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Notes on symplectic geometry PDF

109 Pages·2015·0.437 MB·English
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University of Crete Department of Mathematics Notes on Symplectic Geometry Konstantin Athanassopoulos Iraklion, 2015 ii Contents 1 Introduction 3 1.1 Newtonian mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Lagrangian mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 The equations of Hamilton . . . . . . . . . . . . . . . . . . . . . . . 16 2 Basic symplectic geometry 19 2.1 Symplectic linear algebra . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 The symplectic linear group . . . . . . . . . . . . . . . . . . . . . . . 24 2.3 Symplectic manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.4 Local description of symplectic manifolds . . . . . . . . . . . . . . . 36 2.5 Lagrangian submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.6 Hamiltonian vector (cid:12)elds and Poisson bracket . . . . . . . . . . . . . 43 3 Examples of symplectic manifolds 49 3.1 The geometry of the tangent bundle . . . . . . . . . . . . . . . . . . 49 3.2 The manifold of geodesics . . . . . . . . . . . . . . . . . . . . . . . . 54 3.3 K(cid:127)ahler manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.4 Coadjoint orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.5 Homogeneous symplectic manifolds . . . . . . . . . . . . . . . . . . . 70 3.6 Poisson manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4 Symmetries and integrability 79 4.1 Symplectic group actions . . . . . . . . . . . . . . . . . . . . . . . . 79 4.2 Momentum maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.3 Symplectic reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.4 Completely integrable Hamiltonian systems . . . . . . . . . . . . . . 90 4.5 Hamiltonian torus actions . . . . . . . . . . . . . . . . . . . . . . . . 95 1 2 CONTENTS Chapter 1 Introduction 1.1 Newtonian mechanics In newtonian mechanics the state of a mechanical system is described by a (cid:12)nite number of real parameters. The set of all possible positions, of a material point for example, is a (cid:12)nite dimensional smooth manifold M, called the con(cid:12)guration space. A motion of the system is a smooth curve (cid:13) : I ! M, where I (cid:26) R is an open interval. The velocity (cid:12)eld of (cid:13) is smooth curve (cid:13)_ : I ! TM. The (total space of the) tangent bundle TM of M is called the phase space. According to Newton, the total force is a vector (cid:12)eld F that acts on the points of the con(cid:12)guration space. Locally, a motion is a solution of the second order differential equation F = mx(cid:127), where m is the mass. Equivalently, (cid:13)_ is locally a solution of the (cid:12)rst order differential equation ( ) ( ) v x_ = 1 : v_ F(x;v;t) m Consider a system of N particles in R3 subject to some forces. If x denotes the i position of the i-th particle then the con(cid:12)guration space is (R3)N and Newton’s law of motion is d2x m i = F (x ;:::;x ;x_ ;:::;x_ ;t); 1 (cid:20) i (cid:20) N; i dt2 i 1 N 1 N wherem isthemassandF istheforceonthei-thparticle. Relabelingthevariables i i settingq3i,q3i+1 andq3i+2 thecoordinatesofx inthisorder,thecon(cid:12)gurationspace i becomes Rn, n = 3N, and the equations of motion take the form d2qj m = F (q1;:::;qn;q_1;:::;q_n;t); 1 (cid:20) j (cid:20) n j dt2 j Suppose that the forces do not depend on time and are conservative. This means that there is a smooth function V : Rn ! R such that @V F (q1;:::;qn;q_1;:::;q_n) = (cid:0) ; 1 (cid:20) j (cid:20) n: j @qj 3 4 CHAPTER 1. INTRODUCTION For instantce, this is the case if N particles interact by gravitational attraction. Rewritting Newton’s law as a system of (cid:12)rst oder ordinary differential equations dqj d2vj @V = vj; m = (cid:0) ; 1 (cid:20) j (cid:20) n; dt j dt2 @qj or changing coordinates to p = m vj we have j j dqj 1 d2p @V = p ; j = (cid:0) ; 1 (cid:20) j (cid:20) n; dt m j dt2 @qj j The solutions of the above system of ordinary differential equations are the integral curves of the smooth vector (cid:12)eld ∑n ∑n 1 @ @V @ X = p (cid:0) : m j@qj @qj @p j j j=1 j=1 Note that the smooth function ∑n 1 H(q1;:::;qn;p ;:::;p ) = p2+V(q1;:::;qn) 1 n 2m j j j=1 is constant along solutions, because ∑n ∑n @V 1 dH = dqj + p dp @qj m j j j j=1 j=1 and so dH(X) = 0. Actually, H completely determines X in the following sense. Let ∑n ! = dqj ^dp : j j=1 Then, ∑n ∑n ∑n ∑n 1 @V i ! = dqj(X)dp (cid:0) dp (X)dqj = p dp + dqj = dH: X j j m j j @qj j j=1 j=1 j=1 j=1 The smooth 2-form ! is closed and non-degenerate. The latter means that given any smooth 1-form (cid:17) the equation i ! = (cid:17) has a unique solution Y. Indeed, for Y any smooth vector (cid:12)eld Y we have ∑n ∑n @ @ @ @ Y = !(Y; ) (cid:0) !(Y; ) @p @qj @qj @p j j j=1 j=1 and so i ! = 0 if and only if Y = 0. Y Returning to Newtonian mechanics, we give the following de(cid:12)nition. 1.1. NEWTONIAN MECHANICS 5 De(cid:12)nition 1.1. An (autonomous) newtonian mechamical system is a triple (M;g;X), where M is a smooth manifold, g is a Riemannian metric on M and X is a smooth vector (cid:12)eld on TM such that (cid:25)(cid:3)X = id. A motion of (M;g;X) is a smooth curve (cid:13) : I ! M such that (cid:13)_ : I ! TM is an integral curve of X. The 1 smoothfunctionT : TM ! Rde(cid:12)nedbyT(v) = g(v;v)iscalledthekineticenergy. 2 Examples 1.2. (a) Let M = R, so that we may identify TM with R2 and (cid:25) with the projection onto the (cid:12)rst coordinate. If g is the euclidean riemannian metric on R and @ 1 @ X = v + ((cid:0)k2x(cid:0)(cid:26)v) ; k > 0; (cid:26) (cid:21) 0; @x m @v then obviously (cid:25)(cid:3)X = id and a motion is a solution of the second order differential equation mx(cid:127) = (cid:0)k2x(cid:0)(cid:26)x_: This mechanical system describes the oscillator. (b) The geodesic vector (cid:12)eld G of a Riemannian n-manifold (M;g) de(cid:12)nes a newtonian mechanical system. Locally it has the expression ∑n ∑n @ @ G = vk (cid:0) (cid:0)kvivj @xk ij @vk k=1 i;j;k=1 where (cid:0)k are the Christofell symbols. ij Often a mechanical system has potential energy. This is a smooth function V : M ! R. Let gradV be the gradient of V with respect to the Riemannian metric. If for every v 2 TM we set (cid:12) (cid:12) d (cid:12) gradV = (cid:12) (v+tgradV((cid:25)(v)); dt t=0 then gradV 2 X(TM) and (cid:25)(cid:3)gradV = 0, since (cid:25)(v + tgradV((cid:25)(v))) = (cid:25)(v), for every t 2 R. Locally, if g = (g ) and (g )(cid:0)1 = (gij), then ij ij ∑n ∑n @V @ @V @ gradV = gij and gradV = gij : @xi@xj @xi@vj i;j=1 i;j=1 De(cid:12)nition 1.3. A newtonian mechanical system with potential energy is a triple (M;g;V), where (M;g) is a Riemannian manifold and V : M ! R is a smooth function called the potential energy. The corresponding vector (cid:12)eld on TM is Y = G (cid:0) gradV, where G is the geodesic vector (cid:12)eld. The smooth function E = T +V ◦(cid:25) : TM ! R is called the mechanical energy. 6 CHAPTER 1. INTRODUCTION Proposition 1.4. (Conservation of energy) In a newtonian mechanical system with potential energy (M;g;V) the mechanical energy is a constant of motion. Proof. We want to show that Y(E) = 0. We compute locally, where we have ∑n 1 E = g vivj +V and ij 2 i;j=1 ( ) ∑n ∑n ∑n ∑n @ @V @ Y = vk (cid:0) (cid:0)kvivj + gik : @xk ij @xi @vk k=1 k=1 i;j=1 i=1 Recall that ∑n ( ) 1 @g @g @g (cid:0)k = gkl il + jl (cid:0) ij : ij 2 @xj @xi @xl l=1 We can now compute ( ) ∑n ∑n ∑n ∑n ∑n @V 1 @g @V @V Y(E) = vk + ijvivjvk (cid:0) (cid:0)kvivj + gik @xk 2 @xk ij @xi @vk k=1 i;j;k=1 k=1 i;j=1 i=1 ( )( ) ∑n ∑n ∑n ∑n @V (cid:0) (cid:0)kvivj + gik g vi ij @xi ik k=1 i;j=1 i=1 i=1 ( )( ) ∑n ∑n ∑n ∑n @V @V 1 @g = vk (cid:0) gik g vi + ijvivjvk @xk @xi ik 2 @xk k=1 i=1 i=1 i;j;k=1 ( )( ) ∑n ∑n ∑n ∑n ( ) 1 @g @g @g (cid:0) g vr gkl il + jl (cid:0) ij vivj rk 2 @xj @xi @xl k=1 r=1 i;j=1 l=1 ∑n ∑n ( ) 1 @g 1 @g @g @g = ijvivjvk (cid:0) il + jl (cid:0) ij vivjvl = 0: □ 2 @xk 2 @xj @xi @xl i;j;k=1 i;j;l=1 A smooth curve (cid:13) : I ! M is a motion of a newtonian mechanical system on M with potential energy V if and only if (cid:13) satis(cid:12)es the second order differential equation ∇ (cid:13)_ = (cid:0)gradV (cid:13)_ where ∇ is the Levi-Civita connection on M. In case V has an upper bound, then a motion is a geodesic with respect to a new Riemannian metric on M, possibly reparametrized. So, suppose that there exists some e > 0 such that V(x) < e for every x 2 M. On M we consider the new Riemannian metric g(cid:3) = (e(cid:0)V)g. Let (cid:13) be a montion with mechanical energy e, that is 1 g((cid:13)_(t);(cid:13)_(t))+V((cid:13)(t)) = e 2 1.1. NEWTONIAN MECHANICS 7 for every t 2 I. Since e > V((cid:13)(t)), we have (cid:13)_(t) ̸= 0 for every t 2 I. The function s : I ! R with ∫ p t s(t) = 2 (e(cid:0)V((cid:13)((cid:28)))d(cid:28); t0 where t 2 I, is smooth and strictly increasing. Let (cid:13)(cid:3) = (cid:13) ◦s(cid:0)1. 0 Theorem 1.5. (Jacobi-Maupertuis) If (cid:13) is a motion of the mechanical system (M;g;V) and V(x) < e for every x 2 M, then its reparametrization (cid:13)(cid:3) is a geodesic with respect to the Riemannian metric g(cid:3) = (e(cid:0)V)g. Proof. It suffices to carry out the computation locally. The Christofell symbols of (cid:3) the metric g are given by the formula ( ) ∑n 1 @V @V @V ∆k = (cid:0)k + (cid:0) (cid:14) (cid:0) (cid:14) + glkg : ij ij 2(e(cid:0)V) @xi jk @xj ik @xl ij l=1 If in the local coordinates we have (cid:13) = (x1;x2;:::;xn), then dxk dxk dt 1 dxk = (cid:1) = p (cid:1) ds dt ds 2(e(cid:0)V) dt and so d2xk 1 d2xk 1 dxk ∑n @V dxl = (cid:1) + (cid:1) : ds2 2(e(cid:0)V)2 d2t 2(e(cid:0)V)3 dt @xl dt l=1 Since d2xk ∑n dxidxj ∑n @V = (cid:0) (cid:0)k (cid:0) gkl d2t ij dt dt @xl i;j=1 l=1 and ∑n dxidxj g = 2(e(cid:0)V) ij dt dt i;j=1 substituting we get d2xk ∑n dxidxj + ∆k = ds2 ij ds ds i;j=1 1 ∑n dxidxj 1 ∑n @V 1 dxk ∑n @V dxl (cid:0) (cid:0)k (cid:0) gkl + 2(e(cid:0)V)2 ij dt dt 2(e(cid:0)V)2 @xl 2(e(cid:0)V)3 dt @xl dt i;j=1 l=1 l=1 1 ∑n dxidxj 1 dxk ∑n @V dxi 1 dxk ∑n @V dxj + (cid:0)k (cid:0) (cid:0) 2(e(cid:0)V)2 ij dt dt 4(e(cid:0)V)3 dt @xi dt 4(e(cid:0)V)3 dt @xj dt i;j=1 i=1 j=1 ( )( ) 1 ∑n @V ∑n dxidxj + gkl g = 4(e(cid:0)V)3 @xl ij dt dt l=1 i;j=1 ( )( ) 1 ∑n @V 1 ∑n @V ∑n dxidxj (cid:0) gkl + gkl g = 0: □ 2(e(cid:0)V)2 @xl 4(e(cid:0)V)3 @xl ij dt dt l=1 l=1 i;j=1 8 CHAPTER 1. INTRODUCTION 1.2 Lagrangian mechanics Let (M;g;V) be a newtonian mechanical system with potential energy V and let L : TM ! R be the smooth function L = T (cid:0)V ◦(cid:25), where T is the kinetic energy and (cid:25) : TM ! M is the tangent bundle projection. Theorem 2.1. (d’Alembert-Lagrange) A smooth curve (cid:13) : I ! M is a motion of the mechanical system (M;g;V) if and only if ( ) d @L @L ((cid:13)_(t)) = ((cid:13)_(t)) dt @vi @xi for every t 2 I and i = 1, 2..., n, where n is the dimension of M. Proof. Suppose that in local coordinates we have (cid:13) = (x1;x2;:::;xn). Recall that (cid:13) is a motion of (M;g;V) if and only if ∑n ∑n @V x(cid:127)k = (cid:0) (cid:0)kx_ix_j (cid:0) glk: ij @xl i;j=1 l=1 Since ∑n 1 L((cid:13)_) = g x_ix_j (cid:0)V((cid:13)); ij 2 i;j=1 for every i = 1, 2,..., n we have ( ) ( ) ∑n ∑n d @L @L d 1 @g @V ((cid:13)_(t)) (cid:0) ((cid:13)_(t)) = g x_j (cid:0) mlx_mx_l + ((cid:13)(t)) = dt @vi @xi dt ij 2 @xi @xi j=1 m;l=1 ∑n ∑n ∑n ∑n @g 1 @g @V ijx_lx_j + g x(cid:127)j (cid:0) mlx_mx_l + ((cid:13)(t)) = @xl ij 2 @xi @xi j=1 l=1 j=1 m;l=1 ( ) ∑n ∑n @g 1@g @V im (cid:0) ml x_mx_l + g x(cid:127)j + ((cid:13)(t)): @xl 2 @xi ij @xi m;l=1 j=1 Taking the image of the vector with these coordinates by (g )(cid:0)1 = (gij), we see ij that the equations in the statement of the theorem are equivalent to ( ( ) ) ∑n ∑n ∑n ∑n @g 1@g @V 0 = gik im (cid:0) ml x_mx_l + gikg x(cid:127)j + gik = @xl 2 @xi ij @xi i=1 m;l=1 j=1 i=1 ( ) ∑n ∑n ∑n @g 1@g @V x(cid:127)k + gik im (cid:0) ml x_mx_l + gik = @xl 2 @xi @xi m;l=1i=1 i=1 ∑n ∑n @V x(cid:127)k + (cid:0)k x_mx_l + gik: □ ml @xi m;l=1 i=1

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