Notes on symplectic geometry

109 Pages·2015·0.437 MB·English

by  Konstantin Athanassopoulos

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