Case 1: Suppose that F (x, v) does not depend on v.

The Euler-Lagrange equations become

‚F dγ0

= 0 ⇐’ the curve γ0 sits on the critical set of F .

γ0 (t), (t)

‚xi dt

For generic F , the critical points are isolated, hence γ0 (t) must be a constant

curve.

Case 2: Suppose that F (x, v) depends a¬nely on v:

n

F (x, v) = F0 (x) + Fj (x)vj .

j=1

n

‚F0 ‚Fj dγj

LHS of (E-L) : (γ(t)) + (γ(t)) (t)

‚xi ‚xi dt

j=1

n

d ‚Fi dγj

RHS of (E-L) : Fi (γ(t)) = (γ(t)) (t)

dt ‚xj dt

j=1

The Euler-Lagrange equations become

n

‚F0 ‚Fi ‚Fj dγj

’

(γ(t)) = (γ(t)) (t) .

‚xi ‚xj ‚xi dt

j=1

n—n matrix

112 APPENDIX C. VARIATIONAL PRINCIPLES

‚Fj

‚Fi

If the n — n matrix ’ has an inverse Gij (x), then

‚xj ‚xi

n

dγj ‚F0

(t) = Gji (γ(t)) (γ(t))

dt ‚xi

i=1

is a system of ¬rst order ordinary di¬erential equations. Locally it has a

unique solution through each point p. If q is not on this curve, there is no

solution at all to the Euler-Lagrange equations belonging to P(a, b, p, q).

Therefore, we need non-linear dependence of F on the v variables in order to have

appropriate solutions. From now on, assume that the

‚2F

Legendre condition: det =0.

‚vi ‚vj

’1

‚2F

Letting Gij (x, v) = (x, v) , the Euler-Lagrange equations become

‚vi ‚vj

d2 γj ‚2F

‚F dγ dγ dγk

’

= Gji γ, Gji γ, .

dt2 ‚xi dt ‚vi ‚xk dt dt

i i,k

This second order ordinary di¬erential equation has a unique solution given initial

conditions

dγ

γ(a) = p and (a) = v .

dt

To check whether the above solution is locally minimizing, assume that

2

‚F

0, ∀(x, v), i.e., with the x variable frozen, the function v ’

‚vi ‚vj (x, v)

F (x, v) is strictly convex.

Suppose that γ0 ∈ P(a, b, p, q) satis¬es (E-L). Does γ0 minimize Aγ ? Locally,

yes, according to the following theorem. (Globally it is only critical.)

Proposition C.3 For every su¬ciently small subinterval [a1 , b1 ] of [a, b], γ0 |[a1 ,b1 ]

is locally minimizing in P(a1 , b1 , p1 , q1 ) where p1 = γ0 (a1 ), q1 = γ0 (b1 ).

Proof. As an exercise in Fourier series, show the Wirtinger inequality: for

f ∈ C 1 ([a, b]) with f (a) = f (b) = 0, we have

2

b b

π2

df

|f |2 dt .

dt ≥

(b ’ a)2

dt

a a

Suppose that γ0 : [a, b] ’ U satis¬es (E-L). Take ci ∈ C ∞ ([a, b]), ci (a) =

ci (b) = 0. Let c = (c1 , . . . , cn ). Let γµ = γ0 + µc ∈ P(a, b, p, q), and let Aµ = Aγµ .

113

C.4. LEGENDRE TRANSFORM

dAµ

(E-L) ⇐’ dµ (0) = 0.

b

d2 Aµ ‚2F dγ0

(0) = γ0 , ci cj dt (I)

dµ2 ‚xi ‚xj dt

a i,j

b

‚2F dγ0 dcj

+2 γ0 , ci dt (II)

‚xi ‚vj dt dt

a i,j

b

‚2F dγ0 dci dcj

+ γ0 , dt (III) .

‚vi ‚vj dt dt dt

a i,j

‚2F

Since ‚vi ‚vj (x, v) 0 at all x, v,

2

dc

≥ KIII

III

dt L2 [a,b]

|I| ¤ KI |c|2 2 [a,b]

L

dc

|II| ¤ KII |c|L2 [a,b]

dt L2 [a,b]

where KI , KII , KIII > 0. By the Wirtinger inequality, if b ’ a is very small, then

III > |I|, |II|. Hence, γ0 is a local minimum.

C.4 Legendre Transform

The Legendre transform gives the relation between the variational (Euler-Lagrange)

and the symplectic (Hamilton-Jacobi) formulations of the equations of motion.

Let V be an n-dimensional vector space, with e1 , . . . , en a basis of V and

v1 , . . . , vn the associated coordinates. Let F : V ’ R, F = F (v1 , . . . , vn ), be

n

a smooth function. Let p ∈ V , u = i=1 ui ei ∈ V . The hessian of F is the

quadratic function on V de¬ned by

‚ 2F

(d2 F )p (u) := (p)ui uj .

‚vi ‚vj

i,j

Exercise 49

d2

Show that (d2 F )p (u) = F (p + tu)|t=0 .

dt2

114 APPENDIX C. VARIATIONAL PRINCIPLES

Exercise 50

A smooth function f : R ’ R is called strictly convex if f (x) > 0 for

all x ∈ R. Assuming that f is strictly convex, prove that the following four

conditions are equivalent:

(a) f (x) = 0 for some point x0 ,

(b) f has a local minimum at some point x 0 ,

(c) f has a unique (global) minimum at some point x 0 ,

(d) f (x) ’ +∞ as x ’ ±∞.

The function f is stable if it satis¬es one (and hence all) of these conditions.

De¬nition C.4 The function F is said to be strictly convex if for every pair

of elements p, v ∈ V , v = 0, the restriction of F to the line {p + xv | x ∈ R} is

strictly convex.

Exercise 51

Show that F is strictly convex if and only if d2 Fp is positive de¬nite for all

p∈V.

Proposition C.5 For a strictly convex function F on V , the following are equiv-

alent:

(a) F has a critical point, i.e., a point where dFp = 0;

(b) F has a local minimum at some point;

(c) F has a unique critical point (global minimum); and

(d) F is proper, that is, F (p) ’ +∞ as p ’ ∞ in V .

Proof. Exercise. (Hint: exercise above.)

De¬nition C.6 A strictly convex function F is stable when it satis¬es conditions

(a)-(d) in Proposition C.5.

Example. The function ex + ax is strictly convex for any a ∈ R, but it is stable

only for a < 0. (What does the graph look like for the values of a ≥ 0 for which it

is not stable?) The function x2 + ax is strictly convex and stable for any a ∈ R. ™¦

—

V — , for

Since V is a vector space, there is a canonical identi¬cation Tp V

every p ∈ V .

115

C.4. LEGENDRE TRANSFORM

De¬nition C.7 The Legendre transform associated to F ∈ C ∞ (V ; R) is the

map

LF : V ’’ V —

—

V— .

p ’’ dFp ∈ Tp V

Exercise 52

Show that, if F is strictly convex, then, for every point p ∈ V , LF maps a

neighborhood of p di¬eomorphically onto a neighborhood of L F (p).

∈ V — , let

Let F be any strictly convex function on V . Given

F : V ’’ R , F (v) = F (v) ’ (v) .

Since (d2 F )p = (d2 F )p ,

⇐’

F is strictly convex F is strictly convex.

De¬nition C.8 The stability set of a strictly convex function F is

SF = { ∈ V — | F is stable} .

Exercise 53

Suppose that F is strictly convex. Prove that:

(a) The set SF is open and convex.

(b) LF maps V di¬eomorphically onto S F .

(c) If l ∈ SF and p0 = L’1 (l), then p0 is the unique minimum point of the

F

function Fl .

Exercise 54

Let F be a strictly convex function. F is said to have quadratic growth at

in¬nity if there exists a positive-de¬nite quadratic form Q on V and a constant

K such that F (p) ≥ Q(p) ’ K, for all p. Show that, if F has quadratic growth

at in¬nity, then SF = V — and hence LF maps V di¬eomorphically onto V — .

For F strictly convex, the inverse to LF is the map L’1 : SF ’’ V described

F

as follows: for l ∈ SF , the value L’1 ( ) is the unique minimum point p ∈ V of

F

F =F ’ .

Exercise 55

Check that p is the minimum of F (v) ’ dF p (v).

116 APPENDIX C. VARIATIONAL PRINCIPLES

De¬nition C.9 The dual function F — to F is

F — : SF ’’ R , F — ( ) = ’ min F (p) .

p∈V

Exercise 56

Show that the function F — is smooth.

Exercise 57

Let F : V ’ R be strictly convex and let F — : SF ’ R be the dual function.

Prove that for all p ∈ V and all l ∈ SF ,

F (p) + F — (l) ≥ l(p) (Young inequality) .

On one hand we have V — V — T — V , and on the other hand, since V = V —— ,

we have V — V — V — — V T — V — . Let ±1 be the canonical 1-form on T — V and ±2

be the canonical 1-form on T — V — . Via the identi¬cations above, we can think of

both of these forms as living on V —V — . Since ±1 = dβ ’±2 , where β : V —V — ’ R

is the function β(p, l) = l(p), we conclude that the forms ω1 = d±1 and ω2 = d±2

satisfy ω1 = ’ω2 .

L’1 = LF — .

Theorem C.10 We have that F

Proof. Let F : V ’ R be strictly convex. Assume that F has quadratic growth at

in¬nity so that SF = V — . Let ΛF be the graph of the Legendre transform LF . The

graph ΛF is a lagrangian submanifold of V — V — with respect to the symplectic

form ω1 (why?). Hence, ΛF is also lagrangian for ω2 .

Let pr1 : ΛF ’ V and pr2 : ΛF ’ V — be the restrictions of the projection

maps V — V — ’ V and V — V — ’ V — , and let i : ΛF ’ V — V — be the inclusion

map. Then (exercise!)

i— ±1 = d(pr1 )— F .

We conclude that

i— ±2 = d(i— β ’ (pr1 )— F ) = d(pr2 )— F — ,

and from this that the inverse of the Legendre transform associated with F is the

Legendre transform associated with F — .

117

C.5. APPLICATION TO VARIATIONAL PROBLEMS

C.5 Application to Variational Problems

Let M be a manifold and F : T M ’ R a function on T M .

γ—F .

Problem. Minimize Aγ = ˜

At p ∈ M , let

Fp := F |Tp M : Tp M ’’ R .

Assume that Fp is strictly convex for all p ∈ M . To simplify notation, assume also

—

that SFp = Tp M . The Legendre transform on each tangent space

—

LFp : Tp M ’’ Tp M

is essentially given by the ¬rst derivatives of F in the v directions. The dual

— —

function to Fp is Fp : Tp M ’’ R. Collect these ¬berwise maps into

’’ T — M ,

L: L|Tp M

TM = L Fp , and

H : T —M —

’’ R , H|Tp M = Fp .

—

Exercise 58

The maps H and L are smooth, and L is a di¬eomorphism.

Let

γ : [a, b] ’’ M be a curve, and

γ : [a, b] ’’ T M

˜ its lift.

Theorem C.11 The curve γ satis¬es the Euler-Lagrange equations on every co-

ordinate chart if and only if L —¦ γ : [a, b] ’ T — M is an integral curve of the

˜

hamiltonian vector ¬eld XH .

Proof. Let

(U, x1 , . . . , xn ) coordinate neighborhood in M ,

(T U, x1 , . . . , xn , v1 , . . . , vn ) coordinates in T M ,

(T — U, x1 , . . . , xn , ξ1 , . . . , ξn ) coordinates in T — M .

On T U we have F = F (x, v). On T — U we have H = H(u, ξ).

’’ T — U

L: TU

‚F

’’ (x, ξ)

(x, v) where ξ = LFx (v) = (x, v) .

‚v

This is the de¬nition of momentum ξ. Then

—

H(x, ξ) = Fx (ξ) = ξ · v ’ F (x, v) L(x, v) = (x, ξ) .

where

118 APPENDIX C. VARIATIONAL PRINCIPLES

Integral curves (x(t), ξ(t)) of XH satisfy the Hamilton equations:

±

dx = ‚H

(x, ξ)

dt ‚ξ

(H)

dξ

‚H

=’ (x, ξ) ,

dt ‚x

whereas the physical path x(t) satis¬es the Euler-Lagrange equations:

‚F dx d ‚F dx

(E-L) x, = x, .

‚x dt dt ‚v dt

dx

Let (x(t), ξ(t)) = L x(t), dt (t) . We want to prove:

dx

t ’ (x(t), ξ(t)) satis¬es (H) ⇐’ t’ x(t), (t) satis¬es (E-L) .

dt

The ¬rst line of (H) is automatically satis¬ed:

dx ‚H dx

(x, ξ) = LFx (ξ) = L’1 (ξ) ⇐’

= ξ = L Fx

—

Fx

dt ‚ξ dt

‚F

= ’ ‚H (x, ξ).

Claim. If (x, ξ) = L(x, v), then ‚x (x, v) ‚x

This follows from di¬erentiating both sides of H(x, ξ) = ξ · v ’ F (x, v) with

respect to x, where ξ = LFx (v) = ξ(x, v).

‚H ‚H ‚ξ ‚ξ ‚F

·v’

+ = .

‚x ‚ξ ‚x ‚x ‚x

v

Now the second line of (H) becomes

d ‚F dξ ‚H ‚F

=’ ⇐’

(x, v) = (x, ξ) = (x, v) (E-L) .

dt ‚v dt ‚x ‚x

by the claim

since ξ = LFx (v)

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Index

action billiards, 39

adjoint, 64, 104 Birkho¬

coadjoint, 64, 104 Poincar´-Birkho¬ theorem, 31, 42

e

coordinates, 55 bracket

de¬nition, 103 Lie, 50

e¬ective, 61 Poisson, 51, 53

free, 105

C 1 -topology, 42, 43

hamiltonian, 57“59

canonical

locally free, 105

symplectic form on a coadjoint

minimizing, 109

orbit, 52, 62, 78

of a path, 108“110

symplectic form on a cotangent

principle of least action, 109

bundle, 15

smooth, 103

symplectomorphism, 18

symplectic, 57

canonical form on T — X

transitive, 105

coordinate de¬nition, 15, 16

action-angle coordinates, 55

intrinsic de¬nition, 17

adapted coordinates, 22

naturality, 17

adjoint

Cartan

action, 64, 104

magic formula, 9, 92, 98

representation, 103, 104

characteristic distribution, 28

angle coordinates, 54

chart

angular momentum, 64

cotangent, 15

antisymmetry, 51

Darboux, 12

arc-length, 34

Chevalley cohomology, 65

Arnold

Christo¬el

Arnold-Liouville theorem, 54

equations, 111

conjecture, 42, 44, 45

symbols, 111

Atiyah-Guillemin-Sternberg theorem,

classical mechanics, 49

60

coadjoint

action, 62, 64, 104

basis

orbit, 62

for skew-symmetric bilinear maps,

representation, 103, 104

1

cohomology

bilinear map, see skew-symmetric bi-

linear map Chevalley, 65

123

124 INDEX

de Rham, 7, 97 construction, 83

Lie algebra, 65 example of Delzant polytope, 82

coisotropic example of non-Delzant polytope,

embedding, 26, 27 82

commutator ideal, 66 polytope, 81, 89

complement theorem, 82

lagrangian, 23 dual function, 116

complete vector ¬eld, 57, 103 dunce cap orbifold, 79

completely integrable system, 54 dynamical system, 42

complex

e¬ective

projective space, 71, 88, 106

action, 61

conehead orbifold, 79

embedding

con¬guration space, 50, 108

closed, 93

conjecture

coisotropic, 26, 27

Arnold, 42, 44, 45

de¬nition, 93

conjugation, 104

isotropic, 27

connectedness, 60

lagrangian, 26

conormal

energy

bundle, 22

classical mechanics, 50

space, 22

energy-momentum map, 57

conservative system, 107

kinetic, 55, 107

constrained system, 109

potential, 55, 107

constraint set, 109

equations

convexity, 60

Christo¬el, 111

cotangent bundle

Euler-Lagrange, 111, 113, 117

as a symplectic manifold, 15

Hamilton, 76, 118

canonical symplectomorphism, 17,

Hamilton-Jacobi, 113

18

of motion, 107

conormal bundle, 23

equivariant

coordinates, 15

moment map, 60

lagrangian submanifold, 20, 22,

tubular neighborhood theorem,

23

73

zero section, 21

euclidean

D™Alembert distance, 33, 36

variational principle, 108 inner product, 34, 36

Darboux norm, 36

chart, 12 space, 34

theorem, 12 Euler

theorem in dimension two, 13 Euler-Lagrange equations, 110, 111,

de Rham cohomology, 7, 97 113, 117

deformation equivalence, 8 variational principle, 108

deformation retract, 97 evaluation map, 103

Delzant exactly homotopic to the identity, 45

125

INDEX

example curve, 35

coadjoint orbits, 63, 64 ¬‚ow, 36, 37

complex projective space, 88 geodesically convex, 35

Delzant construction, 88 minimizing, 35

Hirzebruch surfaces, 82 Gotay

McDu¬, 9 coisotropic embedding, 27

of Delzant polytope, 82 gradient vector ¬eld, 49

of hamiltonian actions, 57, 58 gravitational potential, 108

of lagrangian submanifold, 21 gravity, 55, 56

of mechanical system, 107 group

of non-Delzant polytope, 82 Lie, 102

of symplectic manifold, 5, 6, 15 of symplectomorphisms, 18, 42

of symplectomorphism, 31 one-parameter group of di¬eomor-

phisms, 101, 102

quotient topology, 106

reduction, 70 product, 77

simple pendulum, 55 Guillemin, see Atiyah-Guillemin-Sternberg

spherical pendulum, 56

Hamilton equations, 33, 49, 50, 76,

weighted projective space, 79

108, 118

exponential map, 91

Hamilton-Jacobi equations, 113

hamiltonian

facet, 83

action, 57“59

¬rst integral, 53

function, 48, 49, 53, 59

¬xed point, 38, 42, 44

¬‚ow, 91 G-space, 60

form moment map, 59

area, 11 reduced, 76

canonical, 15“17 system, 53

de Rham, 5 vector ¬eld, 47“49

Hausdor¬ quotient, 106

Fubini-Study, 71

hessian, 113

symplectic, 5

Hirzebruch surface, 82

tautological, 15, 16

free action, 105 homotopy

Fubini-Study form, 71 de¬nition, 97

function formula, 97

dual, 116 invariance, 97

generating, 38 operator, 97

hamiltonian, 49, 59

immersion, 93

stable, 114

integrable

strictly convex, 114

system, 53, 54, 79

G-space, 60 integral

Gauss lemma, 37 curve, 48, 101, 108

generating function, 22, 32, 33, 38 ¬rst, 53

geodesic of motion, 53, 65

126 INDEX

intersection of lagrangian submani- algebra cohomology, 65

folds, 44 bracket, 50, 51

inverse square law, 108 derivative, 92, 98

isotopy group, 102

Lie-Poisson symplectic form, 52, 78

de¬nition, 91

lift

symplectic, 8

of a di¬eomorphism, 17

vs. vector ¬eld, 91

of a path, 109, 110

isotropic

of a vector ¬eld, 48

embedding, 27

linear momentum, 64

subspace, 4

Liouville

isotropy, 105

Arnold-Liouville theorem, 54

Jacobi torus, 54

Hamilton-Jacobi equations, 113 locally free action, 105

identity, 51

manifold

kinetic energy, 55, 107 riemannian, 110

Kirillov, see Kostant-Kirillov symplectic, 5

Marsden-Weinstein-Meyer

Kostant-Kirillov symplectic form, 52,

quotient, 70

78

theorem, 69, 70

Lagrange Maupertius

Euler-Lagrange equations, 111 variational principle, 108

variational principle, 108 McDu¬ counterexample, 9

lagrangian complement, 23 mechanical system, 107

lagrangian ¬bration, 55 mechanics

lagrangian submanifold celestial, 42

closed 1-form, 22 classical, 49

conormal bundle, 22, 23 metric, 34, 110

de¬nition, 20 Meyer, see Marsden-Weinstein-Meyer

generating function, 22, 32 minimizing

intersection problem, 44 action, 109

of T — X, 20 locally, 109, 112

vs. symplectomorphism, 29 moment map

zero section, 21 de¬nition, 58

lagrangian subspace, 4, 23 equivariance, 60

left multiplication, 103 example, 62, 63

left-invariant, 103 existence, 65

Legendre hamiltonian G-space, 60

condition, 112 origin, 47

transform, 113, 115, 116 uniqueness, 67

Leibniz rule, 51, 52 upgraded hamiltonian function,

Lie 58

algebra, 51, 103 moment polytope, 61

127

INDEX

momentum, 50, 64, 117 dunce cap, 79

momentum vector, 64 examples, 78

reduced space, 78

Morse function, 45

teardrop, 79

Morse theory, 45

orbit

Moser

de¬nition, 105

equation, 10

point-orbit projection, 105

theorem “ local version, 11

space, 105

theorem “ version I, 9

topology of the orbit space, 105

theorem “ version II, 10

unstable, 106

trick, 8“10

motion

pendulum

constant of motion, 53

simple, 55, 108

equations, 107

spherical, 56

integral of motion, 53, 65

periodic point, 37

phase space, 50, 76, 108

neighborhood

Picard theorem, 92

convex, 95

Poincar´e

µ-neighborhood theorem, 95

last geometric theorem, 42

tubular neighborhood, 26, 94

Poincar´-Birkho¬ theorem, 31, 42

e

tubular neighborhood ¬bration,

recurrence theorem, 41

96

tubular neighborhood in Rn , 94 point-orbit projection, 105

Poisson

tubular neighborhood theorem,

algebra, 51

95

bracket, 51, 53

Weinstein lagrangian neighbor-

Lie-Poisson symplectic form, 52,

hood, 23, 25

78

Weinstein tubular neighborhood,

structure on g— , 52

26

polytope

Newton

Delzant, 81, 89

second law, 50, 107“109

example of Delzant polytope, 82

Noether

example of non-Delzant polytope,

principle, 47, 65

82

theorem, 65

facet, 83

nondegenerate

moment, 61

bilinear map, 3

rational, 81

¬xed point, 45

simple, 81

normal

smooth, 82

bundle, 94

positive

space, 26, 94

inner product, 34

one-parameter group of di¬eomorphisms, potential

101, 102 energy, 55, 107

orbifold gravitational, 108

conehead, 79 primitive vector, 83

128 INDEX

principle semisimple, 67

Noether, 47, 65 simple pendulum, 55

of least action, 107, 109 simple polytope, 81

variational, 108 skew-symmetric bilinear map

product group, 77 nondegenerate, 3

proper function, 93, 114 rank, 2

pullback, 7 standard form, 1

symplectic, 3

quadratic growth at in¬nity, 115 skew-symmetry

quadrature, 57 de¬nition, 1

quotient forms, 6

Hausdor¬, 106 standard form for bilinear maps,

Marsden-Weinstein-Meyer, 70 1

symplectic, 70 slice theorem, 73

topology, 105 smooth polytope, 82

space

rank, 2

con¬guration, 50, 108

rational polytope, 81

normal, 26, 94

recipe

phase, 50, 108

for symplectomorphisms, 31

spherical pendulum, 56

recurrence, 41

stability

reduced

de¬nition, 114

hamiltonian, 76

set, 115

phase space, 76

stabilizer, 105

space, 70, 78

stable

reduction

function, 114

examples, 70

point, 56

for product groups, 77

Sternberg, see Atiyah-Guillemin-Sternberg

in stages, 77

Stokes theorem, 7

low-brow proof, 70

strictly convex function, 112, 114

other levels, 78

strong isotopy, 7, 11

reduced space, 70

Study, see Fubini-Study

symmetry, 76

subspace

representation

coisotropic, 4

adjoint, 103, 104

isotropic, 4

coadjoint, 103, 104

lagrangian, 4, 23

of a Lie group, 102

symplectic, 4

retraction, 97

symplectic

riemannian

action, 57

distance, 35

basis, 3

manifold, 34, 36, 110

bilinear map, 3

metric, 34, 99, 110

right multiplication, 103 canonical form on a cotangent

right-invariant, 103 bundle, 15

129

INDEX

tautological form on T — X

canonical symplectic form on a

coadjoint orbit, 52, 62, 78 coordinate de¬nition, 15, 16

intrinsic de¬nition, 16

cotangent bundle, 15

naturality, 17

deformation equivalence, 8

property, 17

duality, 3

teardrop orbifold, 79

equivalence, 7

theorem

form, 5, 7

Arnold-Liouville, 54

Fubini-Study form, 71

Atiyah-Guillemin-Sternberg, 60

isotopy, 8

coisotropic embedding, 26

linear algebra, 4

convexity, 60

linear symplectic structure, 3

Darboux, 12, 13

manifold, 5

Delzant, 82

orthogonal, 4

µ-neighborhood, 95

properties of linear symplectic struc-

Euler-Lagrange equations, 117

tures, 3

implicit function, 33

quotient, 70

Marsden-Weinstein-Meyer, 69, 70

reduction, see reduction

Moser “ local version, 11

strong isotopy, 7

Moser “ version I, 9

subspace, 4

Moser “ version II, 10

toric manifolds, 79

Noether, 65

vector ¬eld, 47, 48, 57

Picard, 92

vector space, 2, 3

Poincar´ recurrence, 41

e

volume, 6, 7

Poincar´™s last geometric theo-

e

symplectomorphic, 3, 7

rem, 42

symplectomorphism Poincar´-Birkho¬, 31, 42

e

Arnold conjecture, 42, 44 slice, 73

canonical, 18 standard form for skew-symmetric

cotangent bundle, 19 bilinear maps, 1

de¬nition, 7 Stokes, 7

exactly homotopic to the iden- symplectomorphism vs. lagrangian

tity, 45 submanifold, 29

¬xed point, 42, 44 tubular neighborhood, 26, 94, 95

generating function, 33 tubular neighborhood in Rn , 94

group of symplectomorphisms, 18, Weinstein lagrangian neighbor-

42 hood, 23, 25

linear, 3 Weinstein tubular neighborhood,

recipe, 31, 32 26

vs. lagrangian submanifold, 28, Whitehead lemmas, 67

29 Whitney extension, 25, 98

system time-dependent vector ¬eld, 91

conservative, 107 topology of the orbit space, 105

constrained, 109 toric manifold

mechanical, 107 4-dimensional, 89

130 INDEX

toric manifolds, 79 Young inequality, 116

transitive action, 105

tubular neighborhood

equivariant, 73

¬bration, 96

homotopy-invariance, 97

in Rn , 94

theorem, 26, 94, 95

Weinstein theorem, 26

twisted product form, 28

twisted projective space, 79

unstable

orbit, 106

point, 56

variational

principle, 108

problem, 107, 117

vector ¬eld

complete, 103

gradient, 49

hamiltonian, 47“49

symplectic, 47, 48, 57

vector space

symplectic, 2, 3

volume, 7

weighted projective space, 79

Weinstein

isotropic embedding, 27

lagrangian embedding, 26

lagrangian neighborhood theorem,

23, 25

Marsden-Weinstein-Meyer quotient,

70

Marsden-Weinstein-Meyer theo-

rem, 69, 70

tubular neighborhood theorem,

26

Whitehead lemmas, 67

Whitney extension theorem, 25, 98

Wirtinger inequality, 112, 113

work, 107