Student Seminar: Classical and Quantum Integrable Systems

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7.3 Seminar 3 Exercise 1. Consider a motion of a one-dimensional system. Let S(E) be the area enclosed by the closed phase curve corresponding to the energy level E. Show that the period of motion along this curve is equal to T = dS(E) dE . Exercise 2. At the entry of the satellite into a circular orbit at a distance 300km from the Earth the direction of its velocity deviates from the intended direction by 1 ◦ towards the earth. How is the perigee of the orbit changed? Exercise 3. Find the principle axes and moments of inertia of the uniform planar plate |x| ≤ q, |y| ≤ b, z = 0 with respect to 0. Exercise 4. Find the inertia tensor of the uniform ellipsoid with the semi-axes a, b, c. Exercise 5. Solve the Euler equations for the symmetric top: I 1 = I 2 . Exercise 6. Consider the mathematical pendulum (of mass M) in the gravitational field of the Earth. Integrate equations of motion in terms of Jacobi elliptic functions. If the second (imaginary) period has any physical meaning? What is the elliptic modulus k 2 ? Consider the limits k = 0 + and k = 1 − . L 01 01 M A pendulum in the gravitational field of the Earth. Here L is its length and G is the gravitational constant. G – 97 –
7.4 Seminar 4 Exercise 1 (K. Bohlin). Consider the Kepler problem. Let x, y be the Cartesian coordinates on the plane of motion. Introduce a complex variable z = x + iy and show that the non-linear change of variables z → u 2 , t → τ given by z = u 2 dt , dτ = 4|u2 | = 4|z| maps the Kepler orbits with the constant energy E < 0 into the ones of the harmonic oscillator with the complex amplitude u (a two-dimensional oscillator). Find the period of oscillations. Exercise 2 (Lissajous figures). Consider the two-dimensional harmonic oscillator. Show that if ω 1 ω 2 = r s , where r, s are relatively prime integers then there is a new additional integral of motion F = ā s 1a r 2 , where ā 1 = 1 √ 2ω1 (p 1 + iω 1 q 1 ) , a 2 = 1 √ 2ω2 (p 2 − iω 2 q 2 ) . The corresponding closed trajectories of the two-dimensional harmonic oscillator are called the Lissajous figures. Find the Poisson brackets between F and F i = 1 2 (p2 i + ωi 2 qi 2 ), i = 1, 2. Exercise 3. Consider the Kepler problem. Show that the components of the angular momentum J i and the components of the Runge-Lenz vector R i form w.r.t. the Poisson bracket the Lie algebra so(4). Recall that a 4 × 4 matrix X belongs to the Lie algebra so(4) if it is skew-symmetric, i.e. X t + X = 0. Express the Kepler Hamiltonian in terms of the conserved quantities J i and R i . Exercise 4. Prove that the Poisson bracket {L 1 , L 2 } = [r 12 , L 1 ] − [r 21 , L 2 ] between the components of the matrix L implies that the quantities I k = trL k are in involution, i.e. that {I k , I m } = 0. Exercise 5 (Calogero model). Consider a dynamical system of n particles with the coordinates q j and momenta p j , where j = 1, . . . n. The Hamiltonian of the – 98 –
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Preprint typeset in JHEP style - HY
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7. Homework exercises 95 7.1 Semina
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which can be written as follows {q
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To get more insight on the Liouvill
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4. The equations of motion with Ham
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Let us show that the variables (I i
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We have H = p2 2 p2 + V (x) = 2 + V
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We can now see how the solution can
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Thus, and, therefore, the half-peri
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This is the so-called focal equatio
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to show that ˙⃗ R = 0. The last
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is the bounded kinetic energy). Acc
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One can study the structure of the
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Substituting these formula into the
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Upper−half plane 01 01 0 0 1 1 01
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Thus, the combination ˙θ 2 − 2m
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Then using the eoms ˙q = p and ṗ
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where we have assumed that the eige
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are called the Euler-Arnold equatio
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Thus, g(λ)Q(λ)g(λ) −1 = ( = g
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We see that the there is one more v
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4.1 General remarks Remarkably, the
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We thus obtain u 2 x = k − 2eu +
Page 47 and 48: Here ɛ 0 = ±1. This solution can
Page 49 and 50: where σ i are the Pauli matrices 9
Page 51 and 52: Let g ≡ g(x, t, λ) be a regular
Page 53 and 54: Indeed, ∂ t L x − ∂ x L t + [
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Page 57 and 58: • Coordinate Bethe ansatz. This t
Page 59 and 60: The first interesting observation i
Page 61 and 62: This state is an eigenstate of the
Page 63 and 64: as the pseudo-particle with the mom
Page 65 and 66: This allows one to determine the ra
Page 67 and 68: The first problem is to find all po
Page 69 and 70: 5.2 Algebraic Bethe Ansatz Here we
Page 71 and 72: Semi-classical limit and quantizati
Page 73 and 74: Thus, the transfer matrix is also p
Page 75 and 76: To write down the fundamental commu
Page 77 and 78: provided W A n + W D n = 0 for all
Page 79 and 80: and if k < j the contribution will
Page 81 and 82: Now we have to specify which of M p
Page 83 and 84: • Pseudo-orthogonal groups SO(p,
Page 85 and 86: 3. The Jacobi identity [[ξ, η],
Page 87 and 88: • Special linear group SL(n, R) o
Page 89 and 90: where A is an element of the corres
Page 91 and 92: Here the Jacobi identity has been u
Page 93 and 94: Note that both ad tz and ad y are d
Page 95 and 96: if α + β is a root, [e α , e −
Page 97: • case 3: • case 4: g2 U(q) =
Page 101 and 102: 7.5 Seminar 5 Exercise 1 (Open Toda
Page 103 and 104: 7.6 Seminar 6 Exercise 1 Prove that
Page 105 and 106: is a representation of the group SU
Page 107 and 108: Exercise 4 Consider the non-linear
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Student Seminar: Classical and Quantum Integrable Systems

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