Dirac Method for Maxwell field theory, Maxwell’s field theory, Electric Limit, Magnetic Limit

1. GALILEAN ELECTROMAGNETISM
and
CONSTRAINED HAMILTONIAN SYSTEMS
A PROJECT REPORT SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENT FOR THE DEGREE OF
M.Sc. IN PHYSICS
Submitted by
SHUBHAM PATEL (17PH40035)
Under the Supervision of
Dr. SANDIPAN SENGUPTA
DEPARTMENT OF PHYSICS
indian institute of technology kharagpur, west bengal 721 302, india
Date of Submission: November 13, 2018
1

2. Abstract
The main goal of this report is to introduce and review of Galilean Electro-
magnetism and other limits, system of unit independence and Carrollian Limit
in electromagnetism. In the second part of report I have talked about Hamil-
tonian formulation and Constrained systems of Electrodynamics (Maxwell Field
Theory). I’m also going to introduce with Lagrangian with higher order field
tensors and its Hamiltonian formulation. Moreover, I’ve given a description of
Ostrogradsky’s Construction for higher order derivative terms in Lagrangian and
their transition into Hamiltonian.
2

3. INTRODUCTION
Galilean electromagnetism is a formal electromagnetic field theory that is con-
sistent with Galilean invariance. The theory is useful for describing the electric and magnetic
fields in the vicinity of charged bodies moving at non-relativistic speeds relative to the frame
of reference.
We are wholesomely aware of relativistic invariance or precisely I should say Lorentz
invariance of Maxwell equations but hardly know about non-relativistic or Galilean
invariance of Maxwell’s Theory. The main purpose for this report will be to depict
Galilean invariance or Galilean limits of Maxwell equations.
There exist two different and perfectly well-defined limits of electromagnetism, not
a single non-relativistic limit. The first is valid when electric effects are dominant (E >>
cB), we call it electric limit; the second one holds when magnetic effects are dominant
(cB >> E), it is called magnetic limit.
We cannot take the limit c → ∞, reason will be apparent(different limit known as
instantaneous limit). Le Bellac and Levy-Leblond formulated the theory using MKS unit
system, but the theory should be invariant from system of units. We will also see this
formulation.
I also described a novel limit named Carrollian limit and its Electromagnetic
behavior in which we take C → ∞. where C is new constant required for redefinition of our
time whose dimension is of velocity.
In the second part of report I have talked about Constrained system. A constraint
is the restriction on the system in which the dimension of the relevant configuration space
is reduced than what we begin with. The dynamical phase space variables of the theories
are not all independent, rather some of these variables have to satisfy constraints following
from the structure of the theory. Such systems are known as constrained systems and the
naive passage to the Hamiltonian description for such a system starting from the Lagrangian
description fails The constrained systems that we will deal with arise in situations where the
choice of Lagrangian do not specify the equations of motion uniquely. In the Lagrangian
framework, these corresponds to Singular Lagrangians. Analysis of such systems is better
carried in Hamiltonian framework where we have an understanding of gauge theories and we
will be essentially focusing on such constrained Hamiltonian systems.
In this report I am going to represent Maxwell’s theory and constraints found in it. And
right now i am working with a higher order term of field tensors in the Lagrangian taken from
the paper of Muller-Hoissen, vol 201 reference [4]. My deed is to solve for the Hamiltonian
and finding constraints for the the theory.
3

4. Content
Part I
1.Introduction to Limits
(i) Ultratimelike condition
(ii) Ultraspacelike condition
(iii) Instantaneous limit
1.1 Electric Limit
1.2 Magnetic Limit
2. Unification of Galilean Limits
3. Equivalence of c
4. Electric and Magnetic limit for an arbitrary chosen system of units
(i) Electric Limit
(ii) Magnetic Limit
(iii) Values of constant parameters in different systems of units
5. Carrollian Limit
6. Carrollian Electromagnetism
6.1 Electric type
6.2 Magnetic type
7. Conclusion
Part II
1. Maxwell’s field theory
2. Equations of Motions
3. Dirac Method for Maxwell field theory
4.Higher order Field tensor Lagrangian 5. Ostrogradsky’s construction for Higher
derivatives
6. Conclusion
Bibliography
4

5. :
Part I GALILEAN ELECTROMAGNETISM
1. INTRODUCTION TO LIMITS: The usual Lorentz transformations of a four
vector (x0, x) are given by:
x0 = γ x0 −
v
c
.x (1)
x = γ x −
v
c
x0 + (γ − 1)
v(v · x)
v2
(2)
where γ = 1 − v2
c2 and x0 = ct Then we can simply transit from Lorentz to Galilean
transformation by setting γ → 1 or we can say v
c
<< 1, along with two different conditions
as follows:
i) x0 >> ˜x (Ultratimelike condition)
Then our Galilean transformations will be
x0 = x0, x = x −
v
c
x0 (3)
ii) x0 << ˜x :(Ultraspacelike condition) then GT,
x0 = x0 −
v.x
c
, x = x (4)
For example our usual Galilean transformations are:
c∆t = c∆t, ∆r = ∆r − v∆t (5)
holds true only if |∆r| << c|∆t| according to (64). The spatio-temporal gradient obeys the
alternate transformation,
= ,
1
c
∂
∂t
=
1
c
∂
∂t
−
v
c
· (6)
If we apply 64 and 65 to the current four-vector we obtain what we will be seen to
be the electric limit if c|ρ| >> |j| and the magnetic limit if c|ρ| << |j|. Obviously, if
there existed only positive (or negative) electric charges, the electric limit alone would be
physically relevant. The existence of two types of electric charges allows |j| to be much larger
than c|ρ| in many cases.
iii) Instantaneous limit:
Other than these there is an another limit known as instantaneous limit which is obtained
by applying c → ∞. This is not a non-relativistic limit. The effect of this limit is same as
electric limit but interpretation is different.
5

6. 1.1 Electric limit:
c|ρ| >> |j| =⇒ E >> cB (7)
then transformation law for (cρ, j) :
ρe = ρe, je = je − vρ. (8)
Where ’e’ subscript corresponds to electric limit. We can prove that continuity equation is
Galilean invariant under transformation 69 and 6.
Continuity equation :
∂ρe
∂t
+ · je =
∂ρe
∂t
− v · ρe + · je − ( · v)ρe =
∂ρe
∂t
+ · je (9)
je is transport of charge.
Lorentz transformation for electric and magnetic fields are:
E = γ E +
v × B
c2
+ (γ − 1)
v(v · E)
v2
cB = γ cB −
v × E
c
+ (γ − 1)
v(v · cB)
v2
(10)
Now if we apply 68 and γ → 1 then
E e = Ee, B e = Be −
v × Ee
c2
(11)
The motion of an electric field ( more generally, any time variation) induces a magnetic field,
while a time-varying magnetic field does not induce any electric field. Thus Faraday’s law
of induction is no longer true in this limit: there can be no Faraday term ∂Be
∂t
in Maxwell’s
Equations,
· Ee =
ρ
0
, · Be = 0
,
× Ee = 0, × Be = 0µ0
∂Ee
∂t
+ µ0je. (12)
From 6 we can check that that 73 are invariant under Galilean transformatiom 72.
i) · E e= · Ee
ii) · B e= · (Be − 0µ0v × Ee)
= · Be- · (v × Ee)
= · Be−Ee · ( × v) + v · ( × Ee) = · Be
6

7. · B e= · Be
iii)
· E e= · Ee
iv)
× Be = 0µ0
∂Ee
∂t
+ µ0je
using 6 and 69
= 0µ0
∂
∂t
− v · Ee + µ0 je − vρ
now using the identity
× (A × B) = A( · B) − B( · A) + (B · )A − (A · )B
along with Gauss Law · E = ρ
0
we get (v · ) = −( · E)v then second and last term of iv) cancels and gives
= 0µ0
∂Ee
∂t
+ µ0j = × Be
=⇒ × B e= × Be
The electric field Ee is derived from a scalar potential φe, the magnetic field Be from a vector
potential Ae. which obey the transformation law
φe = φe
Ae = Ae − 0µ0vφe (13)
The limit of Lorentz Force F = d3
r[ρ(r)E(r) + j(r) × Br]
If we demand Lorentz force invariant under Galilean transformation then, we find that
je × Be = (je − vρe) × (Be − µ0 0v × Ee)
is not invariant i.e., je × Be = je × Be so it is inconsistent with the field transformation.
Hence we can only have electric forces given by
Fe = d3
r[ρe(r)Ee(r)]
. Thus, in the electric limit, the magnetic field Be does exist, but has no effect at all.
7

8. 1.2 Magnetic Limit:
c|ρ| << |j| =⇒ E << cB (14)
This limit provides a phenomenological theory of magnetostatics, and may be
applied to the usual situation at a macroscopic level, where magnetic effects are in general
dominant because of the balance between positive and negative charges. In this limit, the
current four-vector transforms according to equations.
jm = jm, ρm = ρm − 0µ0v · jm (15)
subscript ’m’ corresponds to magnetic limit. Considering condition 75 with 71 keeping in
mind γ → 1 we obtain the field transformation
E m = Em + v × Bm, B m = Bm. (16)
These equations imply that the motion of a magnetic field (or its time variation) in-
duces an electric fiehl, while a time-varying electric field does not produce any magnetic
field, ttence there can be no displacement current in Maxwell’s equations which read in this
limit,
·Em = ρm
0
, ·Bm = 0,
× Em =
∂Bm
∂t
, × Bm = µ0jm (17)
We can again check as earlier that 78 are invariant under 76 and 77. One thing we can
remark here that equations 76 and 78 allows only stationary currents. Taking divergence of
fourth equation of 78 and using vector identity divergence of curl is zero, simply leads to
· ( × Bm) = 0 = µ0 · jm, =⇒ · jm = 0
and the current jm canot be related to a transport of charge. In this limit, there cannot be
any accumulation of charge in a fixed volume. The fields Em and Bm may be derived from
potentials,
Em = − φm −
∂Am
∂t
, Bm = × Am (18)
which obey the transformations
φm = φm − v · Am, , Am = Am. (19)
Again similar to electric limit, finally we can find the force law
Fm = d3
r[jm(r) × Bm(r)]. (20)
since the electric force is inconsistent with Galilean invariance. Thus, the electric field Em
is nonzero, but it does not produce any observable effect.
8

9. We found out the main defect of both limits, if we are to write down a physically
interesting theory of electromagnetism; the fields Be andEm produce no effect at all(68 and
75 ), so that we have no no magnetic force between a current and a moving charge je (because
of only Fe = d3
r[ρe(r)Ee(r)]) and no induced current in presence of time varying magnetic
field (because in magnetic limit displacement current is zero).
In some textbooks the following low-velocity limit for the field transformation law:
E = E + v × B, B = B − 0µ0v × E. (21)
These equations coincide neither with 72 nor with 77 and do not correspond to any kind
of Galilean limit. We think that equations 82 have no well-defined meaning, and should be
avoided altogether.
9

10. 2.Unification of Galilean limits:
As we have problem with Galilean invariance that even after the fields Be and Em are
present but produce no effect at all. We can avoid this difficulty by distinguishing carefully
between electric and magnetic charges and currents in their interaction with the fields. we
can generalize the theory by introducing only one kind of electromagnetic field (E, B). but
since we have to consider simultaneously electromagnetic fields of electric type and magnetic
magnetic type .
According to our earlier concept we know field and corresponding sources are
(Ee, Be) → (ρe, je)and(Em, Bm) → (ρm, jm).
Now if we want to unify them, we must allow Be to interact with magnetic current jm and
Em to interact with electric charge ρe and also opposite ones.
According to our force law:
F = d3
r(ρeEe + jm × Bm + ρeEm + je × Bm + ρmEe + je × Be). (22)
We can check invariance of this force by using transformation 69, 72, 76 and 77.
ρeE e = ρeEe
j m × B m = jm × Bm
third and fourth term combined to give invariance as
ρeE m + j e × B m = ρe(Em + v × Bm) + (je − vρe) × Bm
= ρeEm + je × Bm
now fifth and sixth term together gives rise to
ρmE e + j m × B e = (ρm − 0µ0v · jm)Ee + jm × (Be − 0µ0v × Ee)
= (ρmEe + jm × Be) − 0µ0jm · Ee. (23)
10

11. where I have used identity a × b × c = b(a · c) − c(a · b).
Now using Ee = − φe and (φejm) = jm( φe) + φe · jm and · jm = 0
=⇒ jm( φe) = · (φejm)
So last term becomes 0µ0v d3
r · (φejm) = 0 (applying Divergence theorem : since
vandjm are in same direction).
so equation 84 becomes invariant
ρmE e + j m × B e = (ρmEe + jm × Be)
Thus the force term is invariant.
Shortcoming:
In this theory capacitors do not work in particular with alternating current jm
is necessarily stationary =⇒ no continuity equation ( jm = 0) . Thus there is jo relation
between intensity I in the wire and time derivative dQ
dt
of charge stored in the capacitor. The
behavior of a capacitor is purely relativistic effect. there could be je type electric currents.
But then there would be no magnetic force between two currents and theory would be in-
variant.
Puzzle: The theory which Le-Bellac and Levy-Leblond has formulated is in MKS units.
Is this theory also consistence with other systems of units? Because in CGS system where c
occurs in most of the terms, then applying c → ∞ gives wrong results. That is the reason
we can’t just put c → ∞. Also any physical theory should be independent of any kind of
systems of units apart from some constants.
11

12. Solution of the puzzle: We will find here that the theory is true for any kind of
system of units. We will discuss the dual role of c in Maxwell’s equations and introduce c
equivalence principle, which says that the ’c of units’ is equivalent to ’the c of propagation’,
and then express Maxwell’s eqs in a form independent of specific units.
3. Equivalence of c:
If we specify cu as speed of propagation for system of units and c in any arbitrary system of
units, eventually we will see that both of them are equivalent.
The magnitude of force between two charges q is given by:
F =
α
4π
q2
R2
(24)
and also consider the magnitude of force per unit length between two infinitely long and
parallel currents I separated by R
dF
dl
=
χβ
4π
2I2
R
(25)
where α, βandχ are determined by choice of units and 4π for convenience. from 85 and 86
we find that
dim
αq2
R2
= dim βχI2
so
dim
α
βχ
= dim (velocity)2
= c2
u
If either αor βχ is specified then the value of the other quantity must be determined exper-
imentally.
In SI units we choose β = µ0 = 4π × 10(
− 7)N/A2
and χ = 1
then we experimentally obtain α = 1/ 0N/m. Using these SI values we get
α
βχ
= 2.9986 × 108
m/s2
= c2
u (26)
which is speed of light. this value prompt us to to identify cu with speed of propagation c
in vacuum.
In CGS units, we take α = 4π then experimentally obtain
βχ
4π
=
1
c2
u
12

13. For Heaviside Lorentz units we choose α = 1 and obtain
βχ =
1
c2
u
In all those systems above we got the same cu = c = 2.9986 × 108
m/s2
, which means that
87 is independent of choice of units.
Now if we include these parameters in Maxwell’s eqs, we can express the static limit (time
independent) of Maxwell’s equations as
· E= αρ
· B= 0
× E= 0
× B = βJ
Time dependent generalization
· E(x,t) = αρ
· B(x,t) = 0,
× E(x,t) = 0,
× B(x, t) = βJ(x, t) (27)
These equations define an instantaneous action at a distance theory and are inconsistent
with continuity eqn. We can resolve this problem by adding an extra term in LHS of
fourth eqn as
× B(x, t) + k
∂E(x, t)
∂t
= βJ(x, t)
This equation satisfy the continuity eqn(by taking divergence) as
−
β
(kα)
J(x, t) +
∂ρ(x, t)
∂t
= 0 (28)
when k = −β
α
Still these eqns represents instantaneous limits (no inclusion of c) .
Further we have to modify. we can do that by taking curl of 89 we get
− 2
B −
β
α
∂( × E)
∂t
= β × J (29)
This eqn is equivalent to wave equation for be which is
2
B −
1
c2
∂2
B
∂2t
= β × J (30)
13

14. where c is speed of light. thus comparing 90 and 91 we get
× E = −
α
βc2
∂B
∂t
(31)
Thus we expressed of Our Maxwell’s eqns in a way independent of system of units as
· E = αρ (32)
× E = −
α
βc2
∂B
∂t
(33)
· B = 0 (34)
× B −
β
α
∂E
∂t
= βJ. (35)
In above four eqns only 94 Faraday’s law contains c(speed of propagation). Using the eqn
87 we can writh eqn 94 as
× E = −χ
c2
u
c2
∂B
∂t
(36)
In SI units χ = 1 and If we put cu = c gives us well known Maxwell eqn
× E +
∂B
∂t
= 0
This is the equivalence principle :
cu = c
The speed cu thus obtained in the process of defining electromagnetic units via action at a
distance forces is equivalent to the speed c of electromagnetic waves in vacuum. Thus dual
role of c is eliminated and we have c independent of system of units. Now we can define the
Galilean limits in any unit system without restricting ourselves to MKS.
4. Electric and Magnetic limit for an arbitrary chosen system of units: Relativistic
transformation of electric and magnetic fields and charge and current densities.
E = γ E +
v
c
×
α
βc
+ (γ − 1)
v(v · E)
v2
(37)
α
βc
B = γ
α
βc
+
v
c
× E + (γ − 1)
v(v · αB/βc)
v2
(38)
ρ = γ ρ −
v
c2
· J ,
J = J − γvρ + (γ − 1)
v(v · J)
v2
. (39)
14

15. (i) Electric limit: applying conditions
|E| >>
α
βc
|B|, |v| << c (40)
and c|ρ| >> |J|
E = E
B = B −
β
α
v × E
ρ = ρ
J = J − vρ (41)
Using these transformation we find that eqns 94 and 95 are not invariant which could be
checked, 94
× E +
α
βc2
∂B
∂t
= × E +
α
βc2
∂B
∂t
+
α
βc2
(v · )B −
1
c2
v ×
∂E
∂t
−
1
c2
(v · )v × E (42)
the last three terms are non zero, so it must be modified in the electric limit. This could be
done by using the other eqn which is not invariant and which is 95
· B = · B +
β
α
v · ( × E) (43)
this is not invariant because second term is non zero. to make it invariant we put the second
term of 104 zero which means
× E = 0 (44)
which is invariant because = and E = E in electric limit. Thus we have to replace
eqn 94 by 105. Finally our Maxwell’s eqns modified as
· E = αρ
× E = 0
· B = 0
× B −
β
α
∂E
∂t
= βJ. (45)
these eqns could be checked to be invariant as we did in electric limit case. using vector
identities:
· (a × b) = b · ( × a) − a · ( × b)
and
× (a × b) = a( · b) − b( · a) + (b · )a − (a · )b (46)
15

16. (ii) Magnetic limit
|E| <<
α
βc
|B|, c|ρ| << |J| (47)
and |v| << c Applying this condition our transformation eqns 98 to 99 becomes:
E = E +
α(c × B)
βc2
, B = B
ρ = ρ −
(v · J)
c2
, J = J (48)
Here we see that eqns 93 and 96 are not invariant under transformation 109. We can check
this
× B −
β
α
∂E
∂t
− βJ = × B −
β
α
∂E
∂t
− βJ −
β
α
(v · )E −
v
c2
×
∂B
∂t
−
(v · )
c2
v × B. (49)
this implies that eqn is not invariant, so we have to modify this. we can do this by using the
next eqn which is not invariant,
· E − αρ = · E − αρ −
α
βc2
v · × B − βJ
This eqn is not invariant until the last term is zero. If the last term is zero which means
× B − βJ
then our transformed Maxwell’s equation will be invariant under magnetic limit 109. Our
final eqn for Magnetic limit
· E = αρ
× E = −
α
βc2
∂B
∂t
· B = 0
× B − βJ = 0 (50)
Which ar invariant.
(iii) Values of constant parameters in different systems
CGS unit:
α = 4π β =
4π
c
, χ =
1
c
16

17. SI unit:
α =
1
o
, β = µ0, χ = 1
Heaviside (Lorentz units)
α = 1, β =
1
c
, χ =
1
c
Remark:
I have shown in this section of report the dual role of speed c and verified that Maxwell’s
eqns are independent of any specific system of units.
5. Carrollian Limit
We are mostly interested in non-Minkowskian spacetimes whose structures are invariant
under boost. Upto now we have see one type of Limit Galilean limit which is the part
of a group known as Galilei group. The Galilei group is standard contraction of Poincare
group. There is also one unfamiliar limit which is also a boost invariant.
Here I will introduce from carrollian limit in Electromagnetism. Let I first define two different
times.
Our ususal definition of time coordinate is
x0
= ct
where c is speed of light, by which the Galilean limit is obtained by roughly say c → ∞
limit. Now the other limiting process in which we can define a new time as
s = Cx0
where C is some new constant which again has dimension of velocity is the new time coor-
dinate s has dimension dim[s] = L2
T−1
(action/mass). now we can apply the limit which is
analogous to Galilean limit C → ∞. Again we can write Lorentz transformations as ,
x = x + (γ − 1)
(b · x)b
b2
− bγx0
, x0
= γ(x0
− b · x) (51)
where b could be defined as b = −Cβ have dimension of velocity and γ = (1 − β2
)−1/2
17

18. Now applying the so called Carrollian Limit C → ∞
x = x, s = s − b · x (52)
or
x = x, x0
= x0
−
b · x
C
(53)
We can compare this transformation with Galilei boost using usual definition of time t = x0
c
and considering b = cβ as
x = x + bt, t = t (54)
s and t are different [non-Minkowskian] times in which they have different physical dimen-
sions.
6. Carrollian Electromagnetism: Maxwell’s Equation in vacuum
× E +
∂B
∂t
= 0, c2
× B −
∂E
∂t
= 0
· B = 0, · E = 0
where t is relativistic time.
Imposing c → ∞ will give us Galilean magnetic limit. similarly we can obtain electric
limit by simply redefining fields B → Be = cB, E → Ee = E
c
and letting c → ∞. Let us now
check Carrollian limit of Maxwell’s eqns by considering s as time instead of t.
s = Cx0
and t =
x0
c
=⇒ t =
s
cC
Now redefining of the electromagnetic fields to bring Maxwell’s eqns in the usual form of
after redefinition of time as
E = ˜E B =
˜B
cC
(55)
Thus Maxwell’s eqns
· ˜B = 0, · ˜E = 0
× ˜E +
∂ ˜B
∂s
= 0, × ˜B − C2 ∂ ˜E
∂s
= 0 (56)
18

19. 6.1 Electric type: Imposing Carrollian limit i.e., C → ∞ switches off the Ampere term
× ˜B, giving us Carrollian electromagnetism of the type of the electric type.56
· ˜Be = 0, · ˜Ee = 0
× ˜Ee +
∂ ˜Be
∂s
= 0,
∂ ˜Ee
∂s
= 0 (57)
This theory is also Carroll invariant. Now according to equation 51 applying Carrollian
Constraint i.e., C → ∞ our transformation for Electric type will be,
˜Ee = ˜Ee(x, s − b · x)
˜Be = ˜Be(x, s − b · x) + b × ˜Ee(x, s − b · x) (58)
which keeps Maxwell’s eqns 57 invariant.
6.2 Magnetic type: Again implementation of Carrollian limit we get Magnetic type elec-
tromagnetism by redefining fields as we did earlier.
B → Bm = CB, E → Em =
E
C
(59)
After converting electric and magnetic fields in eqn 56 using eqn 59
× ˜Em +
∂ ˜Bm
∂s
= 0, · ˜Be = 0
,
· ˜Em = 0,
∂ ˜Bm
∂s
= 0 (60)
These eqns are also invariant under magnetic type limitations,
˜Bm = ˜Bm(x, s − b · x)
˜Em = ˜Em(x, s − b · x) + b × ˜Bm(x, s − b · x) (61)
19

20. 7. Conclusion:
All the purposes in the inventory we started with the report are eventually
systematically surmounted. I would represent a summary all I’ve done in this
report briefly. I have shown different type of limits and those implementation
on Maxwell’s Electromagnetic theory. Those limits are Galilean limits,
instantaneous limit and Carrollian limit. All are defined for electric and
magnetic types. I have also shown the dual role of c and its resolution in
a single c, a universal constant (speed of light), along with the Maxwell’s
equations’ independence of system of units.
20

21. Part II CONSTRAINED HAMILTONIAN
First I’ll show that how constraints come into the picture and then will try to find
the solution. We are interested here in Maxwell’s theory, so our main aim is to find con-
straints of Maxwell’s field theory. We begin with Our usual classical Maxwell’s equations
defining Lagrangian density of Maxwell’s theory, conjugate momentum and then enter into
Hamiltonian formulation.
1. Maxwell’s Field Theory:
Maxwell’s Equations
· E = ρ/ε (62)
· B = 0 (63)
× E = −
∂B
∂t
(64)
× B = µ0 J + ε0
∂E
∂t
(65)
In free field these equations will be
· E = 0
· B = 0
× E = −
∂B
∂t
× B = µ0ε0
∂E
∂t
(66)
Now if our Lagrangian is
L = −
1
4
FµνFµν
(67)
where
Fµν
= ∂µ
Aν
− ∂ν
Aµ
(68)
then Conjugate momenta is given by,
πµ
= d4
x
δL
δ∂0Aµ(x )
= −F0µ
(69)
21

22. so, if µ = 0 then π0
= 0 which is the element of field tensor
F00
= 0. (70)
This is a Constraint, which we will discuss soon. Now since
πi
= −F0i
= −∂0
Ai
+ ∂i
A0
The canonical Hamiltonian
Hc = d3
x πµ
∂0Aµ +
1
4
FµνFµν
(71)
After some algebra we get
Hc = d3
x −
1
2
πi
πi +
1
4
FijFij
− A0(∂iπi
) + π0
∂0A0 (72)
here A0 is undetermined and the last term is already zero because π0
= 0 . First two terms
are usual Electromagnetic energy densities
E2
c2
+ B2
(73)
To keep Hamiltonian as well as Action invariant, the equation of motion leads to
∂0π0
= ∂iπi
(74)
If we drop last term of (72)which holds true always then the third term leads to a
constraint which is
∂iπi
= 0 (75)
Namely Gauss Law.
πi
= −F0i
= Ei
, π0
= 0 (76)
is know as primary constraint and Gauss law is secondary constraint. Again we have a
constraint which is known as Bionchi Identity.
∂α
Fµν
+ ∂ν
Fαµ
+ ∂µ
Fνα
= 0 (77)
where Fµν
is defined earlier. Choosing variables α, µ, ν we obtain again Maxwell eqns.
We know that electric and magnetic fields can be defined by scalar and vector potentials as
B = × A, E = − φ −
∂A
∂t
. (78)
which leads to continuity eqn.
J +
∂ρ
∂t
= 0 (79)
which is again a constraint.
22

23. 2. Equations of motion: Here I will discuss about canonical equation of motion for
Aµ and ˙Amu and will see what we get.
We have Lagrangian density defined as (67)
L = −
1
4
FµνFµν
where
Fµν
= ∂µ
Aν
− ∂ν
Aµ
If we use this field tensor in Lagrangian density
L = −
1
4
(∂µ
Aν
− ∂ν
Aµ
)(∂µAν − ∂νAµ)
= −
1
2
(∂µ
Aν
∂µAν − ∂µ
Aν
∂νAµ)
since,
∂L
∂Aµ
= 0
and
∂L
∂∂µAν
= −Fµν
Euler Lagrangian eqn leads to
∂µFµν
= 0 (80)
this compact form represents Maxwell’s equation, where
F0i
= −Ei
, Fij
= −εij
k Bk
and Aµ
= (φ, A) (81)
In Hamiltonian formulation, the canonocal equation of motion is given by
dAµ
dt
= {Hc, Aµ} (82)
where curly braces represent Poisson Bracket. If we put the Hamiltonian described above
then,
dAµ
dt
= d3
x πµ
∂0Aµ +
1
4
FµνFµν
, Aµ (83)
I am solving this by splitting. The first term becomes according to the definition of Poisson
Bracket,
{πµ
∂0Aµ
, Aµ} =
∂(πµ
∂0Aµ
)
∂∂0Aµ
·
∂Aµ
∂Aµ
−
∂(πµ
∂0Aµ
)
∂Aµ
·
∂Aµ
∂∂0Aµ
(84)
23

24. Here the second term vanishes and first term leads to
{πµ
∂0Aµ
, Aµ} = πµ
(85)
Now
FµνFµν
= F0iF0i
+ Fi0Fi0
+ FijFij
= 2F0iF0i
+ FijFij
(86)
Fµν
= −Fνµ
then
Fi0Fi0
= (−F0i)(−F0i
) = F0iF0i
(87)
also
F0i = η0αηiβFαβ
(88)
which retain only if α = 0 and β = i and then ηii = −1 so
F0iF0i
= −(F0i
)2
(89)
Thus 2nd term of Poisson Bracket
1
4
FµνFµν
, Aµ = −
1
2
(∂µ
Aν
∂µAν − ∂µ
Aν
∂νAµ), Aµ
=
∂(−1
2
(∂µ
Aν
∂µAν − ∂µ
Aν
∂νAµ))
∂∂µAν
·
∂Aµ
∂Aµ
−
∂(−1
2
(∂µ
Aν
∂µAν − ∂µ
Aν
∂νAµ))
∂Aµ
·
∂Aµ
∂∂µAν
.
(90)
Again here 2nd term vanishes. and 1st term survives. In 1st term the 1st differentiation
gives us
−Fµν
Thus the Hamiltonian Equation of motion (82) gives us
dAµ
dt
= {Hc, Aµ} = d3
x(πµ
− Fµν
) (91)
where πµ
= −F0µ
⇒
dAµ
dt
= − d3
x(F0µ
+ Fµν
) (92)
In second term of right hand side, actually I did algebra in general that’s why ν is occurring.
we have to put ν = 0 as we are differentiating with respect to ’t’. so
dAµ
dt
= − d3
x(F0µ
+ Fµ0
) (93)
Using antisymmetric property of Field tensor
F0µ
= −Fµ0
24

25. So final result
dAµ
dt
= 0 (94)
This is a constraint which we have already predicted earlier and which implies that Aµ is
a constant of motion. Now if I apply this constraint on the definition of electric field, which
is defined as
E = − φ −
∂A
∂t
then we find
E = − φ (95)
Taking divergence of eqn (95)
· E = − · φ = − 2
φ (96)
again this constraint leads to same eqn of motion which is Gauss Law. For free fields
divergence of electric field is zero, so the expression above represents Laplace Equation,
which is
2
φ = 0 (97)
Thus, in this section we have find equation of motion for canonical variable Aµ
dAµ
dt
= 0, or ∂0A0 = 0 (98)
25

26. 3. Dirac method for Maxwell Field theory
As we have seen in the Maxwell field theory the dynamical phase space variables of these
theories are not all independent, rather some of these variables have to satisfy constraints
following from the structure of the theory. Such systems are known as constrained sys-
tems.
Since lagrangian density
L =
1
4
FµνFµν
where F0i = Ei and Fij = −εijkBk, i,j,k=1,2,3 we also know that Maxwell field theory is
invariant under gauge transformations
Aµ → Aµ = Aµ + ∂µα (99)
where alpha denotes local parameter. As a result of gauge invariance the matrix of
quadratic derivatives become singular and the conjugate momenta
Πµ
= −F0µ
so that we have
Π = E = −(
˙
A + A0) (100)
with
Π0
= 0
this determines that the theory have primary constraint given by
Φ1
= Π0
≈ 0 (101)
we can obtain the canonical Hamiltonian density
H = Πµ ˙Aµ − L = −Π · ˙A +
1
4
FµνFµν
since Π0
= 0
= −Π · (−Π − A0) +
1
2
(−E2
+ B2
)
=
1
2
(Π2
+ B2
) + Π · A0 (102)
Hamiltonian of the theory is now obtained by adding the primary constraint
Hp = Hcan + λ1Φ1
=
1
2
(Π2
+ B2
) + Π · A0 + λ1Π0 (103)
26

27. where λ1 denotes Lagrange multiplier of the theory. Now the equal time canonical Poisson
bracket are given by
{Aµ(x), Aν(y)} = 0 = {Πµ
(x), Πν
(y)} (104)
{Aµ(x), Πν
(y)} = δν
µδ3
(x − y) = − {Πν
(y), Aµ(x)} (105)
, using these we can evaluate time evolution of primary constraint and requiring the con-
straint to time independent we obtain
˙Φ1 ≈ Φ1
, Hp
= Π0
(x),
1
2
(Π2
(y) + B2
(y) + A0(y) · Π(y) + λ1(y)Π0
(y) (106)
only term which survive is
= − Π0
(x), A0(y) · Π(y)
˙Φ1 = · Π(x) ≈ 0 (107)
Therefore we have a secondary constraint in the theory given by,
Φ2
= · Π(x) ≈ 0
now it can be checked that the secondary constraint is time independent,
Φ2
(x) ≈ · Π(x), Hp ≈ 0 (108)
so that the chain terminates.
Thus we see that the Maxwell theory has two constraints,
Φ1
= Π0
≈ 0, Φ2
= · Π(x) (109)
And it is clear that both these constraints are first class constraints. this is consistent with
that not all Lagrange multiplier in the primary Hamiltonian are determined when there are
first class constraint present. In this case as we have seen, the Lagrange multiplier λ1 remains
as yet undetermined (there is one primary constraint which is first class). Furthermore, first
class constraints signal the presence of gauge invariances(local invariances) in the theory
which we know very well in the case of Maxwell field theory. According to Dirac procedure,
in the presence of first class constraint we are supposed to add gauge fixing conditions to
convert them into second class constraints. Let us choose
χ1
= A0(x) ≈ 0, χ2
= · A(x) ≈ 0
Thus, we have determined all the constraints of the theory.
27

28. 4. Higher order Field tensor Lagrangian:
If I have a Lagrangian of higher order
L = (FµνFµν
)2
− 2Fµ
ν Fν
λ Fλ
k Fk
µ
(taken from reference: ”Non-Minimal coupling from dimensional reduction of the Gauss-
Bonnet Action, Folkert Muller-Hoissen, Vol 201, number 3, Received 30 October 1987,
Physical letter ”) I first take 1st term and separate time and spatial parts
FµνFµν
= 2(F0iF0i
)2
+ (FijFij
)2
and then 2nd term
Fµ
ν Fν
λ Fλ
k Fk
µ = F0
j Fj
0 F0
mFm
0 +F0
j Fj
l Fl
mFm
0 +Fi
0F0
l Fl
0F0
i +Fi
0F0
l Fl
mFm
i +Fi
j Fj
l Fl
0F0
i ++Fi
j Fj
0 F0
mFm
i +Fi
j Fj
l Fl
mFm
i
Here the dimension of our Lagrangian is [L]−8
. If we also add usual Lagrangian term FµνFµν
with this Lagrangian whose dimension is [L]−4
then to make action dimensionless we have
to couple that Lagrangian with a mass term whose dimension is [L]−4
or M4
.
Now I will have to calculate the conjugate momenta and then corresponding Hamiltonian.
28

29. Appendix
5. Ostrogradsky’s Construction for Higher Derivatives:
There is a question that ”Why are there only derivatives to the first order in the La-
grangian?” But by Theorem of Ostrogradsky, I came to know that its not true. there are
also Lagrangian which not only depend on position and velocity (possibly on time) but
also depend on acceleration and jerk(third derivatives) Since we already know how to
solve first derivatives from textbooks, I represent here the simplest case of second derivatives.
Lagrangian which depends on the second derivative in an essential manner. Inessential
dependences are terms such as q ˙q which may be partially integrated to give ˙q2
. Mathemati-
cally, this is expressed through the necessity of being able to invert the expression
P2 =
∂L(q, ˙q, ¨q)
∂¨q
and get a closed form for ¨q(q, ˙q, P2) Note that usually we also require a similar statement
for ˙q(q, p), and failure in this respect is a sign of having a constrained system, possibly with
gauge degrees of freedom.
In any case, the non-degeneracy leads to the Euler-Lagrange equations in the usual
manner:
∂L
∂q
−
d
dt
∂L
∂ ˙q
+
d2
dt2
∂L
∂¨q
= 0
Non-degeneracy of L means that the canonical coordinates can be expressed in terms of the
derivatives of q and vice versa.
This is then fourth order in t, and so require four initial conditions, such as q, ˙q, ¨q,
...
q
This is twice as many as usual, and so we can get a new pair of conjugate variables when
we move into a Hamiltonian formalism. We follow the steps of Ostrogradski, and choose our
canonical variables as Q1 = q, Q2 = ˙q which leads to
P1 =
∂L
∂ ˙q
+
d
dt
∂L
∂¨q
P2 =
∂L
∂¨q
The non-degeneracy allows ¨q to be expressed in terms of Q1, Q2, P2 through the second
equation, and the first one is only necessary to define
...
q .
We can then proceed in the usual fashion, and find the Hamiltonian through a Legendre
transform:
H = PiQi − L = P1Q2 + P2 ¨q(Q1, Q2, P2) − L(Q1, Q2, ¨q)
29

30. we can take time derivative of the Hamiltonian to find that it is time independent if the
Lagrangian does not depend on time explicitly, and thus can be identified Hamiltonian as
the energy of the system.
However, we now have a problem: H has only a linear dependence on P1, and so can be
arbitrarily negative. In an interacting system this means that we can excite positive energy
modes by transferring energy from the negative energy modes, and in doing so we would
increase the entropy — there would simply be more particles, and so a need to put them
somewhere. Thus such a system could never reach equilibrium, exploding instantly in an
orgy of particle creation.
This problem is in fact completely general, and applies to even higher derivatives in a similar
fashion.
Boundedness:
Let us consider an example of 2nd time derivative:
L = −
εm
2ω2
¨x2
+
m
2
˙x2
−
mω2
2
x2
Here we apply
P1 =
∂L
∂ ˙q
+
d
dt
∂L
∂¨q
=⇒ P1 = m ˙x +
εm
ω2
...
x =⇒
...
x =
ω2
P1 − mω2
X2
εm
and
P2 =
∂L
∂¨q
=⇒ P2 = −
εm
ω2
¨x =⇒ ¨x = A =
ω2
P2
εm
Thus our Hamiltonian
H = P1X2 −
ω2
P2
2
2εm
−
m
2
X2
2 +
mω2
2
X2
1
Our Hamiltonian is linear in P1 and quadratic in P2. So with respect to P1 it (Hamil-
tonian) is unbounded. It is bounded neither from above nor below. it is unstable. Since
momentum can take either positive or negative values. But with respect to P2 is quadratic,
and it takes only positive values. So Hamiltonian is bounded from below.
We now notice that the Hamiltonian is linear in P1. This is Ostrogradsky’s instability,
and it stems from the fact that the Lagrangian depends on fewer coordinates than there
are canonical coordinates (which correspond to the initial parameters needed to specify the
problem). The extension to higher dimensional systems is analogous, and the extension to
higher derivatives simply means that the phase space is of even higher dimension than the
configuration space, which exacerbates the instability (since the Hamiltonian is linear in even
30

31. more canonical coordinates)
Construction for N derivatives
if our Lagrangian is L = L(x, ˙x, ¨x, ..., xN
) then from Euler Lagrangian Equation
Σ −
d
dt
i
∂L
∂xi
= 0
we have 2N coordinates which are
Xi
≡ xi−1
Pi
= Σ −
d
dt
j−1
∂L
∂xj
where j runs from o to N. We can find the Hamiltonian now in usual way.
6. Conclusion:
Thus I have solved for Hamiltonian from the Lagrangian and found the Con-
straints for our usual Maxwell’s field theory, proposed a method to solve the
theory of constrained systems given by Dirac. I have also shown the method
how to transit from Lagrangian to Hamiltonian formulation in higher order time
derivatives, defined boundedness, stability and instability. The Ostrogradsky in-
stability has been proposed as an explanation as to why no differential equations
of higher order than two appear to describe physical phenomena. Right now I
am dealing with higher order terms in the Lagrangian consisting field tensors. I
will find Hamiltonian and constraints for the Lagrangian.
31

32. Bibliography
Part I
i) On some applications of Galilean electrodynamics of moving bodies: M. de
Montigny, and G. Rousseauxc American Journal of Physics 75, 984 (2007);
10.1119/1.2772289
ii) M. Le Bellac and J.-M. Levy-Leblond, ”Galilean Electromagnetism,” Nuovo
Cimento 14B(1973), 217-233
iii) The Galilean limits of Maxwell’s equations: Jos´e A. Heras Citation:
American Journal of Physics 78, 1048 (2010); doi: 10.1119/1.3442798
iv)Forty year of Galilean Electromagnetism(1973-2013) ,Germain Rousseaux
, Eur. Phys. J. Plus (2013) 128: 81 DOI 10.1140/epjp/i2013-13081-5
v) Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of
time, C Dual, G.W. Gibbons, P.A. Hovathy, P.M. Zhang : Centre de Physique
Theorique, Morseille, France; D.A.M.T.P., Cambridge University,UK,(Dated:March
12, 2014)
link: arxiv:1402.0657v5[gr-qc] , Section v.
Part II
[1] P. A. M. Dirac, Lectures on Quantum Mechanics, Yeshiva University, New
York (1964).
[2] K. Sundermeyer, Constrained Dynamics, Springer-Verlag, Berlin Heidel-
berg New York (1982).
[3] Ashok Das, Lectures on Quantum Field Theory
[4]Lecture on Constrained Systems, Ghanashyam Date
[5] Non-Minimal coupling from dimensional reduction of the Gauss-Bonnet
Action, Folkert Muller-Hoissen, Vol 201, number 3, Received 30 October
1987, Physical letter B
[6] The Theorem of Ostrogradsky, R. P. Woodard
32

33. ACKNOWLEDGEMENTS
A very special gratitude goes out to professor Dr. Sandipan Sengupta
for his best guidance and motivation for the project. I am also grateful to
the faculty of Physics department for their unfailing support and assistance.
33