Vilasi G. Hamiltonian Dynamics
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204
Integration
Theory
In such a case these points do not contribute to the integral.
In this case both sides
of
the
Eq.
(7.5)
are vanishing and the equation
still holds.
X
is
tangent to
av
in an open region
of
it.
So,
summing over all parts
aV
of
aU,
we obtain
from which
By comparing the two expressions,
Eqs.
(7.4)
and
(7.6),
of
the Lie derivative
of
the integral
su
w,
we finally obtain
and since
ixw,
as well
as
w,
is arbitrary, we conclude
Theorem
24
(Stokes)
For
any
(n
-
1)-form
cy
manafold
U,
the
following relation
holds.
that
over
an n-dimensional
If
U
=
[a,b]
is an interval of the real line, and
f
:
U
+
8
a differentiable
function, then the Stokest theorem reduces to
J,”
f’dx
=
f(b)
-
f(a)
I
since
aU
=
{a,
b}.
+George Gabriel Stokes
was
born in Skreen (Ireland) in
1819
and died in Cambridge in
1903.
Physicist and mathematician, he has been
a
professor
of
mathematics at Cambridge
University and
is
universally known
for
the results on the transformations
of
integrals, on
the liquid waves and
for
his theories on optics, founded on the hypothesis
of
dragging ether.
Gradient,
Curl
and Divergence
205
7.6
Gradient,
Curl
and
Divergence
On an n-dimensional orientable manifold
M,
endowed with
a
metric tensor
field
g,
all
properties concerning the volume duality and the Hodge duality can
be point-wise carried over directly. This allows
us
to better understand the
meaning of some familiar concepts in
R3,
such
as
the
gradient,
the
cur1
and
the
divergence.
0
The gradient
Consider
a
function
f
and take its exterior derivative
df.
The vector
field associated to the differential form
df,
by means of the inverse of
the Euclidean metric tensor
17,
is
called the
gradient
of
f:
Vf
=
v-'(df,-)
CL---L.
ivfq
=
df.
0
The curl
Consider
a
vector field
U,
take the associated differential form
a
=
q(U,-)
and its exterior derivative
da.
The volume-dual of
da
is a
vector which is called the
curl
of
U:
V
x
U
=
Q-'(da),
with
R
=
dx Ady Adz.
Consider a vector field
V,
take the associated differential 2-form
via
the
volume form
R
=
dx
A
dy
A
dz.
Its exterior derivative is proportional
to
R
up to
a
multiplicative function called the
divergence
of
V:
0
The divergence
(V
*
V)R
=
diva.
Moreover, if
V
=
V
x
U
=
Q-'(da),
then
V
*
V
x
U
=
div,uR= din-l(dalR
=
d(O(O-'(da)))
=
d(-l)3-'da
=
d2a
=
0.
Exercise
7.6.1.
Use Stokes' theorem to prove that, for every exact differential
2-form
w
on
the
sphere
S2,
206
Integmtaon
Theory
Proof.
In
order for
w
to be exact, a diflerential
1-form
Q
have to exist such that
w
=
da. In this case, Stokes’ theorem gives
s,,.
=
s,,
da
=
s,,,
Q
=
0,
since
S2
has no boundary.
Exercise
7.6.2.
Use Stokes’ theorem to show that for the differential
2-form
w
=
x’dx2
A
dx3 on
S3,
where
S2
is the
unit
sphere considered
as
a
submanifold
of
S3.
Answer.
The differential3-form
dw
=
dx’
A
dx2
A
dx3
is
the usual volume form,
so
that when integrated on the volume
V
of
the sphere,
gives
The result then follows by Stokes’ theorem.
The above exercise gives an example
of
a closed differential
2-form
on
S2
which is not exact, since it does not satisfy the criterion of the first exercise.
7.7
A
Primer
for
Cohomology
Let
P(M)
and
BP(M)
be the set
of
all
closed differential p-forms
and
the set
of
all
exact differential p-forms, respectively. Both sets have
a
natural structure
of
vector space and, moreover,
BP(M)
is a subspace
of
Zp(M).
Then, we can
introduce in
P(M)
an
equivalent relation,
namely
M
by declaring
a
M
p
H
(a
-
p)
E
BP(M)
;
A
Primer
for
Cohomology
i.e.,
207
The set of all equivalence classes is denoted with
W(M)
and is called the
It is easy to show that, if
M
is any connected manifold, then
pth de Rham cohomology vector space
of
M.
Ho(M)
=
Zo(M)
=
%.
Indeed, a zero-differential form is just a function,
so
that
Zo(M)
is the space
of functions
f
for which
df
=
0;
i.e.
Zo(M)
=
8.
Moreover,
Bo(M)
=
(0);
i.e. the zero function,
so
that constants
f
and
g
are equivalent if they coincide.
If
M
is
not connected, then an element in
Zo(M)
will be constant on each
connected component
of
M,
but the value
of
the constant can be different on
different components,
so
that
Ho(M)
=
Zo(M)
=
R",
where
m
is the number of components of
M.
Exercise
7.7.1.
Show that
for
the n-dimensional open ball
or
any region
U
difleomorphic to it,
W(U)
=
0,
p
2
1.
(Hint: All closed differential
p-forms
are equivalent to one another, and hence
to the zero differential p-form).
Exercise
7.7.2.
Show that
H*(S*)
#
0,
H*-l(P)
=
0.
It
can be proven that51
H"(Sn)
=
8,
HP(Sn)=O,
O<p<n,
HO(SD)
=
8.
208
Integration
Theory
Remark
15
Remark
16
The given definition
of
HP(M)
relied on the differential
structure
of
M.
However, it can be proven (see,
for
instance Ref.
55)
that
the cohomology groups depend, only on the topology
of
M
and not its
differentiability.
The dimension
of
Hp(M)
is called the pth-Bettit number.
7.8
Scalar Product
of
Differential p-Forms
Let
M
be
an
orientable n-dimensional compact differential manifold and let
a,
p
be differential p-forms
hp(M).
The Hodge dual
*p
of
p
is
a
differential
(n
-
p)-form:
*p
E
An-p(M),
so
that
a
A
*p
is
a
differential
n-form.
This
allows
us
to define the scalar product
(a,
p)
of
a
and
p,
by
Exercise
7.8.1.
Show that the previous formula defines a scalar product on
AP(M).
7.8.1
Exterior codifferential
By using the above scalar product, we can define
a
new operator
6,
the adjoint
of
d,
by
Clearly,
6
:
hP(M)
+
AP-'(M),
1
5
p
5
n,
6f
=
0
for
every function
f
.
The operator
6
is called the
codifferential.
It is worth to observe that
it
can
be introduced only by using a metric tensor field defined on
M.
SEnrico Betti, born in Pistoia in
1823
and died in Pisa in
1892,
has been
a
professor
of
mathematical physics at Pisa University and Director
of
the Scuola Normale Superiore
in Pisa. He gave deep contributions to algebra, topology, elasticity theory, and potential
theory. An excellent teacher, and among his students were Luigi Bianchi and Vito Volterra.
Scalar
Product
of
Dafierential
p
-Forms
209
Exercise
7.8.2.
Show that
for
every daflerential
p-form
a
and that
6
is
not
a
derivation.
Let
us
finally observe that, from the Eq.
(7.8),
it follows that
J2
=
0.
The Laplace-Beltrami operator
The Laplace-Beltrami§ operator
A
is
defined by the relation
A=do6+60d=(d+d)2,
and it
is
a
self-adjoint operator, since
Exercise
7.8.3.
Give the expression
of
A
in
!R3
by
using the Cartesian coor-
dinates and the spherical-polar ones.
A
differential form
w
that satisfies the differential equation
Aw=O
is called
harmonic.
Clearly,
Aw=O.-t.dw
=0,6~
=O.
Indeed
(Aw,w)
=
(ddw,~)
+
(6dw,w)
=
(6w,
aw)
+
(dw,
dw)
,
with
(6w,
6w)
2
0,
(dw,
dw)
2
0.
5Eugenio Beltrami, born in Cremona in
1835
and died in Rome in
1900,
has been
a
profes-
sor
of
algebra and analytical geometry
at
Bologna University and, after,
at
Pisa,
Pavia and
Rome universities. His research activity on the Newtonian potential and on the differential
parameters can be considered of basic importance, and
his
Saggio sulb anterpretuzione della
Geometria
non
Eucladea
is now considered
as
classical
work.
210
Integmtion Theory
7.8.2
Hodge
theorem
Hodge theorem
is
an important
decomposition theorem
that we quote without
proof.
Theorem
25
(Hodge)
Every differential p-form
w
can
be written as
w=da+6P+y,
where
a
is
a
differential (p
-
1)-form,
P
is a differential
(p
+
l)-form, and
y
a
harmonic form. Moreover, the differential forms da,
6P
and
y
are unique.
Chapter
8
Lie
Groups
and
Lie
Algebras
8.1
Lie
Groups
A
finite-dimensional Lie group
is
a
CF
manifold
G
of dimension
n,
endowed
with
a
group structure, such that the product
(g,h)
E
G
x
G
Hgh
E
G
(8-1)
and the inverse
are
Cm
maps. The diffeomorphisms
L,
;
h
E
G
+gh
E
G,
R,
:
h
E
G
+
hg
E
G
are called the
left
translation
by
g
and the
right translation
by
h,
respectively.
Let
(U,cp)
be a chart in
G
such that
e
E
U
and
cp'(e)
=
0,
where
e
is the
identity
of
the group. For every open set
U
containing
e,
there exists an open
set
V
c
U,
to which
e
belongs,
such that
V
.
V
c
U,
where
V
-
V
=
{gh
:
g,
h
E
Then, the product
p(V)
x
cp(V)
is an open set in
Rn
x
Rn
containing the
point
(0,O).
Since
G
is a Lie group, the map
(8.1)
is differentiable,
so
that,
if
g
and
h
are two elements in
V,
the coordinates
cpi(gh)
=
(gh)i
of
their product
V)*
211
212
Lie
Groups
and
Lie
Algebras
are differentiable functions of the coordinates
xi
=
pi(g)
of
g
and
yi
=
pi(h)
of
h,
and we can set
(gh)i
=
fi(x1,.
. .
,
x",yl,
*
.
.
,
y")
or
shortly,
(&)a
=
fi@,
Y)
'
The group structure implies that the functions
fi
must satisfy the following
properties:
The first property follows from the associativity of the group product; i.e.
((sh).)i
=
(S(Wi
>
Qg,
h,u
E
G
=j
fi(f(x,
Y),
.)
=
fih
f(Y,
2))
7
for every choice of
x,
y,
z
in
p(V),
with
zi
=
pi(.).
The second follows from
(geli
=
(eg)*
=
xi
=+
fi(x,
0)
=
fi(0,
x)
=
xi.
Moreover, for the
fi's
the following expansion can be performed:
fi(x,y)
=
xi
+
yi
+
c
x:,x*yp.
(8.3)
*2l,P>1
To build the inverse
9-l
of an element
g
it
is
sufficient to solve the following
system of equations with respect to
y':
(8.4)
1
fi(Z1,.
. .
,Z",
y
,
. .
.
,
y")
=
0.
From
Eq.
(8.3),
we have
so
that the Jacobian determinant
a(f',
.
. .
,
f")/a(y',
.
. .
,
y")
at the point
(0,O)
in
R"
x
!Xn
is
1.
Therefore, by continuity, the Jacobian does not vanish in
a
neighborhood of the origin and,
by
the implicit functions theorem, there exists
an open set
V'
c
V
in
e,
such that
for
every
g
E
V'
the system
(8.4)
has
a
unique solution
(y',
.
.
.
,
y").
Building
of
a
Lie
Algebra
from
a
Lie
Group
213
8.1.1
Local
Lie
groups
A
local
Lie
group
is
a
local version of a Lie Group. Then, it is
a
pair
(A,
f),
where
A
is
an
open set in
Rn
containing the origin of the coordinates, and
f
a
differentiable map
f:AxA+R"
satisfying,
Vz,
y,
z
E
A,
the following conditions:
(4
f(2,
f(y,
2))
=
f(f(2,
Yh
z>;
(b)
f(0,
.)
=
f(.,
0)
=
z;
(c) there exists a differentiable map
E
:
A
+
W,
such that
I(.,
4.))
=
f(E(ZC),
2)
=
0
*
Thus,
given
a
Lie group
G
it
is
always possible to build
a
local Lie group,
with the identification
A
f
yQ')
c
an.
Two local Lie groups,
(A1,
fl)
and
(A2, f2),
are isomorphic
if
there exists
any two neighborhoods,
A{
c
A1
and
A',
c
A2,
of the origin of
R"
and
a
diffeomorphism
2c,
:
A',
Ah
such that the diagram
I
1
is
commutative,
as
to say
+(fl(.,
9))
=
f2((2c,
x
1cl)(z,
Y))
=
f2(lL(Z),
Ilr(Y))
1
v
(2,
Y>
E
A',
x
A:
Of course, all local Lie groups obtained from the same Lie group
G,
with
the previous procedure, are isomorphic among themselves.
8.2
Building
of
a Lie Algebra
from
a
Lie
Group
8.2.1
Lie algebras
The algebraic definition of Lie algebra has been already given in Part
I.
Here
we are going to give just a mention about three types
of
algebras with great