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Introduction to Poisson Geometry” Lecture Notes “Introduction to Poisson Geometry” This file contains: 1- Slides from the lectures…………………………………………………………………..…….. page 2 2- Notes taken by Giovanni Bazzoni…………………………………………………………..page 38 3- Main reference for lecture 4 (proof of action-angle theorem in the regular Poisson setting)………………….………………………………………………………………….page 53 Introduction to Poisson Geometry Eva Miranda and Geoffrey Scott UPC-Barcelona and University of Toronto Poisson 2014 Summer School University of Illinois at Urbana-Champaign Miranda and Scott (UPC and UT) Poisson 2014 July 28-31, 2014 1 / 12 Outline 1 Sim´eon-Denis Poisson 2 Jacobi, Lie and Lichnerowicz 3 Motivating examples 4 Plan for lectures 1 and 2 5 Plan for the remaining lectures Miranda and Scott (UPC and UT) Poisson 2014 July 28-31, 2014 1 / 12 Poisson, 1809 Figure: Poisson bracket Miranda and Scott (UPC and UT) Poisson 2014 July 28-31, 2014 2 / 12 Don’t miss the Poisson exhibit! Figure: Young Poisson and Yvette Kosmann-Schwarzbach Figure: Sim´eon-Denis Poisson, Les math´ematiques au service de la science. Exhibit in Paris (March-June 2014). Miranda and Scott (UPC and UT) Poisson 2014 July 28-31, 2014 3 / 12 Jacobi, Lie and Lichnerowicz Figure: Jacobi, Lie and Lichnerowicz Miranda and Scott (UPC and UT) Poisson 2014 July 28-31, 2014 4 / 12 Example 1: Lie algebras of matrix groups The operation on matrices [A, B] = AB − BA is antisymmetric and satisfies [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0, (Jacobi). Example: SO(3, R) = {A ∈ GL(3, R),AT A = Id. det(A) = 1} and T so(3,R) := TId(SO(3, R)) = {A ∈ M(3, R),A + A = 0. T r(A) = 0}. The brackets are determined on a basis 0 1 0 0 0 1 0 0 0 e1 := −1 0 0 , e2 := 0 0 0 , e3 := 0 0 1 0 0 0 −1 0 0 0 −1 0 by [e1, e2] = −e3, [e1, e3] = e2, [e2, e3] = −e1. ∗ Define the (Poisson) bracket using the dual basis x1, x2, x3 in so(3,R) {x1, x2} = −x3, {x1, x3} = x2, {x2, x3} = −x1 It satifies Jacobi {xi, {xj, xk}} + {xj, {xk, xi}} + {xk, {xi, xj}} = 0 Miranda and Scott (UPC and UT) Poisson 2014 July 28-31, 2014 5 / 12 From Lie algebras to Poisson structures (Exercise 4) Another way to write the Poisson bracket ∂ ∂ ∂ ∂ ∂ ∂ −x3 ∧ + x2 ∧ − x1 ∧ ∂x1 ∂x2 ∂x1 ∂x3 ∂x2 ∂x3 Using the properties of the Poisson bracket, 2 2 2 2 2 2 {x1 + x2 + x3, xi} = 0, i = 1, 2, 3 and the function f = x1 + x2 + x3 is a constant of motion. Each sphere is endowed with an area form (symplectic structure). Miranda and Scott (UPC and UT) Poisson 2014 July 28-31, 2014 6 / 12 Example 2: Determinants in R3 (Exercise 12) Dynamics: Given two functions H, K ∈ C∞(R3). Consider the system of differential equations: (x, ˙ y,˙ z˙) = dH ∧ dK (1) H and K are constants of motion (the flow lies on H = cte. and K = cte.) Geometry: Consider the brackets, {f, g}H := det(df, dg, dH) {f, g}K := det(df, dg, dK) They are antisymmetric and satisfy Jacobi, {f, {g, h}} + {g, {h, f}} + {h, {f, g}} = 0. The flow of the vector field {K, ·}H := det(dK, ·, dH) and {−H, ·}K is given by the differential equation (1) and {H, K}H = 0, {H, K}K = 0 Miranda and Scott (UPC and UT) Poisson 2014 July 28-31, 2014 7 / 12 Example 3: Hamilton’s equations The equations of the movement of a particle can be written as Hamilton’s equation using the change pi =q ˙i, There is a geometrical structure behind this formula symplectic form ω (closed non-degenerate 2-form). Non-degeneracy for every smooth function f, there exists a unique vector field Xf (Hamiltonian vector field), iXf ω = −df Miranda and Scott (UPC and UT) Poisson 2014 July 28-31, 2014 8 / 12 Example 4: Coupling two simple harmonic oscillators ∗ 2 The phase space is (T (R ), ω = dx1 ∧ dy1 + dx2 ∧ dy2). H is the sum of potential and kinetic energy, 1 1 H = (y2 + y2) + (x2 + x2) 2 1 2 2 1 2 3 H = h is a sphere S . We have rotational symmetry on this sphere the angular momentum is a constant of motion, L = x1y2 − x2y1, XL = (−x2, x1, −y2, y1) and XL(H) = {L, H} = 0. Miranda and Scott (UPC and UT) Poisson 2014 July 28-31, 2014 9 / 12 Example 5: Cauchy-Riemann equations and Hamilton’s equations 2 Take a holomorphic function on F : C −→ C decompose it as 4 F = G + iH with G, H : R −→ R. Cauchy-Riemann equations for F in coordinates zj = xj + iyj, j = 1, 2 ∂G ∂H ∂G ∂H = , = − ∂xi ∂yi ∂yi ∂xi Reinterpret these equations as the equality {G, ·}0 = {H, ·}1 with {·, ·}j the Poisson brackets associated to the real and imaginary part of the symplectic form ω = dz1 ∧ dz2 (ω = ω0 + iω1). Check {G, H}0 = 0 and {H, G}1 = 0 (integrable system). Miranda and Scott (UPC and UT) Poisson 2014 July 28-31, 2014 10 / 12 Plan for the first 2 lectures Definition and examples. Weinstein’s splitting theorem and symplectic foliation. Figure: Alan Weinstein and Reeb foliation Normal form theorems. Figure: Marius Crainic, Rui Loja Fernandes and Ionut Marcut Miranda and Scott (UPC and UT) Poisson 2014 July 28-31, 2014 11 / 12 Plan for the remaining lectures Global aspects: Poisson cohomology and modular class. Compatible Poisson structures and integrable systems. Hamiltonian group actions and integrable systems. Action-angle coordinates on Poisson manifolds. Miranda and Scott (UPC and UT) Poisson 2014 July 28-31, 2014 12 / 12 Introduction to Poisson Geometry- Lecture 3 Eva Miranda and Geoffrey Scott UPC-Barcelona and University of Toronto Poisson 2014 Summer School University of Illinois at Urbana-Champaign Miranda and Scott (UPC and UT) Poisson 2014 July 28-31, 2014 1 / 5 Schouten Bracket of vector fields in local coordinates Case of vector fields, A = P a ∂ and A = P b ∂ . Then i i ∂xi i i ∂xi ∂b ∂ ∂a ∂ [A, B] = X a (X j ) − X b (X j ) i ∂x ∂x i ∂x ∂x i j i j i j i j ∂ Re-denoting ∂x as ζi (“odd coordinates ”). P i P Then A = i aiζi and B = i biζi and ζiζj = −ζjζi Now we can reinterpret the bracket as, ∂A ∂B ∂B ∂A [A, B] = X − X ∂ζ ∂x ∂ζ ∂x i i i i i i Miranda and Scott (UPC and UT) Poisson 2014 July 28-31, 2014 2 / 5 Schouten Bracket of multivector fields in local coordinates We reproduce the same scheme for the case of multivector fields. ∂A ∂B ∂B ∂A [A, B] = X − (−1)(a−1)(b−1) X ∂ζ ∂x ∂ζ ∂x i i i i i i is a (a + b − 1)-vector field. where X ∂ ∂ X A = Ai1,...,ia ∧ · · · ∧ = Ai1,...,ia ζi1 . ζia ∂xi1 ∂xia i1<···<ia i1<···<ia and X ∂ ∂ X B = Bi1,...,ib ∧ · · · ∧ = Bi1,...,ib ζi1 . ζib ∂xi1 ∂xib i1<···<ib i1<···<ib ∂(ζi1 ...ζip ) (p−k) with := (−1) ηi1 ... ηik ηip−1 ∂ζik b Miranda and Scott (UPC and UT) Poisson 2014 July 28-31, 2014 3 / 5 Theorem (Schouten-Nijenhuis) The bracket defined by this formula satisfies, Graded anti-commutativity [A, B] = −(−1)(a−1)(b−1)[B, A]. Graded Leibniz rule [A, B ∧ C] = [A, B] ∧ C + (−1)(a−1)bB ∧ [A, C] Graded Jacobi identity (−1)(a−1)(c−1)[A, [B, C]]+(−1)(b−1)(a−1)[B, [C, A]]+(−1)(c−1)(b−1)[C, [A, B]] = 0 If X is a vector field then, [X, B] = LX B. Miranda and Scott (UPC and UT) Poisson 2014 July 28-31, 2014 4 / 5 Example 5: Cauchy-Riemann equations and Hamilton’s equations 2 Take a holomorphic function on F : C −→ C decompose it as 4 F = G + iH with G, H : R −→ R. Cauchy-Riemann equations for F in coordinates zj = xj + iyj, j = 1, 2 ∂G ∂H ∂G ∂H = , = − ∂xi ∂yi ∂yi ∂xi Reinterpret these equations as the equality {G, ·}0 = {H, ·}1 {H, ·}0 = −{G, ·}1 with {·, ·}j the Poisson brackets associated to the real and imaginary part of the symplectic form ω = dz1 ∧ dz2 (ω = ω0 + iω1). Check {G, H}0 = 0 and {H, G}1 = 0 (integrable system). Miranda and Scott (UPC and UT) Poisson 2014 July 28-31, 2014 5 / 5 Introduction to Poisson Geometry Lecture 4 (Integrable Systems) Eva Miranda and Geoffrey Scott UPC-Barcelona and University of Toronto Poisson 2014 Summer School University of Illinois at Urbana-Champaign Miranda and Scott (UPC and UT) Poisson 2014 July 28-31, 2014 1 / 15 Example 5: Cauchy-Riemann equations and Hamilton’s equations 2 Take a holomorphic function on F : C −→ C decompose it as 4 F = G + iH with G, H : R −→ R. Cauchy-Riemann equations for F in coordinates zj = xj + iyj, j = 1, 2 ∂G ∂H ∂G ∂H = , = − ∂xi ∂yi ∂yi ∂xi Reinterpret these equations as the equality {G, ·}0 = {H, ·}1 {H, ·}0 = −{G, ·}1 with {·, ·}j the Poisson brackets associated to the real and imaginary part of the symplectic form ω = dz1 ∧ dz2 (ω = ω0 + iω1). Check {G, H}0 = 0 and {H, G}1 = 0 (integrable system). Miranda and Scott (UPC and UT) Poisson 2014 July 28-31, 2014 2 / 15 Example 2: Determinants in R3 (Exercise 12) Dynamics: Given two functions H, K ∈ C∞(R3). Consider the system of differential equations: (x, ˙ y,˙ z˙) = dH ∧ dK (1) H and K are constants of motion (the flow lies on H = cte.
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