Vilasi G. Hamiltonian Dynamics
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294
The
Orbits
Method
The symplectic structure on
V
is given by Eq.
(10.33).
In terms of the
bracket
1.
,
.(
here introduced, it can be written
as
~~(~~~~~),
~d~~~~))
=
(It,
dT1,
J>
=
or,
4,
dT)
=
d~(u~~~~))
>
where
H
denotes the restriction
of
T
to an orbit
of
the coadjoint representation.
From
the above equation, it follows
L~~~~~~
=
0
>
which
says
that the
Euler
equation
is
a
Hamiltonian equation,
H
being
the
Hamilton function.
Chapter
11
Classical Electrodynamics
Particles and fields are the fundamental concepts
of
classical physics. Particles
are identified
as
point particles and fields are tensor-valued functions
on
space-
time. Given sources and initial cond~t~ons, fields are ruled
by
the Maxwell*
equations.
11.1
Maxwell's
Equations
The
pheno~enolo~ical equations of the ele~trodynami~s are given by the
following MaxwelI equations:
iu
8
*
<do
=
0
c dt
L,
d-
7idG
=
Q
magnetic
poles
do
not
exist
Farada~'s
law
Gauss' law
Js
2.
ZdD
=
-
s,,
I?.
ti
*James
Clerk Maxwell was born in Edin~urgh in
1831
and died in Cambridge
iR
1979.
He
has
been
a
professor
of
physics
at
Cambridge University from the year
1871.
Famous
physiciat and mathematician, he gave a mathematical expreasion to Faraday's intuitive and
experimental views in the classic
ktise
on
Electricity
and
Magnetism
(London,
1873)
in
which,
as
a
consequence
of
his electromagnetic theory
of
the light, he foresaw electromagnetic
waves
later
detected
by
Hertz
and
applied by Marconi.
295
296
Classical Electrodynamdcs
where
0
c
is the light velocity in vacuum,
0
S
is
a
regular surface in
!R3,
whic,. may change in time, with
a
given
0
&”
is the boundary
of
S
with the orientation induced by the
one
of
S,
0
U
is
a regular submanifold (a volume) of
!R3
with
a
given orientation
and
aU
is
the boundary (a surface)
of
U
with the orientation induced
by the one
of
U.
orientation defining the exterior normal
5,
E
and
are the electric vector field and the magnetic induction vector
field that can also be defined by the Lorentz force
%,
which acts on
a
particle
with electric charge
e
and velocity
ii:
(11.1)
Maxwell’s equations represent the synthesis of discoveries by Faraday,
Gauss and Amp&re.t
$Michael Faraday
was
born at Newington, United Kingdom in
1791.
In
1813
he was
engaged
as
laboratory assistant by
H.
Davy. Against the current idea on the
action
of
forces, he introduced the concept
of
force lanes
to explain the propagation
of
electric
or
magnetic effects. Today they are known
as
integral curves
of
electric
or
magnetic fields.
In the year
1831,
he discovered the
electromagnetic indzlction phenomenon
and constructed
the Erst electric generator. He also discovered the effects of the magnetic field on the light
plane polarization and,
in
chemistry, two fundamental laws on the propagation of the electric
current in chemical solutions. In the first, he established the direct proportionality between
the amount of transformed matter and the amount
of
electric charge passing trough the
electrolyte; in the second, he established the proportionality between the amounts
of
different
substances and their equivalent weights. In order to describe the experiments and to explain
the results, Faraday invented the words
ion, cathode, anode, electrolyte.
He died
at
Hampton
Court in
1867.
Karl F’rederick Gauss was
born
at
Braunschweig in
1777
and died in Gottingen in
1855.
From
the year
1807
he
has
been professor
at
Gottingen University and Director
of
the
Gottingen Astronomic Observatory. Founder of the differential geometry
of
surfaces, math-
ematician, physicist and astronomer, he was called
princeps mathematicorum.
In hie
works,
he adopted the motto
pauca
sed
maturn
(little and deep). Indeed, his
works
are celebrated
also
for
the excellence of the form. However, the
pauca
fill up eleven big volumes.
Anarc5 Marie Amphre
was
born
at
Lyons
in
1775,
and died
at
Marseilles on June
10,1836.
Mathem~tician, chemist, physicist,
man
fascinated by the mystery, Amp-ire attempted
to
And
in nature an answer to his need of universality.
From
the year
1809
he
has
been
professor
of
mathematics at the
&oEe Polytechnique
in Paris. His papers, concerning the connection
between electricity and magnetism, were written in
1820.
Geometrical
Identification
of
Fields
on
R3
297
When
a
charge density
p
can be introduced, such that
the Maxwell equations take the form:
l,
2.
fidu
=
0
(11.2)
where
J’=
pv’
is the current density.
When
S
is
a
stationary surface and fields are sufficiently regular, we can
use Stokes’ theorem to get the Maxwell equations in the following differential
form:
1
aB’
rotz
=
---
c
at
I
divd
=
47rp
(11.3)
11.2
Geometrical Identification
of
Fields on
R3
Phenomenological Eqs.
(11.2)
are important to understand the geometrical
meaning of fields
($,a,fi,B).
Indeed, Eqs. (11.2) show that
8,
b
and
J’
have to define differential 2-forms on
S3,
since they are integrated on
a
2-
dimensional manifold
aU,
while
2
and have to define differential 1-forms
on
R3,
since they are integrated on
a
1-dimensional manifold
as.
Finally,
p
has to define
it
differentiable 3-form) since it is integrated on
a
volume
U.
298
Classical
Electrodynamics
Actually, because of transformation properties
of
electromagnetic fields,
have to define
t~isted~~t~~
rather
Indeed, the transformation laws
of
the fields
E’
and
B,
under space trans-
determined on physical grounds,
8
and
than
even
differential forms.
formations
t+t’=t,
xi
-+
X’a
=
fi(x),
or
time inversion
t-+t’=-t,
xi
-+
x/i
- -
x2,
’
may be obtained from the expression
(11.1)
of the Lorentz force.
It is clear that, owing to the presence of
a
vector product in the expression
of
the Lorentz force,
8
must change sign under time inversion. Therefore,
will be effected, by such
a
transformation, even if it does not depend on time.
Clearly, if we consider only coordinate transformations with positive Jaco-
bian determinant, the identification of electromagnetic fields with even differ-
ential forms would be possible.
We can resume the above discussion saying that:
Fields entering Ampkre’s law must be identified with twisted diflerential
forms while fields entering Faraday’s law will be identified with even differential
forms.
From
Eqs.
(11.2),
we can try the following identification:
E
=
E,dx
+
E,dy
+
Ezdz
B
=
Bxdy
A
dz
+
B,dr
A
dx
+
B,dx
A
dy
H=
Hxdx
+
H,dy
+
HZdz
D=
D,dy
A
dz
+
D,dr
A
dx
+
D,dX
A
dy
J
=
Jxdy
A
dz
+
J,dz
A
da:
+
Jzdx
A
dy
R=edxAdyAdt
electric field differential 1-form
magnetic induction
differential 2 -form
magnetic field
differential 1 -form
electric induction
differential 2-form
it current differential 2-form
charge diflerential3-fom
Geometrical Identification
of
Electromagnetic Field in Space- Time
299
which allows
us
to rewrite Maxwell’s equations in the following form:
S,
B
=
-
S,,
E
c
dt
Id
J,
D
+
p
S,
J
=
J,,
H
.
c
dt
If
S
is
stationary and fields are regular, we can apply Stokes’ theorem to get
Maxwell’s equations in the following differential form:
dB=O
1.
dE
=
-B
dD
-
4nR
=
0
C
1
C
dH
-
-(B
+
47rJ)
=
0
As
a
consequence of transformation properties, in particular
of
the behavior
of
8
under time inversion, it seems more appropriate to consider electromag-
netic fields directly
on
spacetime.
11.3
Geometrical
Identification
of
Electromagnetic
Field
in
Space-Time
By introducing the following differential forms:
F
=
dxo
A
E
-
B
Faraday’s
2-form
G
=
dxo AH
+
D
Ampire’s
2-form
I
=
dxo
A
J
-
R
charge-current
3-form
300
Classical
Electdynamics
with
zo
=
ct,
~axweIl’s equations can be written simply as follows:
dF
=O,
dG=I.
From the last equation, since
d2
=
0,
we have the
continuity equation
dI=O.
(11.4)
11.3.1
The vector potential and the gauge tranBfonnation
The first equation in Eqs.
(11.4)
says that
F
is
a
closed differential
2-form
and,
then, that there exists, locally, a differential
1-form
A
such that
F=dA.
With the differential
1-form
A
we can associate, by using the metrics,
a
vector
field, which is called the
vector potential.
The differential
1-form
A
is not uniquely defined, since
we
can add
to
A
any exact differential
1-form
df:
F
=
dA
=
d(A
+
df)
.
The map
A
-+
A‘
=
A
+
df
is
called a
gauge transformation.
Under space inversion
P
:
(2O,2
123
,2
,It:
)
-+
(20,--&
-22,
-23),
we have
we have
Geometrical Identification
of
Electromagnetic Field in
Space-
Time
301
Thus,
the potential
A
behaves like
an
ordinary differential l-form under
parity tr~formation and
like
a
twisted ~fferential 1-form under time reversal.
These
properties are in agreement with
CPT
theorem
according to which
photons
are charge-odd particles;
that
is,
the potential
A
is
a
twisted differential
l-form
under
charge conjugation:
C
:
Aodx’
+
Aidxi
-+
A&I?s”
+
A:ddi
=
-(AodzO
+Aid&).
11.3.2
Const&diue
equations
Fields
(J??,B,
I?,
6)
are not independent entities but are connected through
pheno~enolo~ical relations
D=B(2,B],
H
=
H[#,d],
which depend on the specific media under consideration.$ When conducting
media are also considered, there
is
a generalized Ohm’s law
J
=
Jp3,dJ
I
If
we restrict ourselves to linear response media, the constitutive equations
have the following
form:
(11.5)
where
E
and
p
are invertible linear maps called
dielectric map
and
magnetic
~~~e~~~l~t~
map,
r~pectively,
In the case
of
specific isotropic media,
as
the vacuum,
Eqs.
(11.5)
takes the
very simple form:
(11.6)
where
Z
is the identity map.
HfE,
81
may
not
be
local (hysteresis) and may
not
be linear.
*The
square bracket has been used
to
remind that the relations
D
=
D[e,
81
and
H
=
302
Classical
Electrodynamics
In order to put Eqs.
(11.6)
in a provisional geometrical form, we need
a
linear map between differential 1-forms and differential Zforms,
from
E
to
D
and from
H
to
B.
If
we
think to
E,
B,
H,
D
as
differential forms
on
the
manifold
M
=
R3,
the linear map, we are speaking about, can be given by the Hodge dual
*
:
A(!R3)
--+
A2(R3).
Thus, the experimental relations
(11.6)
determine the
Euclidean metric in
B3
and can be written in the following geometrical form:
where the Hodge dual
*
is
constructed out from the volume form
Sl
=
dx
A
dy
A
dz
and the Euclidean metric
gij
=
Sjj.
However, special relativity forces
to
think
F
and
G
as
differ~ntial 2-forms
on
the space-time manifold
M
=
S4.
Thus, the linear map
is
still given
by a Hodge dual but, this time, constructed out from the volume form
SZ
=
cdt
A
dx
A
dy
A
dz
and the Minkowski metric
gaj
=
qij.
Therefore, we obtain
the constitutive equations in the following form:
In
four dimensions, the Hodge dual
*
:
A2(R4)
--+
A2(R4)
determines the metrics up to
a
scalar function,
so
that the constitutive pro-
perties of the vacuum
fix
up the conformal Lorentzian structure of space-time.
Remark
22
It
is
advisable to observe that Maxwell's equations,
written
an
the form
dF=0,
dG=I,
are invariant for any transformation
in
space-time.
If
we require that such
t~~sjo~at~ons preserve the const~~utzve equations
then the symmetry transformation group reduces
to
the conformal group.
Geometrical
Identification
of
Electromagnetic Field in Space- Time
303
Finally,
by
using the codifferential operator
6,
which, in
M
=
R4,
can be
expressed
as
follows:
6
=
-
*d*,
we have
PO
PO
so
that Maxwell’s equations in the vacuum can be written in the following
form:
dF=0,
6~
=
-p*
I,
&O
or
dF
=0,
11.3.3
The
wave
equation
By
replacing
F
=
dA
in the second Eq.
(11.7),
we have
or
We can also write
(11.7)
where
0
is the Laplace-Beltrami operator which in the Minkowski space-time
is given
by
a2
82
a2
a2
ax;
ax:
ax;
ax;.
0
=
-
-
-
-
-
-
-