposes, this ebook synthesizes the various ways to the topic, delivering a collection of interconnected equipment for fixing difficulties in mathematical physics. The authors introduce and clarify every one procedure, and exhibit the way it will be utilized to specific examples. instead of featuring an exhaustive record of many of the integrable structures, they specialise in classical gadgets that have famous quantum opposite numbers, or are the semi-classical limits of quantum gadgets. They therefore allow readers to appreciate the literature on quantum integrable platforms.

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The variables r, θ, φ are then called separated variables. The 1-form α restricted on Mf is then obviously closed. 8 The Euler top and have simple time evolution with respective frequencies (1, 0, 0) by eq. 4). Hence ψJ 2 and ψJ3 remain constant, while ψH = t − t0 . This gives the standard formula for the Kepler motion: t − t0 = r dr 2 H − V (r) − J2 r2 Note that the constancy of ψJ3 implies: J3 φ˙ = 2 sin θ J2 − J32 sin2 θ θ˙ This, in turn, implies the conservation of J1 , J2 : J1 = −J3 cot θ cos φ − sin φ J2 − J32 sin2 θ J2 = −J3 cot θ sin φ + cos φ J2 − J32 sin2 θ so that the motion takes place in the plane perpendicular to J, as expected.

Example. Let us give an example of this construction in the simple example of the harmonic oscillator. The Lax matrix L is given in eq. 8) and we introduce the action–angle coordinates ρ, θ as in eq. 3). In these coordinates the matrix L is diagonalized by: U = U −1 = cos 2θ sin 2θ sin 2θ − cos 2θ Since {U1 , U2 } = 0, r12 = q12 , which is easily computed to be: r12 = ω 2ρ2 0 −1 1 0 ⊗L It is easy to verify that this r-matrix indeed satisﬁes eq. 10). Let us notice that it is a dynamical r-matrix, which means that it depends explicitly on the dynamical variables.

20). The Lax matrix can thus be interpreted as belonging to the coadjoint orbit of the element A− (λ) of G ∗ under the loop group G. 21) This shows that the equation of motion is a ﬂow on the coadjoint orbit. 3 Coadjoint orbits and Hamiltonian formalism 43 Coadjoint orbits in G ∗ are equipped with the canonical Kostant–Kirillov symplectic structure. Choosing two linear functions h1 (Ξ) = Ξ(X) and h2 (Ξ) = Ξ(Y ) with X, Y ∈ G, so that dh1 = X and dh2 = Y , the Kostant–Kirillov Poisson bracket reads: {Ξ(X), Ξ(Y )} = Ξ([X, Y ]) where the right-hand side is the linear function Ξ → Ξ([X, Y ]).

### Introduction to classical integrable systems by Babelon O., Bernard D., Talon M.

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