Vilasi G. Hamiltonian Dynamics
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154
~a~~f~l~s
and
~angen~
Spaces
The Lie derivative of
a
differentiable function
f
on the manifold
M
has the
following expression:
where
f=
f
o
@-l
is the function representing
f
in the chart
(U,$)
in which
p
is
represented.
5.12
Submanifolds
Examples of submanifoIds are given by a sphere
S2
or
a
curve
y
in the space
g3.
In some neighborhood
U
C
%I3
of any point
p
E
S2,
a
coordinate system
(2,
y,
z)
can be introduced for such that the points of
S2
n
U
are characterized by
z
=
0.
Similarly, in some neighborhood
U
C
g3,
of any point
p
E
y,
a coordinate
system
(x,y,z)
can be introduced for
IR3,
such that the points of
y
nU
are
characterized by
y
=
z
=
0.
A
sphere
S2
(or a curve
y)
is said to be 2-dimensional (l-dimensional)
submanifold
S
of the manifold
M
=
%I3.
Thus, it
is
natural to say that
an
dimensional
s~~~u~~fo~d
S,
of
an
n-dimensional manifold
M,
is
a
set of points of
M
such that, in some neigh-
borhood
U
C
M
of any point
p
E
S,
a
coordinate system
(XI,.
,
.
,
xn)
can be
introduced for
M
in
which the points of
S
n
U
are characterized
by
xm+l
=
p+2
=.
.,
=
x"
=
0.
More formally, a one-to-one map
f
:
Q
-+
M
is
said to be an
embedding
of
the m-dimensional manifold
Q
in an n-dimensional manifold
M,
(m
5
n),
if
at every
q
E
Q
there is
a
neighborhood
V
5
Q
of
q
and
a
chart
(U,p)
of
M
at
p
=
f
(g),
such that
(Y,
p
o
flv)
is
a
chart of
&;
that is,
p
o
flv
:
Q
-+
Rm
are
coordinates on
Y
for
Q.
The manifold
Q
is
said to be
embedded
in the manifold
M.
The image
S
=
f(Q)
is called a
s~~mu~z~o~d
of the manifold
M,
provided
with the manifold structure for which
f
:
Q
--+
S
5
M
is
a
diffeomorphism.
If
f
is
not one-to-one, we shall speak
of
immersion.
In other words, a map
f
:
Q
-+
M
is
said to be an
immersion
of
the manifold
Q
in
a manifold
M,
if
at every
p
E
Q,
there
is
a
neighborhood
V
G
Q
of
p
and a chart
(U,
p)
of
M
at
f(p),
such that
(V,
po
f)
is achart of
Q;
that is,
po
f
:
Q
-+
8"
are coordinates
on
V
for
Q.
The manifold
Q
is said to be
immersed
in the manifold
M.
Submanifolds
155
By recalling what has been said in Sec,
5.6,
concerning maps between man-
ifolds, a vector field defined on
a
submanifold
S
is
also a vector field on
M,
and
a
wvector field
on
M
is also
a
covector field
on
S.
A
suggested reading
on the subject and its applicatio~
is
given by the Marmo, Sabtan, Simoni,
Vi
tale
book.
41
5.12.1
The
fiobeni2~s
theorem
It has been shown that, given
a
smooth vector field
X
on an n-dimensional
manifold
M,
one can find
a
curve
(integral curve)
that, at every point
p
E
M,
the value
X,
of the vector field
X
coincides with the tangent vector to the
curve
at
the same point.
In other words, since
a
vector field
X
is an assignment at every point
p
E
M
of
a vector
X,
in the tangent space
7,M,
we
can paraphrase the
previous statement saying:
Given, at every point p
E
M,
a 1-dimensional subspace
D,
of the tangent
space
7,M,
one can find a 1-dimensional submanifold
N
such that
Dp
=
T~,~p
E
M.
It
is
interesting to have an answer to the analogous problem:
Given, at every point p
E
M,
a 2-dimensional subspace
D,
of
the tangent
space
7,M
(i.e.
a
pdrsne), does a 2-~~mensional s~bman~~ld
N,
such that
LIP
=
7pN,
exist
Vp
E
M?)
The
answer
is
generally:
No.
In order
to
discuss the general case, it
is
advisable to introduce the f~l~~wing
a
An assignment
D
at every point
p
G
M,
of a h-dimensional sub-
space
LIP
of
the tangent space
7,M,
that
is,
a hyperplane,
is
called
a
h-d~~ens~#nal dis~T~but~on on
M,
or
also,
a
di~e~ntial systems
of
h-planes on
M.
A
h-dimensional distribution
D
is said to be
C"
if, at every point
p
6
M,
there exists
a
neighborhood
U
of
p
and h Co3-vector fields,
namely
XI,.
.
.
,
xh,
defined in
U
and defining, at every point
q
E
U,
a
basis
X,(q),
. .
.
,Xh(q)
for
D,.
The vector fields
XI,,
,
.
,
Xh
are then
called
a
local basis
for
D.
0
A
vector field
X
is said
to belong
to
D
if
X, E
LIP
at every point
PEM.
useful definitions:
156
Man~~ol~
and
Tangene
§paces
A
CM
d~stribution
D
is called
invol~tive
if
x
E
D,Y
E
D
*
[X,Y]
E
D.
The above relation is equivalent to say that
a
local basis
{XI,
.
.
.
,
Xh}
of
a involutive distribution
has
the following property:
[Xi,
Xj]
=
CfjXk
,
since the Lie bracket of any two vector fields
X
and
Y,
which are their
f
-linear (i.e. the coefficients are f~ctions) combinations
x
=
fi(P)Xi,
Y
=
gi(p)x~
f
will be linear combinations of
Xi:
[X,
Y]
=
[f”xi,gjxj]
=
figqxi,xj]
+
fiyi(gk)X,
-
giE(f”xk
=
(figjcFj
-t-
fiXi(gk)
-
giXi(fk))Xk
=
d$Xk.
A
connected submanifold
N
of
M
is
called
an
integral manifotd of the
d~st~~utio~
D
if
f,(T&)
=
Dq
for
all
p
E
N,
where
f
is
the embedding
of
N
into
M.
The subma~foid
N
is
called
a
m~imal inte~a~ ~u~ifo$~
of
D,
when no other integral manifold of
D,
cont~ning
N,
exists.
It can be proven (see for instance Refs.
11,
29
and
50)
that
Theorem
22
(Frobeniw)
If
D
is an involutive distribution on a difleren-
tiat
manifold
M,
through every
point
p
E
M,
there passes
a
unique
maximal
integral manifold
N(p)
of
D.
Any
integral manifold through
p
is
an
open
sub-
rnanafold of N(p).
In other words, if
XI,.
. .
,
Xh
are
h(<
n)
vector fields defined on
a
region
U
of an ~-dimensional manifold
M
such that
[Xi,
Xjj
=
CtXk,
the integral curves
of
vector fields mesh
to
form a family
of
submanifolds.
Each submanifold has dimension equal to the dimension of the vector space
these fields define
at
any point, which
is
at
most
h.
Each point
of
U
belongs to
one and only one submanifold, provided that the dimension of the vector space
Submanifolds
157
defined
by
the fields
is
the same everywhere
in
U.
This family of submanifolds
is called
a
~0~~~~~0~
of
U,
and each submanifold
a
leaf
of
U.
The central idea underlying the F’robenius theorem
is
that, if the integral
curves,
of
the vector fields
XI,.
.
.
,
Xh
defining a distribution, are to define a
submanifold
to
which the vector fields must be tangent, they have to mesh
one another
as
cotton threads in
a
web. In other
words
the
flows
&,
of
the vector field
Xd
have to transform an integral curve
of
a
vector field
Xj,
in the integral curve
(of
the image) of
a
vector field constructed
as
linear
combination (with functions)
of
XI,.
.
.
,
Xh.
This will be guaranteed
if
all their
Lie brackets
[Xi,
Xj]
are themselves tangent; that is, belong
to
the distribution
[Xi,
Xj]
=
ctjXk.
This just means that the distribution has
to
be involutive.
Chapter
6
Differential
Forms
6.1
The
Tensors
In previous sections
it
has
been shown how to construct the dual space
E*
of
a given vector space
E.
The elements
of
such spaces constitute the simplest
examples of
tensors.
A
more interesting example is given by the
area
A(U,
V)
of
a
given pa-
rallelogram constructed by two vectors
U,
V.
Its most important property is
expressed
as
follows:
A(X
+
Y,
2)
=
A(X,
2)
+
A(Y,
Z)
Thus, the area of a parallelogram is a rule
A:
(U,V)
E
Ex
E
-+
A(U,V)
E
%,
which associates a real number with two vectors
linearly
in the entries
U,
V.
tensor
of
(0,
P)-type.
Any
bilinear map
T,
from the Cartesian product
E
x
E
to
R,
is called
a
The space of all such tensors
is
denoted with
C(E)
E
Lin(E
x
E,
R)
,
159
160
Dafferential
Fons
and can be endowed naturally with
a
vector space structure, defined
by
(Tl
f
T2)(X,
Y)
=
Tl(X,
Y)
f
T2(X,
Y)
,
(~T)~X,Y)
=
~~~(X,Y~),
Vk
E
32.
A
basis
of
such vector space can be easily constructed by using
a
basis
{ei}
of
E
and its dual basis
{di}.
In the given basis
{ei},
the vectors
X
and
Y
can be written
as
x
=
Xiei,
Y
=
Yjej
,
and we have
T(X,
Y)
=
T(Xiei,
Yjej)
=
XiYjT(e,,
ej)
=
T!jXiYj
,
with
Tij
_=
T(e+,ej)
E
R.
also be written in the form
Since, by definition, for all
X
E
E,
Si(X)
=
Xi,
the previous relation can
T(X,Y)
=
~~~~~{~~~(Y).
(6.1)
Thus, by introducing the
tensor
product
@
of
two covectors,
a!
E
E*,
p
E
E*,
by
(a!
Qd
P)(X,
Yf
:=
a!(X)P(Y)
>
the reIation
(6.1)
becomes
T(X,
Y)
=
(Ti$@
Qd
&j){X,
Y)
,
or
for
the arbitrariness
of
X,
Y,
T
=
Tij.tsi
Qd
.tsj,
Since the tensor
T
is an arbitrary element
of
the vector space
q(E),
the
last relation
shows
that
a
basis
for
this space
is
given by the
n2
elements
(6*
rip
@}.
Thus,
a
basis
in
E
will
fix
a
basis in
its
dual space,
E*,
and
dso
a
basis in the vector space
of
(0,2)
tensors.
For
this reason the
n2
elements
of
R
are called the
components
of
the tensor
T
in the given
basis.
Similarly, any
~~l~neur
map
R,
from
the Cartesian product
E*
x
E*
to
92,
R:
(cr,/3)
E
E'
x
E*
-+
R(a,P)
E
R,
The
Tensors
161
is
called
a
tensor
of
(2,O)-tgpe.
The space
of
all such tensors
is
denoted with
c(E)
=
Lin(E*
x
E*,
3)
,
and
can
be endowed naturally with
a
vector space structure defined
by
(R1
+
Rz)(X,
Y)
=
&(X, Y)
+
MX,
Y)
1
(kR)(X,Y)
=
k(R(X,Y)),
Vk
E
R.
By defining the tensor product
X
CED
Y,
of
two vectors
X,
Y
of
El
to be the
(2,O)
tensor given by
(X
@
Y)(a,P)
=
a(x)p~Y)
f
Va
E
E*,
p
E
E*
1
a
basis
{q
(&,
ej}
of
q(E)
is
fixed in terms
of
a
chosen basis
(ei}
of
E.
Once
more,
any
~~~~n~~~
map
S,
from
the Cartesian product
E*
x
E
to
92,
S
:
(a,X)
E
E*
x
E
--+
S(a,X)
E
W,
is
called
a
tensor
of
(l,l)-type.
The space
of
all such tensors
is
denoted with
7,'IEf)
=
Lin(E*
x
E,
3)
,
and
c&n
be
endowed naturally with a vector space structure defined
by
(4
+
Sz)(a,
X)
=
$1
(a1
X)
+
Sz(a,
X)
1
(kS)(a,
X)
=
k(S(a,
X))
,
Vk
E
R
*
Exercise
6.1.1
an
obvious
~e~n~t~on for
this
tensor
product.
Previous examples exhaust the concepts
of
tensor of
rank
2.
More generallyl any
~~lt~~~~e~r
map
Show
that
a
basis
of
T1(E)
can
be
giuen
bg
(8'
@
ej},
with
T
:
E
x
E:
x
e.1
x
E
x
E*
x
E*
x
-**
x
E*
+
Rl
-v
q
times
p
times
is called
a
tensor
of
(p,
+type.
The tensor
T
is also said to be
of rank
p
c
q.
The space
of
all such tensors
is
denoted by
T(E)
=
Lin(E
x
Ex,...
x
5
x
,E*
x
E*
;
x
dT13),
q
times
ptimes
and
cm
be endowed naturally with a vector
space
structure
defined
by
(TI
+
T2)(X,
Y,.
.
.
,
Z,a,
P,
=
Ti(X,
Y,.
Z,a,P,
*.
,Y)
CT2(X,Y,.
.
-12,
a,
P,
f
*
.
,TI
,
(kT)(X,
Y,
.
*.,
2,
CX,
8,.
. .
,y)
=
k(T(X,Y,
.
.
.
,
2,
a,
PI.
,.
,y))
,
Vk
E
9,
A
basis of
<P(E)
is
easily found by using the same procedure
used
for
Q(E),
Q(E),
or
A1(E).
Indeed,
let
T
be a
(p,q)
tensor,
(e,)
be
a
basis
of
E
and
{$I*)
be its dual basis. We have
T(X,Y,.
.
.
,Z,
cl~,P,
. . .
,Y)
=
X'Y'
**
*
ZkQa,Pq
*
*
*rrT
x
(e%> e,
1
*
*
>
ekt
a',
Pp
I
*
*
-
pq::yiy3
.
*
.
p(-@,
.
. .
TT
-
-
~~~:~$I~(x)~~(Y)
*
..d'(Z)e,(a)e,(p)
*
..e,(r>
=
(~~q~~~F~~
@
1~7
@
* *
-
@ @
ep
QD
eq
ri~
**
*
e,)
i~.
k
-
x
(X,Y,.
*
*
,
Z,a,P,.
. .
$7)
*
Since
XI,
Y,
. . .
2,
a,
p,
.
.
.
,
y
are arbitrary vectors and covectors, we can
write
T
=
T~q~:~-S~
@-Sj
@
@-Sk
@
ep
09
eq
@
-
QD
e,
,
is
given by
(64
which shows that
a
basis
for
the vector space
P~~PQ~..,QD~~
rpep~egQD-..@er.
"
'-
g
times
ptimes
Remark
13
According to the previous definition we can say that
e
A
covector
is
a
tensor
of
(o,l)-tgpe. The co~espond~ng vector spuce
E*
is
also denoted, besides
v(E),
with
A(&),
or simply
A.
So
A(E)
=
A
vector
is
a tensor
of
(l,O)-type. The co~espon~2ng vector
spuce
E
The elements
of
8
are called tensors
of
(0,O)-type.
~(~~
=
E'.
could
be
also
denoted with
$'(E).
The
Tensors
163
A
tensor
T
of
(0,
2)-type
is
said to be
symmetric
if
T(X,Y)
=
T(Y,X)
0
~n~~a~m~e~~c
if
T(X,
Y)
=
-T(Y,
X)
The same definition can be given
for
tensors
of
(2,O)-type and, more
ge-
nerally,
for
tensors
of
(0,p)
or (q,O)-type.
As
for
a
tensor
of
(1,l)-type, no
meaning can be given to the interch~ge of
a
vector with
a
covector.
The
set
of
ail
antisymmetric tensor
of
(0,2)
type
is,
of
course,
a
vector
subspace
A2(E)
of
the vector space
Q(E).
A
basis can be easily found by
considering
a
generic element
A
of
A2(E).
In
a
given basis
{ei)
of
E,
the
antisymmetric
(0,2)
tensor
A
can be written
as
where
the
4n(n
-
1)
distinct numbers
Aij
=
A(ei,ej)
are antisymmetric for
the interchange
i
+)
j.
Thus
Then, by introducing the
exterior (or wedge) product
di
A
03,
of the basis
elements
Gi
and
dj
by
the an~~sy~etr~c (0,2) tensor
A
can
also
be written
in
the form
1
2
A
=
-Aijd‘
Ad’.
Thus,
a
basis for
A2(E)
is
given by the
an(n
-
1) elements
{tJi
A
@}.
More generally, a tensor
T
of
(0,
q)-type
is
said to be
sy~~e~~c
if
T(Xta,Xbr, ..Xc)
=
T(XI,X2,..
.X,)
for ail permuta-
antisymmetric
if
T(Xal Xb,
. . .
x,)
=
-T(Xl,
x2,.
. .
x,)
for all odd
tions
(a,
b,.
.
.
,
c)
of (1,2,.
. .
,q)
permutations
(a,b,,
. .
,c)
of
(l,2,.
.
.
,g)
A
similar definition can be given for tensors of
(q,
0)-type.