<< стр. 5(всего 5)СОДЕРЖАНИЕ

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

The Euler-Lagrange equations become

в€‚F dОі0
= 0 в‡ђв‡’ the curve Оі0 sits on the critical set of F .
Оі0 (t), (t)
в€‚xi dt

For generic F , the critical points are isolated, hence Оі0 (t) must be a constant
curve.

Case 2: Suppose that F (x, v) depends aп¬ѓnely on v:
n
F (x, v) = F0 (x) + Fj (x)vj .
j=1

n
в€‚F0 в€‚Fj dОіj
LHS of (E-L) : (Оі(t)) + (Оі(t)) (t)
в€‚xi в€‚xi dt
j=1

n
d в€‚Fi dОіj
RHS of (E-L) : Fi (Оі(t)) = (Оі(t)) (t)
dt в€‚xj dt
j=1

The Euler-Lagrange equations become
n
в€‚F0 в€‚Fi в€‚Fj dОіj
в€’
(Оі(t)) = (Оі(t)) (t) .
в€‚xi в€‚xj в€‚xi dt
j=1
nГ—n matrix
112 APPENDIX C. VARIATIONAL PRINCIPLES

в€‚Fj
в€‚Fi
If the n Г— n matrix в€’ has an inverse Gij (x), then
в€‚xj в€‚xi

n
dОіj в€‚F0
(t) = Gji (Оі(t)) (Оі(t))
dt в€‚xi
i=1

is a system of п¬Ѓrst order ordinary diп¬Ђerential 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
в€‚2F
Legendre condition: det =0.
в€‚vi в€‚vj
в€’1
в€‚2F
Letting Gij (x, v) = (x, v) , the Euler-Lagrange equations become
в€‚vi в€‚vj

d2 Оіj в€‚2F
в€‚F dОі dОі dОіk
в€’
= Gji Оі, Gji Оі, .
dt2 в€‚xi dt в€‚vi в€‚xk dt dt
i i,k

This second order ordinary diп¬Ђerential equation has a unique solution given initial
conditions
dОі
Оі(a) = p and (a) = v .
dt
To check whether the above solution is locally minimizing, assume that
2
в€‚F
0, в€Ђ(x, v), i.e., with the x variable frozen, the function v в†’
в€‚vi в€‚vj (x, v)
F (x, v) is strictly convex.
Suppose that Оі0 в€€ P(a, b, p, q) satisп¬Ѓes (E-L). Does Оі0 minimize AОі ? Locally,
yes, according to the following theorem. (Globally it is only critical.)

Proposition C.3 For every suп¬ѓciently small subinterval [a1 , b1 ] of [a, b], Оі0 |[a1 ,b1 ]
is locally minimizing in P(a1 , b1 , p1 , q1 ) where p1 = Оі0 (a1 ), q1 = Оі0 (b1 ).

Proof. As an exercise in Fourier series, show the Wirtinger inequality: for
f в€€ C 1 ([a, b]) with f (a) = f (b) = 0, we have
2
b b
ПЂ2
df
|f |2 dt .
dt в‰Ґ
(b в€’ a)2
dt
a a

Suppose that Оі0 : [a, b] в†’ U satisп¬Ѓes (E-L). Take ci в€€ C в€ћ ([a, b]), ci (a) =
ci (b) = 0. Let c = (c1 , . . . , cn ). Let ОіОµ = Оі0 + Оµc в€€ P(a, b, p, q), and let AОµ = AОіОµ .
113
C.4. LEGENDRE TRANSFORM

dAОµ
(E-L) в‡ђв‡’ dОµ (0) = 0.
b
d2 AОµ в€‚2F dОі0
(0) = Оі0 , ci cj dt (I)
dОµ2 в€‚xi в€‚xj dt
a i,j
b
в€‚2F dОі0 dcj
+2 Оі0 , ci dt (II)
в€‚xi в€‚vj dt dt
a i,j
b
в€‚2F dОі0 dci dcj
+ Оі0 , dt (III) .
в€‚vi в€‚vj dt dt dt
a i,j

в€‚2F
Since в€‚vi в€‚vj (x, v) 0 at all x, v,

2
dc
в‰Ґ KIII
III
dt L2 [a,b]

|I| в‰¤ KI |c|2 2 [a,b]
L

dc
|II| в‰¤ KII |c|L2 [a,b]
dt L2 [a,b]

where KI , KII , KIII > 0. By the Wirtinger inequality, if b в€’ a is very small, then
III > |I|, |II|. Hence, Оі0 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 в†’ R, F = F (v1 , . . . , vn ), be
n
a smooth function. Let p в€€ V , u = i=1 ui ei в€€ V . The hessian of F is the
quadratic function on V deп¬Ѓned by

в€‚ 2F
(d2 F )p (u) := (p)ui uj .
в€‚vi в€‚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 в†’ R is called strictly convex if f (x) > 0 for
all x в€€ 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) в†’ +в€ћ as x в†’ В±в€ћ.
The function f is stable if it satisп¬Ѓes one (and hence all) of these conditions.

Deп¬Ѓnition C.4 The function F is said to be strictly convex if for every pair
of elements p, v в€€ V , v = 0, the restriction of F to the line {p + xv | x в€€ R} is
strictly convex.

Exercise 51
Show that F is strictly convex if and only if d2 Fp is positive deп¬Ѓnite for all
pв€€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) в†’ +в€ћ as p в†’ в€ћ in V .

Proof. Exercise. (Hint: exercise above.)

Deп¬Ѓnition C.6 A strictly convex function F is stable when it satisп¬Ѓes conditions
(a)-(d) in Proposition C.5.

Example. The function ex + ax is strictly convex for any a в€€ R, but it is stable
only for a < 0. (What does the graph look like for the values of a в‰Ґ 0 for which it
is not stable?) The function x2 + ax is strictly convex and stable for any a в€€ R. в™¦

в€—
V в€— , for
Since V is a vector space, there is a canonical identiп¬Ѓcation Tp V
every p в€€ V .
115
C.4. LEGENDRE TRANSFORM

Deп¬Ѓnition C.7 The Legendre transform associated to F в€€ C в€ћ (V ; R) is the
map
LF : V в€’в†’ V в€—
в€—
Vв€— .
p в€’в†’ dFp в€€ Tp V

Exercise 52
Show that, if F is strictly convex, then, for every point p в€€ V , LF maps a
neighborhood of p diп¬Ђeomorphically onto a neighborhood of L F (p).

в€€ V в€— , let
Let F be any strictly convex function on V . Given

F : V в€’в†’ R , F (v) = F (v) в€’ (v) .

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

в‡ђв‡’
F is strictly convex F is strictly convex.

Deп¬Ѓnition C.8 The stability set of a strictly convex function F is

SF = { в€€ V в€— | 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 diп¬Ђeomorphically onto S F .
(c) If l в€€ SF and p0 = Lв€’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
inп¬Ѓnity if there exists a positive-deп¬Ѓnite quadratic form Q on V and a constant
K such that F (p) в‰Ґ Q(p) в€’ K, for all p. Show that, if F has quadratic growth
at inп¬Ѓnity, then SF = V в€— and hence LF maps V diп¬Ђeomorphically onto V в€— .

For F strictly convex, the inverse to LF is the map Lв€’1 : SF в€’в†’ V described
F
as follows: for l в€€ SF , the value Lв€’1 ( ) is the unique minimum point p в€€ V of
F
F =F в€’ .

Exercise 55
Check that p is the minimum of F (v) в€’ dF p (v).
116 APPENDIX C. VARIATIONAL PRINCIPLES

Deп¬Ѓnition C.9 The dual function F в€— to F is

F в€— : SF в€’в†’ R , F в€— ( ) = в€’ min F (p) .
pв€€V

Exercise 56
Show that the function F в€— is smooth.

Exercise 57
Let F : V в†’ R be strictly convex and let F в€— : SF в†’ R be the dual function.
Prove that for all p в€€ V and all l в€€ SF ,
F (p) + F в€— (l) в‰Ґ l(p) (Young inequality) .

On one hand we have V Г— V в€— T в€— V , and on the other hand, since V = V в€—в€— ,
we have V Г— V в€— V в€— Г— V T в€— V в€— . Let О±1 be the canonical 1-form on T в€— V and О±2
be the canonical 1-form on T в€— V в€— . Via the identiп¬Ѓcations above, we can think of
both of these forms as living on V Г—V в€— . Since О±1 = dОІ в€’О±2 , where ОІ : V Г—V в€— в†’ R
is the function ОІ(p, l) = l(p), we conclude that the forms П‰1 = dО±1 and П‰2 = dО±2
satisfy П‰1 = в€’П‰2 .

Lв€’1 = LF в€— .
Theorem C.10 We have that F

Proof. Let F : V в†’ R be strictly convex. Assume that F has quadratic growth at
inп¬Ѓnity so that SF = V в€— . Let О›F be the graph of the Legendre transform LF . The
graph О›F is a lagrangian submanifold of V Г— V в€— with respect to the symplectic
form П‰1 (why?). Hence, О›F is also lagrangian for П‰2 .
Let pr1 : О›F в†’ V and pr2 : О›F в†’ V в€— be the restrictions of the projection
maps V Г— V в€— в†’ V and V Г— V в€— в†’ V в€— , and let i : О›F в†’ V Г— V в€— be the inclusion
map. Then (exercise!)
iв€— О±1 = d(pr1 )в€— F .

We conclude that

iв€— О±2 = d(iв€— ОІ в€’ (pr1 )в€— F ) = d(pr2 )в€— F в€— ,

and from this that the inverse of the Legendre transform associated with F is the
Legendre transform associated with F в€— .
117
C.5. APPLICATION TO VARIATIONAL PROBLEMS

C.5 Application to Variational Problems
Let M be a manifold and F : T M в†’ R a function on T M .
Оів€—F .
Problem. Minimize AОі = Лњ
At p в€€ M , let
Fp := F |Tp M : Tp M в€’в†’ R .
Assume that Fp is strictly convex for all p в€€ M . To simplify notation, assume also
в€—
that SFp = Tp M . The Legendre transform on each tangent space

в€—
LFp : Tp M в€’в†’ Tp M

is essentially given by the п¬Ѓrst derivatives of F in the v directions. The dual
в€— в€—
function to Fp is Fp : Tp M в€’в†’ R. Collect these п¬Ѓberwise maps into

в€’в†’ T в€— M ,
L: L|Tp M
TM = L Fp , and

H : T в€—M в€—
в€’в†’ R , H|Tp M = Fp .
в€—

Exercise 58
The maps H and L are smooth, and L is a diп¬Ђeomorphism.

Let
Оі : [a, b] в€’в†’ M be a curve, and
Оі : [a, b] в€’в†’ T M
Лњ its lift.

Theorem C.11 The curve Оі satisп¬Ѓes the Euler-Lagrange equations on every co-
ordinate chart if and only if L в—¦ Оі : [a, b] в†’ T в€— M is an integral curve of the
Лњ
hamiltonian vector п¬Ѓeld XH .

Proof. Let
(U, x1 , . . . , xn ) coordinate neighborhood in M ,
(T U, x1 , . . . , xn , v1 , . . . , vn ) coordinates in T M ,
(T в€— U, x1 , . . . , xn , Оѕ1 , . . . , Оѕn ) coordinates in T в€— M .

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

в€’в†’ T в€— U
L: TU
в€‚F
в€’в†’ (x, Оѕ)
(x, v) where Оѕ = LFx (v) = (x, v) .
в€‚v

This is the deп¬Ѓnition of momentum Оѕ. Then
в€—
H(x, Оѕ) = Fx (Оѕ) = Оѕ В· v в€’ F (x, v) L(x, v) = (x, Оѕ) .
where
118 APPENDIX C. VARIATIONAL PRINCIPLES

Integral curves (x(t), Оѕ(t)) of XH satisfy the Hamilton equations:
пЈ±
пЈґ dx = в€‚H
пЈґ (x, Оѕ)
пЈґ
пЈІ dt в€‚Оѕ
(H)
пЈґ dОѕ
пЈґ в€‚H
пЈґ
пЈі =в€’ (x, Оѕ) ,
dt в€‚x
whereas the physical path x(t) satisп¬Ѓes the Euler-Lagrange equations:

в€‚F dx d в€‚F dx
(E-L) x, = x, .
в€‚x dt dt в€‚v dt
dx
Let (x(t), Оѕ(t)) = L x(t), dt (t) . We want to prove:

dx
t в†’ (x(t), Оѕ(t)) satisп¬Ѓes (H) в‡ђв‡’ tв†’ x(t), (t) satisп¬Ѓes (E-L) .
dt

The п¬Ѓrst line of (H) is automatically satisп¬Ѓed:

dx в€‚H dx
(x, Оѕ) = LFx (Оѕ) = Lв€’1 (Оѕ) в‡ђв‡’
= Оѕ = L Fx
в€—
Fx
dt в€‚Оѕ dt

в€‚F
= в€’ в€‚H (x, Оѕ).
Claim. If (x, Оѕ) = L(x, v), then в€‚x (x, v) в€‚x

This follows from diп¬Ђerentiating both sides of H(x, Оѕ) = Оѕ В· v в€’ F (x, v) with
respect to x, where Оѕ = LFx (v) = Оѕ(x, v).

в€‚H в€‚H в€‚Оѕ в€‚Оѕ в€‚F
В·vв€’
+ = .
в€‚x в€‚Оѕ в€‚x в€‚x в€‚x
v

Now the second line of (H) becomes

d в€‚F dОѕ в€‚H в€‚F
=в€’ в‡ђв‡’
(x, v) = (x, Оѕ) = (x, v) (E-L) .
dt в€‚v dt в€‚x в€‚x
by the claim
since Оѕ = LFx (v)
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Index

action billiards, 39
coadjoint, 64, 104 PoincarВґ-Birkhoп¬Ђ theorem, 31, 42
e
coordinates, 55 bracket
deп¬Ѓnition, 103 Lie, 50
eп¬Ђective, 61 Poisson, 51, 53
free, 105
C 1 -topology, 42, 43
hamiltonian, 57вЂ“59
canonical
locally free, 105
minimizing, 109
orbit, 52, 62, 78
of a path, 108вЂ“110
symplectic form on a cotangent
principle of least action, 109
bundle, 15
smooth, 103
symplectomorphism, 18
symplectic, 57
canonical form on T в€— X
transitive, 105
coordinate deп¬Ѓnition, 15, 16
action-angle coordinates, 55
intrinsic deп¬Ѓnition, 17
naturality, 17
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
Christoп¬Ђel
Arnold-Liouville theorem, 54
equations, 111
conjecture, 42, 44, 45
symbols, 111
Atiyah-Guillemin-Sternberg theorem,
classical mechanics, 49
60
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 п¬Ѓeld, 57, 103 dunce cap orbifold, 79
completely integrable system, 54 dynamical system, 42
complex
eп¬Ђective
projective space, 71, 88, 106
action, 61
embedding
conп¬Ѓguration space, 50, 108
closed, 93
conjecture
coisotropic, 26, 27
Arnold, 42, 44, 45
deп¬Ѓnition, 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
Christoп¬Ђel, 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
DвЂ™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 п¬‚ow, 36, 37
complex projective space, 88 geodesically convex, 35
Delzant construction, 88 minimizing, 35
Hirzebruch surfaces, 82 Gotay
McDuп¬Ђ, 9 coisotropic embedding, 27
of Delzant polytope, 82 gradient vector п¬Ѓeld, 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 diп¬Ђeomor-
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, 57вЂ“59
п¬Ѓrst integral, 53
function, 48, 49, 53, 59
п¬Ѓxed point, 38, 42, 44
п¬‚ow, 91 G-space, 60
form moment map, 59
area, 11 reduced, 76
canonical, 15вЂ“17 system, 53
de Rham, 5 vector п¬Ѓeld, 47вЂ“49
Hausdorп¬Ђ quotient, 106
Fubini-Study, 71
hessian, 113
symplectic, 5
Hirzebruch surface, 82
tautological, 15, 16
free action, 105 homotopy
Fubini-Study form, 71 deп¬Ѓnition, 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 п¬Ѓrst, 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
deп¬Ѓnition, 91
lift
symplectic, 8
of a diп¬Ђeomorphism, 17
vs. vector п¬Ѓeld, 91
of a path, 109, 110
isotropic
of a vector п¬Ѓeld, 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 McDuп¬Ђ counterexample, 9
lagrangian complement, 23 mechanical system, 107
lagrangian п¬Ѓbration, 55 mechanics
lagrangian submanifold celestial, 42
closed 1-form, 22 classical, 49
conormal bundle, 22, 23 metric, 34, 110
deп¬Ѓnition, 20 Meyer, see Marsden-Weinstein-Meyer
generating function, 22, 32 minimizing
intersection problem, 44 action, 109
of T в€— X, 20 locally, 109, 112
vs. symplectomorphism, 29 moment map
zero section, 21 deп¬Ѓnition, 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
deп¬Ѓnition, 105
equation, 10
point-orbit projection, 105
theorem вЂ“ local version, 11
space, 105
theorem вЂ“ version I, 9
topology of the orbit space, 105
theorem вЂ“ version II, 10
unstable, 106
trick, 8вЂ“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
PoincarВґe
Оµ-neighborhood theorem, 95
last geometric theorem, 42
tubular neighborhood, 26, 94
PoincarВґ-Birkhoп¬Ђ theorem, 31, 42
e
tubular neighborhood п¬Ѓbration,
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 gв€— , 52
26
polytope
Newton
Delzant, 81, 89
second law, 50, 107вЂ“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
п¬Ѓxed point, 45
simple, 81
normal
smooth, 82
bundle, 94
positive
space, 26, 94
inner product, 34
one-parameter group of diп¬Ђeomorphisms, potential
101, 102 energy, 55, 107
orbifold gravitational, 108
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 inп¬Ѓnity, 115 skew-symmetry
quotient forms, 6
Hausdorп¬Ђ, 106 standard form for bilinear maps,
Marsden-Weinstein-Meyer, 70 1
symplectic, 70 slice theorem, 73
topology, 105 smooth polytope, 82
space
rank, 2
conп¬Ѓguration, 50, 108
rational polytope, 81
normal, 26, 94
recipe
phase, 50, 108
for symplectomorphisms, 31
spherical pendulum, 56
recurrence, 41
stability
reduced
deп¬Ѓnition, 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
isotropic, 4
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 в€— X
canonical symplectic form on a
coadjoint orbit, 52, 62, 78 coordinate deп¬Ѓnition, 15, 16
intrinsic deп¬Ѓnition, 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
Оµ-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 вЂ“ local version, 11
strong isotopy, 7
Moser вЂ“ version I, 9
subspace, 4
Moser вЂ“ version II, 10
toric manifolds, 79
Noether, 65
vector п¬Ѓeld, 47, 48, 57
Picard, 92
vector space, 2, 3
PoincarВґ recurrence, 41
e
volume, 6, 7
PoincarВґвЂ™s last geometric theo-
e
symplectomorphic, 3, 7
rem, 42
symplectomorphism PoincarВґ-Birkhoп¬Ђ, 31, 42
e
Arnold conjecture, 42, 44 slice, 73
canonical, 18 standard form for skew-symmetric
cotangent bundle, 19 bilinear maps, 1
deп¬Ѓnition, 7 Stokes, 7
exactly homotopic to the iden- symplectomorphism vs. lagrangian
tity, 45 submanifold, 29
п¬Ѓxed 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 п¬Ѓeld, 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
п¬Ѓbration, 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 п¬Ѓeld
complete, 103
hamiltonian, 47вЂ“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