Vilasi G. Hamiltonian Dynamics
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224
Lie
Groups
and
Lie
Algebras
The composition law, defined by
Eq.
(8.17),
provides the vector space
7,G
with
a
Lie algebra structure.
In
conclusion, given
a
Lie group
G,
it is always possible to build from it,
a
Lie algebra. Now we can ask whether, given
a
Lie algebra
A,
there exists
a
Lie group of which,
vice versa,
A
is the algebra. The answer to this question
is given by the following theorem:
Theorem
26
(Cartan)
Every Lie algebra is the Lie algebra
of
some Lie
group.
In
the previous section
we
spoke about
local
Lie groups.
They
are also
related
to
the Lie algebras because every Lie algebra is
a
Lie algebra of some
local Lie group. Moreover, the local Lie groups are isomorphic if and only if
the corresponding algebras are isomorphic
as
well.
Let
A1
and
A2
be two Lie algebras, and
GI
and
G2
the corresponding Lie
groups. By the last we can build two local Lie groups
Gi
and
G;,
which will
be isomorphic if
A1
and
A2
are isomorphic. However, the fact that
G\
and
GL
are isomorphic does not imply that
GI
and
G2
are
so.
In
this case, we speak
of
local isomorphism
between
GI
and
G2.
Theorem
27
(Monodromy)
If
G
is a simply connected Lie group and
F
any Lie
group,
every local homomorphismt
of
G
in
F
is
uniquely prolonged
in
a
global homomorphism
of
G
in
F.
Let us denote with
GL(n,
R)
the Lie group of
n
x
n
invertible real matrices
and with
GL(n,
R)
the corresponding Lie algebra, which is given by the vector
space of
n
x
n
real matrices with the commutator
as
Lie bracket.
A
very
important result is the following:
Theorem
28
(Ado)
Every Lie algebra
of
a Lie group is a subalgebra
of
GL(n,R)
for
some vaEue
of
n.
For
a
simply connected Lie group
G,
the following theorem holds:
For Lie groups, the analogous statement holds only locally; i.e.
Every Lie group is locally isomorphic
to
a subgroup
of
GL(n,R)
for
some
By this theorem the local isomorphism between
GI
and
G2
is prolonged to
value
of
n.
a
global isomorphism.
*A
local
homeomorphism
is
a
homeomorphism
of
the correspondent
local
groups.
Budding
of
a
Lie
Algebmfim
a
Lie
Group
225
Thus, we can conclude that just one simply connected Lie group
G
corre-
sponds
to
a
Lie algebra
A.
8.2.4
We can introduce translation operators also in the dual space
TG
of
7,G.
As
for
the left translation, we can use relations (8.9) and (8.10); the right
tr~slatio~ are defined in
a
perfectly analogous way.
The
~~~jo~~~
~~~aen~a~~on
of
a
Lie
g~uF
(V,
dR;;
(4)
=
(dR,
(V)
I
4
dR,
:
XG
+
x,G,
dfi;
:
TgG
-+
TG.
It
is
also possible to define the operator
Adz,
dual of the operator
Ad,,
as
follows:
(Vt
Ad~(a!)}
=
(Adg(V),
af
-
(8.18)
By
Eq.
(8.18)
and
by
the properties
of
Ad,,
we argue that
Ad:
:
TG
-+
rG
is an invertible linear map
of
7,G
into itself.
The map
Ad*:gEG-+Ad*(gj=Ad3;
~AutrG,
(8.19)
as
the one in
Eq,
(8.13),
is
also a representation
of
the Lie group
G.
It
is
called
co~$joi~t ~p~~e~tat~o~
of
the
Lie
grogp
G
The map (8.19)
is
differentiable.
Its
derivative
(Ad*)*,
at
the unit, denoted
by
ad*,
is
the map
ad*
:
V
E
7,G
-+
ad*@')
=
ad;
f
End
TG.
The operator
ad?
is
the conjugate of
a&,
and
(W,ad;(~))
=
(adv(W),a!),
Va!
E
rG,
VW
E
722.
The vector spaces
7,G
and
TG,
endowed with
a
bracket giving them
a
Lie algebra structure, are usually denoted by the symbols
G
and
G*.
The coadjoint representation
of
a Lie group has an important role in
clas-
sical mechanics.
As
we will see, the orbits
of
the group under the coadjoint
representation are symplectic manifolds,
226
Lie
Groups
and
hie
Algebms
This will be shown
in
Part
111,
in the chapter
Orbits method$
after the
in~roduction
of
some preliminary concepts, An exhaustive discussion
an
this
subject can be found
in
Ref.
41.
The second part of this
book
is
in
fact
completely devoted to
Reduction, Actions
of
Group
and Algebras.
Further
Readings
R,
Abraham,
J.
E.
Marsden, and
T.
Ratiu,
~angfo~~, Tensor An~ys~, and
e
B.
Dubrovin,
S,
Novikov,
A.
Fomenko,
Gkomktrie Co~~e~~~~ne~
.&&ions
Mir
*
C.
J.
Isharn,
Modem D~~erential Geometry
for
Physicasts
(World Scientific,
1989).
*
J.
L.
KOSZUI,
Lectzlres on
Fibre
3undles and D~~e~tz~ Geometry
(Tata
Institute
of
~nd~ent~ Research, Bombay,
1960).
A.
"kautrnan,
~affere~~~a~ Geometry
for
Physicists
(Bibliopolis, Naples,
1984).
A~~~~c~~~o~
~
Addison-~esley,
1983).
(Moscow
1979, 1982).
Part
I11
Geometry and
Physics
Part
111
is
devoted to
a
revisiting of analytical mechanics in terms
of
geomet-
rical structures. Chapter
9
is devoted to the intrinsic formulation of Maxwell’s
differential equations in terms of differential
forms,
so
that it
can
be considered
as
an
introduction for
Gauge
Theories.
229
Chapter
9
Symplectic Manifolds and Harniltonian
Systems
9.1
Symplectic Structures
on
a
Manifold
If
M
is
a
2n-dimensional differentiable manifold, a
symplectic structure
on
M
is
a differential 2-form
w,
required to be
a
closed
&=O,
and
not degenerate
(wP(X,
Y)
=
0
VY
E
7,M)
+
(X
=
0)
Vp
E
M
.
(9.1)
A
pair
(M,w),
with
M
a
2n-dimensional differentiable manifold and
w
a
In a given basis
{ei}
for vector fields on
M,
we
may write
X
=
Xiei,
symplectic structure,
is
called a
syrnplectic manafold
Y
=
Y'ei
,
so
that, with
wij(p)
=
wp(eir
ej),
the relation
(9.1)
becomes
(XiYYiwij(p)
=
O
V
Yi)
+
(Xi
=O)
Vp
E
M,
or
equivalently,
(Xiwg
=
0)
*
(Xi
=
0).
231
232
Symplectic Manifolds and Hamiltonian Systems
Thus, a differential 2-form is not degenerate iff
A
generic differential 2-form
w
on a manifold defines a homomorphism
w
:
7,M
+
TM
of
the vector space
7,M,
of tangent vectors at the point
p
E
M,
into
GM,
the
vector space of differential 1-forms to the manifold
M
at the point
p
E
M,
since
with the vector
Xp
GM,
w
associates the differential
1-form
cyp,
defined
as
ap
=
iXpW(P).
As
a
consequence, with the vector field
X,
w
associates the differential
1-form
a,
defined point-wise
as
When the differential 2-form is not degenerate, the above relation
can
be
Then, a not degenerate differential 2-form
w
defines an isomorphism, be-
point-wise solved with respect to the vector field
X.
tween the vector spaces
7,M
and
TdM,
given
by
X
=
A(a,
*)
,
where the 2-vector field
A:
TM
+
7,M
,
is the inverse of
w,
The above relation in a given coordinate basis, in which
1
1
..a
a
w
=
-wijdxi
A
dx'
,
A
=
-Az3-
A
-
2 2
ax* dxj
'
is
simply written
as
follows:
(9.3)
Locally
and
Globally
Hamiltonian
Vector
Faelds
233
9.2
Locally
and
Globally
Hamiltonian
Vector
Fields
A
vector field
X
on symplectic manifold
(M,
w)
is called a
(locally) Hamiltonian
vector field
if
Lxw=O,
that is,
if
the symplectic structure is invariant under the flow generated by
X.
Since
a
symplectic form is closed, the above relation can also be written,
by using the Cartan identity, in the following form:
Thus,
a
locally Hamiltonian vector field on
M
is a vector field satisfying the
requirement that the differential l-form
CY,
defined by
(Y=ixw,
is
closed.
exists such that
If
the
differential 1-form
CY
=
ixw
is also exact; that is,
a
function
H
on
M
ixw
=
-dH
,
(9.4)
the vector field
is
called a
globally Hamiltonian vector field,
or simply a
Hamil-
tonian vector field,
and the function
H
is called the
Hamiltonian function cor-
responding
to
X.
The minus sign in
Eq.
(9.4)
is introduced just for historical
reasons.
Vice
versa,
any differentiable function on a symplectic manifold
M,
defines
a
Hamiltonian vector field
Xf
by the relation
ix,
w
=
df
.
9.2.1
In
a
coordinate basis, we may write
Integral curves
of
a Hamiltonian vector
field