Theory of systems with Constraints
Javier De Cruz Pe?rez
Facultat de F??sica, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Spain.?
Abstract: We discuss systems subject to constraints. We review Dirac?s classical formalism of
dealing with such problems and motivate the definition of objects such as singular and nonsingular
Lagrangian, first and second class constraints, and the Dirac bracket. We show how systems with
first class constraints can be considered to be systems with gauge freedom. We conclude by studing
a classical example of systems with constraints, electromagnetic field
I. INTRODUCTION
In the standard case, to obtain the equations of motion,
we follow the usual steps. If we use the Lagrangian for-
malism, first of all we find the Lagrangian of the system,
calculate the Euler-Lagrange equations and we obtain the
accelerations as a function of positions and velocities. All
this is possible if and only if the Lagrangian of the sys-
tems is non-singular, if not so the system is non-standard
and we can?t apply the standard formalism. Then we say
that the Lagrangian is singular and constraints on the ini-
tial data occur. Basically a constrained system is one in
which there exists relations between the system?s degrees
of freedom that holds for all time. Constraints should
not be confused with constants of motion. Constant of
the motion arise as a result of the equations of motions
and constraints to be restrictions on the dynamics be-
fore equation of motion are even solved. For example in
the electromagnetic field (which we will see later in more
detail ), the time derivative of the A0 component of the
vector potential does not appear in the action of the sys-
tem, therefore the momentum conjugate to A0 is always
zero, which is a constraint . The first systematic discus-
sion of singular Lagrian systems was given by Dirac, who
developed a standard technique to ?Hamiltonize? a sin-
gular Lagrangian and the new Hamiltonian formalism to
obtain equations of motion for these systems.[2]
In the following sections we will describe in more detail
the concepts mentioned in this introduction and describe
the Dirac method. Finally we apply this formalism, to
solve a classic example, the electromagnetic field. One
of the most important feature of these systems is that
they present gauge invariance. Therefore it can be de-
scribed by a gauge theory. These are a kind of field
theory in which the Lagrangian is invariant under a con-
tinious group of local transformations. We will see the
relation between first class constraints and the gauge free-
dom. Finally we will find the Dirac brackets between the
field variables (potential 4-vector) and their conjugate
momenta.
?Electronic address: jdecrupe7@alumnes.ub.edu
II. GENERAL CONSIDERATIONS
Consider a system with a finite number, k , of degrees
of freedom is described by a Lagrangian L = L(qs, q?s)
that don?t depend on t. The action of the system is [3]
S =
?
dtL(qs, q?s) (1)
If we apply the least action principle action ?S = 0, we
obtain the Euler-Lagrange equations
?L
?qs
?
d
dt
(
?L
?q?s
)
= 0 s = 1, ..., k (2)
Which can be rewritten as[1]
?2L
?q?r q?s
q?s =
?L
?qr
?
?2L
?qsq?r
q?s (3)
Define the matrix Wrs = ?
2L
?q?r?q?s
. Expression (3) can be
solved for the accelerations q?s if and only if
det |Wrs| 6= 0 (4)
If this determinant vanishes, then the Lagrangian is sin-
gular and the accelerations can?t be solved in terms of
the positions and velocities. In Hamiltonian formalism,
this implies the existency of relations between p?s and
q?s.
III. HAMILTONIAN TREATMENT-THE DIRAC
THEORY
The set up of a Hamiltonian formalism for a La-
grangian starts with the Legendre transformation wich
defines the momenta [3]
ps =
?L
?q?s
s = 1, ..., k (5)
Relation (5) is invertible if and only if det |Wrs| 6= 0.
Then they can be solved to obtain the velocities q?s as
functions of coordinates and momenta. If the determi-
nant vanishes the rank Wrs = R < k and only R veloci-
ties can be obtained from (5) as functions of q?s , p?s and
the remaining (k?R) velocities and also that there exist
Systems with constraints Javier de Cruz Pe?rez
(k ?R) independent relations among p?s and q?s. These
relations, wich are direct consequences of the definitions
of p?s and the structure of the lagrangian.
Supose that we solve the k ? R constraints for the
momenta [1]
p? = ??(pj , qs) s = 1, ..., k, j = 1, ..., R, (6)
? = R+ 1, ..., k
This relations are called ?primary constraints?, the word
primary meaning that the equations of motion were not
used to obtain them. If we started with 2k-dimensional
phase-space defined by 2k indpendent coordinates q?s
and p?s. The motion is going to be confined to a surface
of lower dimensionality , defined by constraint equation.
Dirac introduced the concept of ?weak? and ?strong?
equations. Let the constrained submanifold in phase
space be called U, let f(q, p), g(q, p) be two functions
defined in a neighborhood of U. The values of f and g
on U are obtained by replacing p? by ??. If after this
replacment f and g become equal, then we say that they
are ?weakly equal? and write [1]
f(qs, ps) ? g(qs, ps)
Both functions f and g have a 2k-dimensional ?gradient
vector? at each point in phase space, with components
( ?f?qs ,
?f
?ps
) and ( ?g?qs ,
?g
?ps
) respectively. If f equals g on U
and also the gradient of f agrees with that of g when the
arguments are restricted to U, we say that f and g are
?strongly equal? and write
f(qs, ps) ? g(qs, ps)
The submainfold U is defined by a set of weak equation.
Let us define a set of functions
??(qs, ps) = p? ? ?(qs, pj) (7)
then U can be defined by ?? ? 0. At this point adding
the primary constraints via Lagrange mulipliers [2]
L = pj q?j ?H(q, p) + u
??? j = 1, ..., R ? = R+ 1, ..., k
The action of the system defined by
S =
?
dt
(
pj q?j ?H(q, p) + u
???
)
We apply the principle of least action ?S = 0 and obtain
[1]
q?i =
?H
?pi
+ u?
???
?pi
p?i = ?
?H
?qi
+ u?
???
?qi
u? =
?H
?p?
= q??
?? = 0
We can identify arbitrary functions with velocities that
we can?t obtain of (5). To write the equation of motion
more compactly. We introduce a mathematical opera-
tor called Poisson bracket. For two arbitrary functions
f(q, p) and g(q, p) the Poisson bracket is defined by
{f, g} =
?f
?qi
?g
?pi
?
?f
?pi
?g
?qi
(8)
With this definition the equations of motion can be writ-
ten as
q?i ? {qi, H}+ u
?{qi, ??} (9)
p?i ? {pi, H}+ u
?{pi, ??} (10)
and, for an arbitrary function of the phase space,
f? ? {f,H}+ u?{f, ??}
Now let us examine the consequences of these equation
of motion. We have the quantities ??? wich have to be
zero throughout all time. We can apply the equation of
motion, taking f to one of the ??s.
{??, H}+ u
?{??, ??} ? 0 (11)
We must examine these conditions to see what they lead
to. It is possible for then to lead directly to an incon-
sistency. If that happens, it would mean that original
Lagrangian is such that the equation of motion are in-
consistent. If the equation(11) does not have an incon-
sistencies, can be divided in three kinds.
One kind of equation reduces to 0 = 0 and it is
identically satisfied with the help of the primary con-
straints. We have an another kind of equation if and
only if det{??, ??} 6= 0 then the matrix is invertible and
we can find the arbitrary functions
u? ? ?C??{??, H} (12)
Where C?? is the inverse matrix of {??, ??}. In this case
we can define the following expression [1]
{g,H}? = {g,H} ? {g, ??}C
??{??, H} (13)
This expression is called Dirac bracket and is a general-
ization of the Poisson bracket, allowing us to write the
equations of motion as
d
dt
g(q, p) ? {g,H}? (14)
No arbitrary function appears in the solution of the equa-
tion of motion. Let us now return to the analysis of equa-
tion (11) In general the matrix {??, ??} is singular and
the arbitrary functions u? are not all determinate. If the
rank of the matrix is M < (k?R) there are (k?R?M)
null eigenvectors ?a?
?a?{??, ??} ? 0 a = 1, 2, ..., (k ?R?M) (15)
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Systems with constraints Javier de Cruz Pe?rez
Combining equations (11) and (15) we find the following
further conditions on q?s and p?s
?a?{??, H} ? 0 a = 1, 2, ..., (k ?R?M) (16)
These equations may or may not produce restrictions on
the arbitrary functions. If they do not, we have produced
more constraints ? ? 0 that restrict the motion in phase
space. The new constraints and all others that may yet
appear, are called secondary constraints. They arise only
after the equations of motion are used at least once. We
apply the same procedure for secondary constraints that
we apply for the primary constraints. The process ends
after a finite number of steps when we have (k ?R) pri-
mary constraints and A secondary constraints defining a
submanifold U ?in phase space
?? ? 0 ? = R+ 1, ..., k (17)
?a ? 0 a = 1, .., A (18)
These A secondary constraints arise from the require-
ment that the primary ones are preserved in time. We
must now add the requirement that these secondary con-
straints, are preserved in time as well
{??, H}+ u
?{??, ??} ? 0 (19)
{?a, H}+ u
?{?a, ??} ? 0 (20)
And we have the matrix
D =
?
?
?
?
{??, ??}
{?a, ??}
?
?
?
? . (21)
And the condition about eigenvectors
??{??, H}+ ?
a{?a, H} ? 0 (22)
Now we examine the D-matrix. If det |D| 6= 0,then again
we have the situation of the second kind. We can find the
arbitrary functions using equation (12), and have the fi-
nally equation of motion. But if det |D| = 0 we can?t do.
To resolve this problem first of all we should classify the
constraints. A constraint is called first class if its Poisson
brackets with all constraints vanish weakly. The rest of
all we call second class constraints. Note that the clasi-
fication into primary and secondary constraints refers to
the origin of the constraint. And the classification into
first and second class constraints refers to this properties.
We call ?a first class constraints ( primary or secondary
) and ?b second class constraints(primary or secondary).
We can express the constraints as a linear combination
of these two types [2]
um?m = ?
a?a + ?
b?b
To write this expression we need a theorem that says,
for an arbibtrary function F that F ? 0 we can express
it as a linear combination of the constraints. Putting
this linear combination on the equation and calculate the
Dirac bracket for a ?a and ?b, obtain
{?a, H}+ ?
a?{?a, ?a?}+ ?
b?{?a, ?b?} ? 0
{?b, H}+ ?
a?{?b, ?a?}+ ?
b?{?b, ?b?} ? 0
For the properties of the first class constraints the terms
which survive are
{?a, H} ? 0
{?b, H}+ ?
b?{?b, ?b?} ? 0
Now we can say that the matrix {?b, ??b} is nonsingu-
lar and therefore its inverse exists. Finally we write the
equation of motion as
d
dt
g(q, p) ? {g,H}+?a{g, ?a}?{g, ?b}?
bb?{??b, H} (23)
An important feature of the first class constraints is that
generate a gauge transformation. Imagine that our sys-
tem has only first class constraints, in this case we can?t
eliminate the arbitrary functions u? from the equation
of motion. To resolve this situation we need fixing the
gauge. This is procedure to remove the non-pysical de-
grees of freedom of the system. We will see an example
of this in the next section
IV. EXAMPLE OF THE ELECTROMAGNETIC
FIELD
In this section we finally apply the developed theory
to a physicall system, electromagnetic field. The electric
and the magnetic field obey Maxwell equations. These
fields are expressed in terms of the 4-vector potential
A? = (?, ~A) (we use natural units h = c = 1).
Ei = ?iA0 ? ?
0Ai
Bk = klm?lAm
Sumation convention understood. We can describe this
field as an antisymmetric tensor with two indices [4]
F?? = ??A? ? ??A? (24)
The Lagrangian describing this systems is
L = ?
1
4
F??F?? (25)
An it is invariant under the following local transformation
A
?
?(x) = A?(x) + ??(x)
Where (x) is an aribitrary function (x = (t, ~x)) . This
means that the system has gauge invariance, and hence
it may be described by a gauge theory. At this point
we want to pass to the Hamiltonian formalism, for this
purpose we need the Legendre Transformation, but first
compute the momentum, with the following expresion [5]
pi? =
?L
?(?0A?)
(26)
The momentum for A0 (pi0 ? 0)and this is the primary
constraint of the theory and the proof that the electro-
magnetic field is a constrained system. Additionally, we
identify the canonical momentum with the electric field
pii = ?F 0i = Ei (27)
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Systems with constraints Javier de Cruz Pe?rez
We apply the Legendre transformation H = pi??0A??L
and the Hamiltonian is
H =
?
d3x
(
1
4
F ijFij ?
1
2
piipii ? ?ipi
iA0
)
(28)
It has a functional form, for this reason we need introduce
the Poisson brackets for two arbitrary functionals [5]
{F (x), G(y)} =
?
d3z
(
?F (x)
?A(z)?
?G(y)
?pi(z)?
?
?F (x)
?pi(z)?
?G(y)
?A(z)?
)
(29)
It is, where we have introduce the functional derivative
as
?
???
=
?
???
? ?i
?
?(?i??)
Now, as we have described in previous sections we im-
pose the stability of the primary constraint and apply
the expression (11):
{pi0(x), H(y)}+ u1{pi0(x), pi0(y)} = {pi0(x), H(y)}
=
?
d3z
(
?pi0(x)
?A(z)?
?H(y)
?pi(z)?
?
?pi0(x)
?pi(z)?
?H(y)
?A(z)?
)
= ?
?
d3z
?H(y)
?A?(z)
?0??
3(x? z)
= ?
?H(y)
?A0(x)
= ?ipi
i ? 0
Therefore we obtain a secondary constraint ?ipii ? 0.
This constraint is the Gauss? law in terms of the momen-
tum field. Now we impose agains its stability. We will
use the properties of the delta function to express ?ipii
as a functional
?ipi
i(x) = ?
?
d3x?pii(x?)
?
?x(i?)
?3(x? x?) (30)
and use this expression to calculate
{?ipi
i(x), H} ? 0
{?ipi
i(x), pi0(y)} ? 0
The system has two first class constraint. Previously we
have discussed the relationship between the first class
constraints and the gauge freedom. To continue we need
to fix the gauge, for instance choosing the Coulomb gauge
?iAi ? 0 (31)
An repeat the process previously used
{?iAi, H} = ?
i?iA0 ? 0 (32)
If we impose that the fields decay fast enough when r ?
? (A0 ? 0) Now we have four constraints .
?1 = pi
0 ? 0 ?2 = ?ipi
i ? 0
?3 = A0 ? 0 ?4 = ?
iAi ? 0
In principle we had one primary constraint of the first
class and one secondary constraint of the first class. The
gauge fixing convert the first class into a second class.
And now we have one primary constraint of second class
and three secondary constraints of second class. Now let?s
make a change of coordinates, using a Fourier transform,
and work in the momentum space [4]
Ai(~x, t) =
1
(2pi)
3
2
?
d3kA?(~k, t)iei
~k~x
(33)
pii(~x, t) =
1
(2pi)
3
2
?
d3kp?ii(~k, t)e
i~k~x
(34)
It is straightforward to prove that with this change of
coordinates the constraints take the following form
?1(~k) = 0?(~k) ? 0 ?2(~k) = ki i?(~k) ? 0
?3(~k) = A?0(~k) ? 0 ?4(~k) = k
iA?i(~k) ? 0
We can make the following decoposition
~?A = A??
~k
|~k|
+ ~?A? (35)
~?pi = p?i?
~k
|~k|
+ ~?pi? (36)
Where A?? = (kiA?i) and p?i? = (kip?ii). Thus rewrite the
second and fourth constraints as ?2(~k) = p?i?(~k) ? 0 and
?4(~k) = A??(~k) ? 0 . Therefore, ~?A? can be written as a
linear combination of two unit vectors perpendicular to
~k
Ai(~x, t) =
1
(2pi)
3
2
?
d3kei
~k~x
(
?
a=1,2
Aa(~k, t)e?ai
)
(37)
Same procedure for the momentum. In position space
the Hamiltonian has the following expression in terms of
~E and ~B
H =
1
2
?
d3x
(
~E2 + ~B2
)
(38)
We want express the Hamiltonian in the momentum
space, for this purpose, we use the Parseval?s identity
[4]
1
2
?
d3x
(
~E2 + ~B2
)
=
1
2
?
d3k
(
~?E ~?E? + ~?B ~?B?
)
(39)
Using the relations E?i = p?ii,B?k = iklmklA?m = we can
write the Hamiltonian as
H =
1
2
?
d3k
(
|~?pi|2 + |~k|2| ~?A|2
)
(40)
Now the Hamiltonian takes the form of the superposition
of an infinity of decoupled oscillators of frequency ? =
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Systems with constraints Javier de Cruz Pe?rez
|~k|. Note that we have not first class constraints, this fact
allows us to write the matrix M(k, k?) = |{??, ??}| of the
constraints and compute his inverse. First compute the
elements of M(k, k?)
{p?i0(k), A?0(k
?)} = ??3(k + k?)
{p?i0(k), A??(k
?)} = 0
{p?i?(k), A??(k
?)} = ?k2?3(k + k?)
{p?i?(k), A?0(k
?)} = 0
And obtain the expression for the M-matrix
M(k, k?) =
?
?
?
0 0 ?1 0
0 0 0 ?k2
1 0 0 0
0 k2 0 0
?
?
? ?3(k + k?) (41)
The problem of findig the inverse of the M-matrix reduces
to calculate the components of theM?1(k, k?) that match
the relationship
?
d3k??M(k, k??)??M
?1(k??, k?)?? = ??? ?
3(k + k?) (42)
The only nonvanishing matrix elements are M?131 =
??3(k+ k?) and k2M?142 = ??
3(k+ k?) recall that M?? is
an antisymetric matrix. So the inverse of the M-matrix
becomes
M?1(k, k?) =
?
?
?
0 0 1 0
0 0 0 k?2
?1 0 0 0
0 ?k?2 0 0
?
?
? ?3(k + k?) (43)
The existence of the inverse matrix proves that the gauge
conditions really fix the gauge completly. Having calcu-
lated the inverse, we are now able to write down the Dirac
brakets using(13)
{F (k), G(k?)}? = {F (k), G(k?)}
?
?
d3k??d3k???{F (k), ??(k
??)}(M?1)??{??(k
???), G(k?)}(44)
We use this expression to calculate the fundamental Dirac
brackets ?, ? = 0, 1, 2, 3 i, j = 1, 2, 3
{A??(k), A??(k
?)}? = {p?i?(k), p?i?(k?)}? = 0 (45)
{A?i(k), p?i
j(k?)}? =
(
?ji ?
kikj
k2
)
?3(k + k?) (46)
For the components of transverse components A??(k) =
A?ie?i? and p?i?(k) = p?i
ie?i? we have
{A??(k), A??(k
?)}? = {p?i?(k), p?i?(k?)}? = 0
{A??(k), p?i
?(k?)}? = ????
3(k + k?)
And calculate the equation of motion for components
that not subjects to restrictions
{A??(k), H}
? = p?i?(k)
{p?i?(k), H}
? = ?|~k|2A??(k)
V. CONCLUSIONS
In this paper we study a systems with constraints and
develop a consistent method for the dynamical evolution
of such systems. To obtain the Hamiltonian of the sys-
tem starting from the Lagrangian, required a Legendre
transformation, where the canonical momentum had to
be introduced. If the Lagrangian is singular, when do
the process, arise a certain relations between the phase
space variables . These relations impose a conditions
abaout dynamical of the system and are therefore con-
straint functions.We have also classified the constraints
according to their origin or their properties and we see
the relation between first class constraint ant the gauge
freedom. To write the equation of motion we use the
Dirac bracket. This operator is built with the Poisson
brackets and constraints matrix. Finally we apply all
this theory to resolve the problem of find the equation
of motion of the electromagnetic field. To eliminate the
non-physical degrees of freedom we have fix the gauge,
this has generated more restriccions abaout the system.
When we apply all the constraints, only two componnts
of the four potential and other two of momentum remain
independent. The final equations of motion are consis-
tent with all constraints.
Acknowledgments
I would like to express my gratitue to my advisor Josep
LLosa for helping me in evertyhing I needed , and dedi-
cate many hours of his time.
[1] E.C.G ShudarshanClassical dynamics : A modern per-
spective (John-Wiley& Sons, New York 1974).
[2] P.A.M.Dirac Lectures on quantum mechanics Belfer
Graduate School of Science,1964.
[3] H.Goldstein Meca?nica Cla?sica Editorial Reverte? ,1987
[4] J.Llosa,A.Molina Relativitat especial amb aplicacions a
l?electrodia`mica cla`ssica , 2a edicio?
[5] Weinberg The Quantum theory of fields volume I , 1995
Treball de F?? de Grau 5 Barcelona, January 2014