Vilasi G. Hamiltonian Dynamics
Подождите немного. Документ загружается.
Part
I1
Basic
Ideas
of
Differential
Geometry
Differential geometry is the differential calculus that does not depend on
the coordinate system and
so,
it
is
the best language
f5r
a
science (Physics~
deputed to describe phenomena, which do not depend on the observer.
Part
I1
is devoted to
a
summary survey
of
useful geometric concepts,
as
differential manifold, tangent space, fiber bundles, Lie derivative, differential
forms,
and
ezterior der~vative.
~i~e~e~t~a~
forms
describe all the relevant physical entities
as
the
or^,
the
heat, the internal energy, the electromagnetic field,
and
so
on.
Fiber bundles
are used, in elementary particle physics, to describe parti-
cles with internal structures (isotopic spin, etc.) and to construct
instantonic
solutions
in
field theory.
~en~forth, the
sum
over repeated, upper and lower indices is understood
according
to
the Einstein convention:
Ca
aibi
=
127
Chapter
5
Manifolds and Tangent
Spaces
5.1
Di~erential
~~ifolds
A
manifold
M
is
a
separable topological space such that every point
of
it
is
representable at least on one chart;
a
chart
(U,
'p)
of
a
manifold
M
is an open
set
U
2
M,
called domain of the chart, with
a
hom~morphism
rp:UcM
-+Ac
sZn
from
U
to
an
open set
A
in
Rn.
A
map
f
:
U
+
V
between two topological spaces
U
e
V
is
called
a
homeomorpfbissm
if
it
is
one-to-one
and
both
f
and
f-l
are continuous maps.
Let
p
be
an
element in
M
representable on the chart
(U,
9).
The coordi-
nates
Cp'(p)
of
the image
of
the point
p
are called coordinates of
p
in the chart
(U,
v),
or
bcal coordinates
of
p.
Thus,
a
chart is essentially
a
local coordinates
system.
A
Ck
atlas on
a
manifold
M
is
a
collection
{(Um,vm)]
of charts such
aa
the domains
U,
form a covering
of
M,
and the homeomorphisms
'pa
0
'pi1
:
P,(up
nUp)
-+
cpp(u,
nu@)
are
Ck
maps between open sets
of
Rn.
Two
Ck
atlases are equivalent if their union
is
a
Ck
atlas.
129
130
Manifolds and Tangent Spaces
A
manifold
M
with a
Ck
atlas equivalence
class
is called
a
Ck
differential
manifold;
its dimension
is
n.
A
manifold
is
said
to
be
o~entu~~e
if an atlas can
be
chosen in
such
a
way
that for
all
a,
p
the Jacobian determinants det(bqh/aq$) have the same sign.
An intrinsic definition will be given in Sec.
2.5.
The
sphere
as a diflerential manifold
The n-dimensional sphere
S"
is
defined by
It
is
;t
separable topological space, and associated with
U
c
S"
and
Y
c
S",
two charts
(U,
cp)
and
(V,
$)
can be introduced
as
follows:
UE~S"-N,
VES"-S,
where
N
and
S
are the
north
and
south poles
defined
by
N
3
(0,.
.
.
,o,
1)
,
s
=
(0,.
.
.
,o,
-1).
By
using
the
notation
5
=
(XI,..
,
,
zn),
a point
p
E
U
will
have coordinates
(3s
xn+l)+
The map
is called the
stereographic projection
of
Sn
with respect
to
the north pole
N.
It
is
a
one-to-one map that is continuous together with its inverse, which
is
given
bY
Similarly, the
ste~ographzc projection
of
S"
with
respect to the south pole
S
is
defined
as
2
1
+
xn+1
E
R",
+.
@:pE V+@(p)=u=
Curves
on
a
Difleerential
Manifold
131
and
it
is
a
one-to-one map continuous together with its inverse given by
Since the images of the intersection
u
n
Y
=
sn
-
{N,
s)
,
for both
(p
and
$,
is
(Rn
-
(0));
i.e.
p(U
n
V)
=
$(U
n
V)
=
Rn
-
(0)
,
the map
-+
iz
cp
0
$--I
:
G
€
(P
-
(0))
3
t
=:
(p
0
.$-1)(d)
=
-
E
(R”
-
(0))
1.12
is
a
C’”
map.
Thus, the two compatible charts
(U,cp)
and
(V,.$)
are
a
C“
atlas for
Sn,
and
(tl,.
. .
tm),
(211,.
.
.
,
un)
represent the local coordinates
of
a
point
p
E
Sn
with respect
to
them.
Exercise
5.1.1.
Show that
the sphere
S2
is
orientable.
5.2
Curves
on
a Differential Manifold
~~~ere~~~a~~~
functions
on
a
manifold
If
f:M-+R
is
a
tion on
defined
on
a
~fferential manifc
representable on the chart
(U,
p),
the map
M
anL
p
is
an element
in
M
f
0
9-1
:
p(U)
c
RZn
-+
SR
transforms open
set
of
!RZn
to open set of
R.
As
the coordinates
p’((p)
of
p(p)
represent the point
p
in the local chart
(U,cp),
so
the map
f”
=
f
o
p-l
represents the function
f
in
the local chart.
The function
f
is
said
to
be diflerentiable
at
p
E
M,
if
in
a
given chart
(U,cp),
f’
=
f
o
p-l
is differentiable at
p(p),
or
equivalently, if expressed in
local coordinates, gives rise to a differentiable function.
132
Manifolds and Tangent
Spaces
The given definition is independent from the choice of the chart. Indeed,
if
the point
p
is
representable
on
two charts
(U,
p)
and
(V,
$),
we have
f
0
q-1
=
(f
0
9-1)
0
(9
0
$-I).
If
(focp-')
is differentiable at
cp(p),
then
f
o$-l
is differentiable at
+(p),
since
p
o
$-l
is differentiable at
cp(p)
and
(p
o
+-l)($(p))
=
cp(p).
Let
M
and
hl
be two differential manifolds, with
m
and
n
dimensions,
respectively, and
fet
f
be
a
map
f
:
P
E
M
+
f
(PI
E
Jff
(5.1)
from
M
to
hl.
the function
If
p
and
f(p)
are representable on the charts
(U,
p)
and
(V,
$),
respectiveIy,
9
0
f
0
p-l
:
p(U)
c
92%
-+
$(f
(U))
c
!I?%
represents the map
f
in the local charts
(U,-p)
and
(V,
$),
and we will say that
f
is
differentiable at the point
p
E
M,
if
f
=
$
o
f
o
p-'
is differentiable at
The map
(5.1)
is said to be
~iffe~nt~ab~e
in
M
if it is differe~tjable
at
every
point
p
E
M;
if
f
is
one-to-one and
f
and
f-'
are differentiable, then
f
is said
to be a
diffeomorphism.
P(Ph
Cvmee
on
o
manifold
A
curve
y
on
a
differential manifold
M
is
a
homeomorphism
y
:
T
E
I
c
!I?
-+y(~)
EM
from an open set
I
C
R
(open interval) to an open set in
M.
The curve
y
is
said to be
d~~e~entiuble at
T
=
0
if
the map
'pay:
7
f
Ic
%z$Q(y(T))
E
92%
is
differentiable at
T
=
0,
where
p
is the homeomorphism
of
the chart on which
p
5
~(0)
is representable.
Two curves,
4
and
y'
on
M
are called
e~uiva~ent
if
in
a
chart
(U,
cp).
Of
course, the relation
(5.2)
is true in any other
Tangent
Space
at a Point
133
5.3
Tangent Space at a Point
5.3.1
Definition
18
It
is called tangent vector
X,
to a differentiable curve
y
=
y(~)
on
a
manifold
M,
at the point
p
=
"(TO),
the directional derivative
operator along the curve
y
=
Y(T),
at the point
p:
Tangent vectors to a
curve
on
a
manifold
If
p
E
Ua
c
M,
T
E
I
E
R
and
f
is a differentiable function on
M,
we have
where
pa
is
the homeomorphism of the chart
(Ua,qa)
on which
p
is
repre-
sented. Since
,f=
(z',
...,
zn)
EA
C
Rn
+-f(dl...lz")
E
R,
Rn,
(Pa
o
:
T
E
%
+
T(T)
=
(z'(T),
.
.
.
,
~"(7))
E
A
we have
where
Xi
=
dq/dTlTO
.
From the above formula it follows that every tangent vector to
a
given
curve, at
a
given point, is
a
linear combination of the
n
partial derivatives
{d/Bxk},
which are linearly independent.
Of course, every tangent vector is tangent to an infinite number of different
curves through
p
for two different reasons. The first is that there are many
curves which are tangent to one another and have the same tangent vector at
p,
and
the second is that the same path may be reparametrized in such a way
as
to
give the same tangent at
p.
So
it
becomes natural to give the alternative following definition.