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. 5
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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

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

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