# Singular Lagrangians, Constrained Hamiltonian Systems and
Gauge Invariance: An Example of the Dirac–Bergmann Algorithm

J. David Brown<EMAIL_ADDRESS>Department of Physics, North Carolina
State University, Raleigh, NC 27695

###### Abstract

The Dirac–Bergmann algorithm is a recipe for converting a theory with a
singular Lagrangian into a constrained Hamiltonian system. Constrained
Hamiltonian systems include gauge theories—general relativity,
electromagnetism, Yang–Mills, string theory, etc. The Dirac–Bergmann algorithm
is elegant but at the same time rather complicated. It consists of a large
number of logical steps linked together by a subtle chain of reasoning.
Examples of the Dirac–Bergmann algorithm found in the literature are designed
to isolate and illustrate just one or two of those logical steps. In this
paper, I analyze a finite-dimensional system that exhibits all of the major
steps in the algorithm. The system includes primary and secondary constraints,
first and second class constraints, restrictions on Lagrange multipliers, and
both physical and gauge degrees of freedom. This relatively simple system
provides a platform for discussing the Dirac conjecture, constructing Dirac
brackets, and applying gauge conditions.

##  I Introduction

In the early 1950’s, Dirac and Bergmann independently developed the
Hamiltonian formalism for systems with singular Lagrangians [1, 2, 3, 4, 5, 6,
7, 8, 9, 10]. These systems, often called “constrained Hamiltonian systems”,
include gauge theories. Gauge freedom is more clearly and more completely
displayed in the Hamiltonian setting, with the generators of gauge
transformations expressed as functions on phase space. Historically, the main
motivation for casting gauge theories in Hamiltonian form was to facilitate
their canonical quantization. Dirac and Bergmann were primarily motivated by
the prospect of developing a quantum theory of gravity based on a Hamiltonian
formulation of general relativity.

Textbook treatments of Lagrangian and Hamiltonian mechanics invariably assume
that the Lagrangian $L(q,\dot{q})$ is nonsingular; that is, that the matrix of
second derivatives of $L(q,\dot{q})$ with respect to the velocities is
invertible. In classical mechanics, the nonsingular case appears to be
sufficient to cover problems of physical interest. However, one might argue
that textbooks avoid certain physically interesting problems simply because
their Lagrangians are singular.

In field theory, the issue of singular Lagrangians and gauge freedom cannot be
avoided. Nearly every field theory of physical interest—electrodynamics,
Yang–Mills theory, general relativity, relativistic string theory—has gauge
freedom.

The Dirac–Bergmann algorithm transforms a singular Lagrangian system into a
Hamiltonian system. The formalism is elegant but at the same time rather
complex. It consists of a large number of logical steps, linked together by a
chain of reasoning that can be difficult to keep straight. Of course, there
are many examples in the literature in which the Dirac–Bergmann algorithm is
applied, converting a singular Lagrangian into Hamiltonian form. However, to
my knowledge, all of these examples are designed to illustrate just one or two
of the logical steps in the algorithm. The student of the subject is faced
with the task of linking these examples together to create a complete picture
of the algorithm.

For those who learn by example, what is needed is a single example that
illustrates all of the major logical steps in the Dirac–Bergmann algorithm and
shows how these steps are linked together. Such a “complete” example is not
easy to identify because there is no obvious way to predict, starting with a
particular Lagrangian, which of the steps in the algorithm will be needed.

The system analyzed in this paper is defined by the Lagrangian

$L(q,\dot{q})=\frac{1}{2}\Bigl{\\{}(q_{1}+\dot{q}_{2}+\dot{q}_{3})^{2}+(\dot{q}_{4}-\dot{q}_{2})^{2}+(q_{1}+2q_{2})(q_{1}+2q_{4})\Bigr{\\}}$
(1)

where the dot denotes a time derivative. The matrix of second derivatives with
respect to the velocities is

$\frac{\partial^{2}L}{\partial\dot{q}_{i}\partial\dot{q}_{j}}=\begin{pmatrix}0&0&0&0\\\
0&2&1&-1\\\ 0&1&1&0\\\ 0&-1&0&1\end{pmatrix}\ .$ (2)

This matrix is singular; it has rank 2.

As we will see, the system defined by the Lagrangian (1) is relatively
complete.111This example does not cover every contingency. In particular, it
does not include redundant constraints [11]. It contains both primary and
secondary constraints, both first and second class constraints, and
restrictions on the Lagrange multipliers. The first class constraints for this
system are not all primary; this allows us to address the Dirac conjecture.
The second class constraints can be eliminated by introducing Dirac brackets.
Finally, this system contains both physical and gauge degrees of freedom. The
gauge freedom can be eliminated with suitable gauge conditions.

One characteristic of any complete example such as (1) is that the
configuration space, the space of $q$’s, must be at least four-dimensional.
Here is why: The number of physical degrees of freedom is equal to the
dimension of the configuration space, minus the number of first class
constraints, minus half the number of second class constraints. If the example
is to have at least one physical degree of freedom, at least two first class
constraints (one primary and one secondary), and at least two second class
constraints (the number of second class constraints must be even), then the
configuration space must be at least four–dimensional.

The study of constrained Hamiltonian systems predates Dirac and Bergmann with
earlier work by Rosenfeld [12, 13]. Like Rosenfeld, Bergmann and his
collaborators [1, 2, 4, 6, 7, 14] were focused on field theories such as
general relativity that are covariant with respect to general four-dimensional
coordinate transformations. (For a historical review, see [15].) Dirac took a
more basic approach to the problem by considering a generic singular
Lagrangian [3, 5, 8, 9, 10]. He developed the algorithm for the case of
systems with a finite number of degrees of freedom. His view was that the
generalization to field theory, with an infinite number of degrees of freedom,
would be “merely a formal matter”.

Although the typical starting point for the Dirac–Bergmann analysis is a
singular Lagrangian, not all gauge theories can be expressed in terms of a
Lagrangian that depends only on the $q$’s. General relativity is one such
example. The Einstein–Hilbert action is a functional of Lagrange multipliers
(the lapse function and shift vector) as well as the configuration variables
(the spatial metric). Nevertheless, the Dirac–Bergmann algorithm provides the
foundation for our interpretation of general relativity as a constrained
Hamiltonian system.

In this paper, I apply the Dirac–Bergmann algorithm to the Lagrangian (1),
following closely the general treatment given by Henneaux and Teitelboim [11].
In turn, the account of Henneaux and Teitelboim closely follows Dirac’s 1964
Lectures on Quantum Mechanics [10]. Presentations of the Dirac–Bergmann
algorithm can also be found in books by Hanson, Regge and Teitelboim [16],
Sundermeyer [17], Rothe and Rothe [18], and Lusanna [19].

Throughout the paper, I attempt to explain the reasoning behind the logical
steps of the Dirac–Bergmann algorithm, but avoid general proofs. The reader is
referred to references [16, 17, 11, 18, 19] for more details.

We begin in Section II with a derivation of Lagrange’s equations for the
singular Lagrangian (1). The general solution is derived, and in Section III,
we discuss the gauge freedom at the Lagrangian level. We begin construction of
the Hamiltonian theory in Section IV with a derivation of the primary
constraints and canonical Hamiltonian. In Section V, we introduce the primary
Hamiltonian and the primary action. Section VI is devoted to a discussion of
the initial value problem and the need to go beyond the primary Hamiltonian.
In Section VII, we apply Dirac’s consistency conditions to derive the
secondary constraints and restrictions on the Lagrange multipliers. The
concept of weak equality is introduced in Section VIII, along with a formal
analysis of the restrictions on the Lagrange multipliers. The total
Hamiltonian is computed in Section IX, and in Section X we sort the
constraints into first and second class. The first class Hamiltonian and gauge
generators are identified in Section XI, where we also introduce the Dirac
conjecture. In Sections XII and XIII, we define the extended Hamiltonian and
extended action. Dirac brackets are used in Section XIV to eliminate the
second class constraints, which yields a partially reduced Hamiltonian. The
corresponding partially reduced action is derived in Section XV, and in
Section XVI we eliminate the momenta to obtain a partially reduced Lagrangian.
Gauge conditions are introduced in Section XVII and Dirac brackets are used to
eliminate the constraints and gauge conditions. This yields a fully reduced
Hamiltonian. The fully reduced action is derived in Section XVIII. Finally,
Section XIX contains a summary of the Dirac–Bergmann algorithm and a
discussion of the Einstein–Hilbert action for general relativity.

##  II Lagrangian Analysis

The action is the integral of the Lagrangian (1):

$S[q]=\int_{0}^{T}dt\,L(q,\dot{q})\ .$ (3)

The notation $S[q]$ indicates that $S$ is a functional of the complete set of
coordinates, $q_{i}=\\{q_{1},q_{2},q_{3},q_{4}\\}$. The equations of motion
are obtained by extremizing the action. For this example, we are not concerned
with boundary conditions and integrate by parts freely. Lagrange’s equations
are

$\displaystyle 0$ $\displaystyle=\frac{\delta S}{\delta
q_{1}}=\dot{q}_{2}+\dot{q}_{3}+2q_{1}+q_{2}+q_{4}\ ,$ (4a) $\displaystyle 0$
$\displaystyle=\frac{\delta S}{\delta
q_{2}}=-2\ddot{q}_{2}-\ddot{q}_{3}+\ddot{q}_{4}-\dot{q}_{1}+q_{1}+2q_{4}\ ,$
(4b) $\displaystyle 0$ $\displaystyle=\frac{\delta S}{\delta
q_{3}}=-\ddot{q}_{2}-\ddot{q}_{3}-\dot{q}_{1}\ ,$ (4c) $\displaystyle 0$
$\displaystyle=\frac{\delta S}{\delta
q_{4}}=\ddot{q}_{2}-\ddot{q}_{4}+q_{1}+2q_{2}\ .$ (4d)

We can rewrite these as follows. First, add Equations (4b) and (4d), then
subtract Equation (4c). This gives

$q_{1}+q_{2}+q_{4}=0\ .$ (5a) Next, subtract this result from Equation (4a) to
obtain $\dot{q}_{2}+\dot{q}_{3}+q_{1}=0\ .$ (5b) The time derivative of this
equation yields Equation (4c). Finally, we find the result
$\ddot{q}_{4}-\ddot{q}_{2}=q_{2}-q_{4}\ .$ (5c)

by solving Equation (5a) for $q_{1}$ and using the equation of motion (4d).

Equations (5) are equivalent to Lagrange’s Equations (4). In particular,
Equation (4a) is the sum of Equations (5a) and (5b); Equation (4b) is the sum
of Equations (5a) and (5c) with the time derivative of (5b) subtracted;
Equation (4c) is the negative of the time derivative of (5b); Equation (4d) is
obtained by subtracting (5c) from (5a).

The equations of motion for this simple linear system are easily solved. Note
that the combination $q_{4}-q_{2}$ is determined by (5c) along with initial or
boundary data; thus, we have

$q_{4}(t)-q_{2}(t)=A\sin t+B\cos t\ ,$ (6)

where $A$ and $B$ are constants. Now, Equation (5a) gives

$q_{1}(t)+2q_{2}(t)=-A\sin t-B\cos t\ .$ (7)

If we knew $q_{2}(t)$, we could solve the previous two equations for
$q_{1}(t)$ and $q_{4}(t)$, then integrate Equation (5b) to obtain $q_{3}(t)$.
Clearly, we do not have enough information to fully determine each of the
$q$’s as functions of time. One of the $q$’s must remain undetermined. For
example, let us choose $q_{2}$ arbitrarily by setting $q_{2}(t)=-\Psi(t)/2$
for some function $\Psi(t)$. We can then use the equations above to solve for
$q_{1}$, $q_{3}$ and $q_{4}$:

$\displaystyle q_{1}(t)$ $\displaystyle=-A\sin t-B\cos t+\Psi(t)\ ,$ (8a)
$\displaystyle q_{2}(t)$ $\displaystyle=-\Psi(t)/2\ ,$ (8b) $\displaystyle
q_{3}(t)$ $\displaystyle=-A\cos t+B\sin t+\Psi(t)/2-\int_{0}^{t}ds\,\Psi(s)+C\
,$ (8c) $\displaystyle q_{4}(t)$ $\displaystyle=A\sin t+B\cos t-\Psi(t)/2\ ,$
(8d)

where $C$ is an integration constant. This is the general solution of the
equations of motion.

##  III Gauge Invariance

The undetermined function $\Psi(t)$ that appears in the general solution (8)
can be freely specified. This is the gauge freedom of the theory. We can
express the gauge freedom in another way: the Lagrangian (1) and the equations
of motion (either (4) or (5)) are invariant under the replacements

$\displaystyle q_{1}(t)$ $\displaystyle\to q_{1}(t)+\Psi(t)\ ,$ (9a)
$\displaystyle q_{2}(t)$ $\displaystyle\to q_{2}(t)-\Psi(t)/2\ ,$ (9b)
$\displaystyle q_{3}(t)$ $\displaystyle\to
q_{3}(t)+\Psi(t)/2-\int_{0}^{t}ds\,\Psi(s)\ ,$ (9c) $\displaystyle q_{4}(t)$
$\displaystyle\to q_{4}(t)-\Psi(t)/2\ ,$ (9d)

where $\Psi(t)$ is an arbitrary function of time.

Although each configuration of the system (that is, each set of $q$ values)
corresponds to a specific physical state of the system, the converse is not
true. Because of the gauge freedom, there are many sets of $q$’s that describe
one and the same physical state.

Let us examine the gauge freedom more closely, in anticipation of the
Hamiltonian description of evolution. To begin, choose the gauge $\Psi(t)=0$
and consider the general solution (8). This solution describes the evolution
of the system from initial data

$\displaystyle q_{1}(0)=-B\ ,$ $\displaystyle\quad\dot{q}_{1}(0)=-A\ ,$ (10a)
$\displaystyle q_{2}(0)=0\ ,$ $\displaystyle\quad\dot{q}_{2}(0)=0\ ,$ (10b)
$\displaystyle q_{3}(0)=C-A\ ,$ $\displaystyle\quad\dot{q}_{3}(0)=B\ ,$ (10c)
$\displaystyle q_{4}(0)=B\ ,$ $\displaystyle\quad\dot{q}_{4}(0)=A\ .$ (10d)

The configuration at some arbitrary final time $t=T$ is

$\displaystyle q_{1}(T)$ $\displaystyle=-A\sin T-B\cos T\ ,$ (11a)
$\displaystyle q_{2}(T)$ $\displaystyle=0\ ,$ (11b) $\displaystyle q_{3}(T)$
$\displaystyle=C-A\cos T+B\sin T\ ,$ (11c) $\displaystyle q_{4}(T)$
$\displaystyle=A\sin T+B\cos T\ .$ (11d)

This configuration corresponds to a particular state of the physical system.

We can choose a different gauge in Equation (8). As long as the new gauge
satisfies $\Psi(0)=\dot{\Psi}(0)=0$, the solution will describe evolution from
the same initial data (10). For example, with

$\Psi(t)=\frac{(\pi^{2}-4)\epsilon}{8}\left[\cos(\pi
t/T)-1\right]+\frac{\pi\epsilon}{4}\left[\pi t/T-\sin(\pi t/T)\right]$ (12)

where $\epsilon={\rm const}$, the configuration at $t=T$ is

$\displaystyle q_{1}(T)$ $\displaystyle=-A\sin T-B\cos T+\epsilon\ ,$ (13a)
$\displaystyle q_{2}(T)$ $\displaystyle=-\epsilon/2\ ,$ (13b) $\displaystyle
q_{3}(T)$ $\displaystyle=C-A\cos T+B\sin T+\epsilon/2\ ,$ (13c) $\displaystyle
q_{4}(T)$ $\displaystyle=A\sin T+B\cos T-\epsilon/2\ .$ (13d)

The configurations (11) and (13) represent the same physical state of the
system, since they evolve from the same initial data.

We can express this result more compactly as

$\displaystyle\delta q_{1}$ $\displaystyle=\epsilon\ ,$ (14a)
$\displaystyle\delta q_{2}$ $\displaystyle=-\epsilon/2\ ,$ (14b)
$\displaystyle\delta q_{3}$ $\displaystyle=\epsilon/2\ ,$ (14c)
$\displaystyle\delta q_{4}$ $\displaystyle=-\epsilon/2\ .$ (14d)

Here, $\delta q_{i}$ denotes the change in $q_{i}$ at the generic time $T$,
due to the change in gauge function $\Psi(t)$.

Here is another example. With

$\Psi(t)=\frac{\pi^{2}\epsilon}{4T}\left[\cos(\pi
t/T)-1\right]+\frac{\pi\epsilon}{2T}\left[\pi t/T-\sin(\pi t/T)\right]$ (15)

we obtain a configuration that differs from Equation (11) by

$\displaystyle\delta q_{1}$ $\displaystyle=0\ ,$ (16a) $\displaystyle\delta
q_{2}$ $\displaystyle=0\ ,$ (16b) $\displaystyle\delta q_{3}$
$\displaystyle=\epsilon\ ,$ (16c) $\displaystyle\delta q_{4}$
$\displaystyle=0\ .$ (16d)

This configuration is also evolved from the initial data (10), and represents
the same physical state as the configurations (11) and (13).

Although the gauge transformation (9) contains a single arbitrary function of
time, the gauge invariance naturally splits into two types. The first consists
of variations subject to $\delta q_{2}=-\delta q_{3}=\delta q_{4}=-\delta
q_{1}/2$. The second consists of arbitrary variations in $\delta q_{3}$, with
$\delta q_{1}=\delta q_{2}=\delta q_{4}=0$. This apparent “doubling” of the
gauge freedom arises because the solution (9c) for $q_{3}(t)$ (unlike the
other variables) includes the integral of $\Psi(t)$. There is enough freedom
of choice in $\Psi(t)$ to allow variations in $q_{3}$ that are independent of
the variations among the other variables. Both types of gauge transformations
leave the physical state of the system unchanged.

The consequences of gauge invariance are most clearly expressed in the
Hamiltonian formalism. The extended Hamiltonian defined in Section XII
includes phase space generators for both types of gauge transformations.

##  IV Primary Constraints and the Canonical Hamiltonian

We now begin construction of the Hamiltonian description of the system. The
conjugate momenta are defined as usual by $p_{i}=\partial
L/\partial\dot{q}_{i}$. For the Lagrangian (1), we have

$\displaystyle p_{1}$ $\displaystyle=0$ (17a) $\displaystyle p_{2}$
$\displaystyle=2\dot{q}_{2}+\dot{q}_{3}-\dot{q}_{4}+q_{1}$ (17b)
$\displaystyle p_{3}$ $\displaystyle=\dot{q}_{2}+\dot{q}_{3}+q_{1}$ (17c)
$\displaystyle p_{4}$ $\displaystyle=\dot{q}_{4}-\dot{q}_{2}$ (17d)

Because the Lagrangian is singular, the matrix of second derivatives
$\partial^{2}L/\partial\dot{q}_{i}\partial\dot{q}_{j}$ is not invertible and
we cannot solve Equations (17) for the velocities as functions of the
coordinates and momenta. The definitions (17) yield two primary constraints,

$\displaystyle\phi_{1}$ $\displaystyle\equiv p_{1}=0\ ,$ (18a)
$\displaystyle\phi_{2}$ $\displaystyle\equiv p_{2}-p_{3}+p_{4}=0\ ,$ (18b)

that restrict the phase space variables $p_{i}$, $q_{i}$. We will denote these
constraints collectively by $\phi_{a}$, where $a=1,2$. Note that in this
simple example, the primary constraints are independent of the $q$’s.

There is freedom in choosing how the constraints are written. For example, we
could replace the $\phi$’s above with

$\displaystyle\tilde{\phi}_{1}$ $\displaystyle\equiv p_{1}\ ,$ (19a)
$\displaystyle\tilde{\phi}_{2}$ $\displaystyle\equiv-p_{1}+p_{2}-p_{3}+p_{4}\
.$ (19b)

In fact, any choice for the constraints is allowed, as long as they follow
from the definitions $p_{i}=\partial L/\partial\dot{q}_{i}$ and satisfy the
regularity conditions. These conditions state that, roughly speaking, the
constraints should have nonzero gradients on the constraint surface. More
precisely, the Jacobian matrix formed from the derivatives of the constraints
with respect to the $p$’s and $q$’s should have maximal rank on the constraint
subspace [10, 11]. For both choices, (18) and (19), the rank of the Jacobian
matrix is $2$. On the other hand, the set

$\displaystyle\bar{\phi}_{1}$ $\displaystyle\equiv p_{1}^{2}\ ,$ (20a)
$\displaystyle\bar{\phi}_{2}$ $\displaystyle\equiv p_{2}-p_{3}+p_{4}\ ,$ (20b)

is not permissible because the gradient of $\bar{\phi}_{1}$ vanishes on the
constraint surface where $p_{1}=0$. Correspondingly, the rank of the Jacobian
matrix is less than $2$ on the constraint surface.

The next step in constructing the Hamiltonian formalism is to compute the
canonical Hamiltonian. The canonical Hamiltonian $H_{C}$ is defined from the
usual prescription by writing $p_{i}\dot{q}_{i}-L(q,\dot{q})$ in terms of
$p$’s and $q$’s. Although we cannot solve for all of the $\dot{q}$’s in terms
of $p$’s and $q$’s, it can be shown that the combination
$p_{i}\dot{q}_{i}-L(q,\dot{q})$ depends only on the phase space variables [10,
11]. For our example problem, the canonical Hamiltonian is

$H_{C}=\frac{1}{2}\Bigl{[}p_{3}^{2}+p_{4}^{2}-2p_{3}q_{1}-(q_{1}+2q_{2})(q_{1}+2q_{4})\Bigr{]}\
.$ (21)

Note that $H_{C}$ is ambiguous. For example, we could use the primary
constraint (18b) to replace the term $-2p_{3}q_{1}$ with
$-2(p_{2}+p_{4})q_{1}$.

##  V Primary Hamiltonian and the Primary Action

The primary Hamiltonian $H_{P}$ is obtained from the canonical Hamiltonian
$H_{C}$ by adding the primary constraints with Lagrange multipliers,

$H_{P}=H_{C}+\lambda_{a}\phi_{a}\ ,$ (22)

where a sum over the repeated index $a$ is implied. The primary action is
built from the primary Hamiltonian in the usual way:
$S_{P}[q,p,\lambda]=\int_{0}^{T}dt\\{p_{i}\dot{q}_{i}-H_{P}\\}$. Explicitly,
we have

$\displaystyle S_{P}[q,p,\lambda]=\int_{0}^{T}dt\biggl{\\{}$ $\displaystyle
p_{i}\dot{q}_{i}-\frac{1}{2}\Bigl{[}p_{3}^{2}+p_{4}^{2}-2p_{3}q_{1}-(q_{1}+2q_{2})(q_{1}+2q_{4})\Bigr{]}$
$\displaystyle-\lambda_{1}p_{1}-\lambda_{2}(p_{2}-p_{3}+p_{4})\biggr{\\}}\ .$
(23)

The primary action is a functional of the complete set of phase space
coordinates, $q_{i}$, $p_{i}$, as well as the Lagrange multipliers
$\lambda_{1}$ and $\lambda_{2}$.

The equations of motion are obtained by extremizing the primary action
$S_{P}$. Extremization with respect to the momenta $p_{i}$ gives

$\displaystyle\dot{q}_{1}$ $\displaystyle=\lambda_{1}\ ,$ (24a)
$\displaystyle\dot{q}_{2}$ $\displaystyle=\lambda_{2}\ ,$ (24b)
$\displaystyle\dot{q}_{3}$ $\displaystyle=p_{3}-q_{1}-\lambda_{2}\ ,$ (24c)
$\displaystyle\dot{q}_{4}$ $\displaystyle=p_{4}+\lambda_{2}\ ,$ (24d) while
extremization with respect to the coordinates $q_{i}$ yields
$\displaystyle\dot{p}_{1}$ $\displaystyle=p_{3}+q_{1}+q_{2}+q_{4}\ ,$ (24e)
$\displaystyle\dot{p}_{2}$ $\displaystyle=q_{1}+2q_{4}\ ,$ (24f)
$\displaystyle\dot{p}_{3}$ $\displaystyle=0\ ,$ (24g)
$\displaystyle\dot{p}_{4}$ $\displaystyle=q_{1}+2q_{2}\ .$ (24h) Extremizing
the action $S_{P}$ with respect to the Lagrange multipliers gives the
constraints, $\displaystyle\phi_{1}$ $\displaystyle\equiv p_{1}=0\ \ ,$ (24i)
$\displaystyle\phi_{2}$ $\displaystyle\equiv p_{2}-p_{3}+p_{4}=0\ .$ (24j)

These equations of motion (24) are equivalent to Lagrange’s Equations (4). To
show this, we first solve Equations (24c,d,i,j) for the momenta to obtain

$\displaystyle p_{1}$ $\displaystyle=0\ ,$ (25a) $\displaystyle p_{2}$
$\displaystyle=\dot{q}_{3}-\dot{q}_{4}+q_{1}+2\lambda_{2}\ ,$ (25b)
$\displaystyle p_{3}$ $\displaystyle=\dot{q}_{3}+q_{1}+\lambda_{2}\ ,$ (25c)
$\displaystyle p_{4}$ $\displaystyle=\dot{q}_{4}-\lambda_{2}\ .$ (25d)

Using these results along with Equations (24a,b) for the Lagrange multipliers,
we find that Equations (24e,f,g,h) agree precisely with Lagrange’s Equations
(4).

##  VI Hamilton’s Equations and the Initial Value Problem

At this point one might ask whether the task of expressing the singular system
(1) in Hamiltonian form is complete. After all, the primary action (V)
provides the correct equations of motion for the phase space variables $q_{i}$
and $p_{i}$. In fact, we can obtain the time evolution of any phase space
function $F$ from $\dot{F}=[F,H_{P}]$ where $H_{P}$ is the primary Hamiltonian
and $[\,\cdot\,,\,\cdot\,]$ denotes Poisson brackets. Hamilton’s equations for
the coordinates and momenta, $\dot{q}_{i}=[q_{i},H_{P}]$ and
$\dot{p}_{i}=[p_{i},H_{P}]$, coincide with Equations (24a–h).

Our task of expressing the singular system in Hamiltonian form is not yet
complete because we still need to interpret Hamilton’s Equations (24a–h) as an
initial value problem. That is, Hamilton’s equations should determine the
future history of the system solely from initial data. In contrast, the
primary action (V) defines a boundary value problem in which the configuration
variables, the $q$’s, are specified at initial and final times.

The key difference between the equations of motion $\delta S_{P}=0$ and
Hamilton’s equations $\dot{F}=[F,H_{P}]$ is that the former include the
primary constraints, Equations (24i,j), whereas the latter do not. Thus, the
phase space trajectories that extremize the action $S_{P}$ must lie entirely
in the primary constraint surface. (The primary constraint “surface” is the
subspace of phase space that satisfies the primary constraints.) In contrast,
the trajectories obtained from Hamiltonian evolution $\dot{F}=[F,H_{P}]$ are
defined throughout the entire phase space. Note that we cannot simply append
the primary constraint equations to Hamilton’s equations, because in that case
the complete system would not be in Hamiltonian form.

Of course, the physically allowed phase space trajectories must satisfy the
primary constraints. With an initial value interpretation of Hamilton’s
equations, we can try to enforce the primary constraints with appropriate
choices of initial data and Lagrange multipliers. In particular, we can choose
initial data that lie on the primary constraint surface $\phi_{1}(0)\equiv
p_{1}(0)=0$ and $\phi_{2}(0)\equiv p_{2}(0)-p_{3}(0)+p_{4}(0)=0$. However,
this is not enough, because the primary constraints are not necessarily
satisfied at later times as the system evolves into the future.

We can describe the situation as follows. The trajectories that extremize the
action $S_{P}$, the physical trajectories, do not necessarily fill the entire
primary constraint surface. Instead they might span only a subspace of the
primary constraint surface. If the initial data lie in the primary constraint
surface but outside the subspace of physical trajectories, then the primary
constraints will not be preserved as the data are evolved.

How should the initial data and Lagrange multipliers be restricted such that
the primary constraints hold throughout the evolution? The primary constraints
will hold for all time if they hold initially and their time derivatives (to
all orders) also vanish initially. In the general case, this leads to a
hierarchy of restrictions on the initial data in the form of secondary,
tertiary, etc., constraints.222Some authors use the term “secondary
constraints” to refer to all higher–order constraints—that is, all constraints
beyond the primary level. It can also lead to restrictions on the Lagrange
multipliers.

The higher order (secondary, tertiary, etc.) constraints and restrictions on
the Lagrange multipliers are not new—imposing them does not change the content
or predictions of the physical theory. This is because the higher-order
constraints and restrictions on the Lagrange multipliers are direct
consequences of the equations of motion (24) that follow from the primary
action (V). They are simply “hidden” in those equations. The process of
identifying the higher-order constraints and restrictions on Lagrange
multipliers reveals these hidden conditions.

##  VII Consistency Conditions, Secondary Constraints and Restrictions on the
Lagrange Multipliers

We can ensure that the primary constraints hold for all time by applying
Dirac’s consistency conditions [10]. Begin by computing the time derivatives
of the primary constraints with the primary Hamiltonian,
$\dot{\phi}_{a}=[\phi_{a},H_{P}]$. Now set these equal to zero:

$[\phi_{a},H_{P}]=0\ .$ (26)

For each value of the index $a$, there are three possibilities.333This assumes
Lagrange’s equations are self–consistent. Otherwise, the consistency
conditions could lead to a contradiction such as $1=0$. First,
$[\phi_{a},H_{P}]$ might vanish on the constraint surface $\phi_{a}=0$, so
that the consistency condition (26) reduces to the identity $0=0$. Second,
$[\phi_{a},H_{P}]$ could be a (non-constant) phase space function that is
independent of the Lagrange multipliers. In this case, Equation (26) is a
secondary constraint. Finally, $[\phi_{a},H_{P}]$ might depend on the Lagrange
multipliers. Then, Equation (26) fixes one of the Lagrange multipliers in
terms of the phase space variables and the other Lagrange multipliers.

The secondary constraints that arise from this process must themselves satisfy
the consistency conditions. This can lead to tertiary constraints and more
restrictions on the Lagrange multipliers. In turn, the tertiary constraints
can lead to quaternary constraints, and so forth. We must continue to apply
the consistency conditions until the process naturally stops.

For our example, the primary constraints are

$\displaystyle\phi_{1}$ $\displaystyle\equiv p_{1}=0\ ,$ (27a)
$\displaystyle\phi_{2}$ $\displaystyle\equiv p_{2}-p_{3}+p_{4}=0\ ,$ (27b)

and their time derivatives are

$\displaystyle\dot{\phi}_{1}$
$\displaystyle=[\phi_{1},H_{P}]=p_{3}+q_{1}+q_{2}+q_{4}\ ,$ (28a)
$\displaystyle\dot{\phi}_{2}$
$\displaystyle=[\phi_{2},H_{P}]=2(q_{1}+q_{2}+q_{4})\ .$ (28b)

Thus, we find the secondary constraints

$\displaystyle\psi_{1}$ $\displaystyle\equiv p_{3}+q_{1}+q_{2}+q_{4}=0\ ,$
(29a) $\displaystyle\psi_{2}$ $\displaystyle\equiv 2(q_{1}+q_{2}+q_{4})=0\ .$
(29b)

These will be denoted collectively by $\psi_{a}$.

Applying the consistency conditions to the secondary constraints gives

$\displaystyle\dot{\psi}_{1}$
$\displaystyle=[\psi_{1},H_{P}]=p_{4}+\lambda_{1}+2\lambda_{2}=0\ ,$ (30a)
$\displaystyle\dot{\psi}_{2}$
$\displaystyle=[\psi_{2},H_{P}]=2(p_{4}+\lambda_{1}+2\lambda_{2})=0\ .$ (30b)

These equations restrict the Lagrange multipliers to satisfy

$p_{4}+\lambda_{1}+2\lambda_{2}=0\ .$ (31)

The process has now terminated. In this example, there are no tertiary or
higher-order constraints.

Recall from the previous section that our goal was to restrict the initial
data and Lagrange multipliers such that the primary constraints vanish for all
times under the Hamiltonian evolution defined by $H_{P}$. We achieve this by
imposing the primary constraints at the initial time,

$\displaystyle\phi_{1}(0)$ $\displaystyle\equiv p_{1}(0)=0\ ,$ (32a)
$\displaystyle\phi_{2}(0)$ $\displaystyle\equiv p_{2}(0)-p_{3}(0)+p_{4}(0)=0\
,$ (32b)

the secondary constraints at the initial time,

$\displaystyle\psi_{1}(0)$ $\displaystyle\equiv
p_{3}(0)+q_{1}(0)+q_{2}(0)+q_{4}(0)=0\ ,\qquad$ (33a)
$\displaystyle\psi_{2}(0)$ $\displaystyle\equiv
2[q_{1}(0)+q_{2}(0)+q_{4}(0)]=0\ ,$ (33b)

and restricting the Lagrange multipliers to satisfy Equation (31) for all time
$t$.

Let us review the reasoning. From Equations (30), the restriction (31) on the
Lagrange multipliers tells us that $\dot{\psi}_{a}=0$ for all time. By
Equations (33), $\psi_{a}$ vanishes initially, so we see that $\psi_{a}$ must
vanish for all time. Now we use Equations (28) to conclude that
$\dot{\phi}_{a}$ must vanish for all time. Since $\phi_{a}$ vanishes
initially, by Equations (32), it follows that the primary constraints
$\phi_{a}=0$ must hold for all time $t$.

##  VIII Weak Equality and Lagrange Multiplier Analysis

It will be useful to follow the general Dirac–Bergmann algorithm closely and
carry out a formal analysis of the restriction (31) on the Lagrange
multipliers [10, 11]. We begin with the concept of weak equality.

Let ${\cal C}_{A}$ denote the complete set of (primary, secondary, tertiary,
etc.) constraints. For our example,

${\cal C}_{A}\equiv\begin{pmatrix}\phi_{1}\\\ \phi_{2}\\\ \psi_{1}\\\
\psi_{2}\end{pmatrix}=\begin{pmatrix}p_{1}\\\ p_{2}-p_{3}+p_{4}\\\
p_{3}+q_{1}+q_{2}+q_{4}\\\ 2(q_{1}+q_{2}+q_{4})\end{pmatrix}$ (34)

where the index $A$ runs from $1$ to $4$.

Two phase space functions $F$ and $G$ are weakly equal if they are equal when
the (primary, secondary, tertiary, etc.) constraints hold. In other words, $F$
and $G$ are weakly equal if they coincide on the constraint surface, the
subspace of phase space defined by ${\cal C}_{A}=0$. Weak equality is written
as $F\approx G$.

Functions $F$ and $G$ are strongly equal if they agree throughout phase space.
Strong equality is written as $F=G$.

Now we turn to the formal analysis of the restriction (31) on the Lagrange
multipliers. This restriction can be expressed as the weak equality $\dot{\cal
C}_{A}\approx 0$. From Equations (28) and (30), we have

$\dot{\cal C}_{A}=[{\cal C},H_{P}]=\begin{pmatrix}p_{3}+q_{1}+q_{2}+q_{4}\\\
2(q_{1}+q_{2}+q_{4})\\\ p_{4}+\lambda_{1}+2\lambda_{2}\\\
2(p_{4}+\lambda_{2}+2\lambda_{2})\end{pmatrix}\approx\begin{pmatrix}0\\\ 0\\\
0\\\ 0\end{pmatrix}$ (35)

which simplifies to

$\begin{pmatrix}0\\\ 0\\\ p_{4}\\\ 2p_{4}\end{pmatrix}+\begin{pmatrix}0&0\\\
0&0\\\ 1&2\\\ 2&4\end{pmatrix}\begin{pmatrix}\lambda_{1}\\\
\lambda_{2}\end{pmatrix}\approx\begin{pmatrix}0\\\ 0\\\ 0\\\ 0\end{pmatrix}\
.$ (36)

This is a system of inhomogeneous linear equations for the Lagrange
multipliers. A particular solution is

$\left.\begin{pmatrix}\lambda_{1}\\\ \lambda_{2}\end{pmatrix}\right|_{\rm
particular}=\begin{pmatrix}0\\\ -p_{4}/2\end{pmatrix}\ ,$ (37)

and the homogeneous solutions are

$\left.\begin{pmatrix}\lambda_{1}\\\ \lambda_{2}\end{pmatrix}\right|_{\rm
homogeneous}=\begin{pmatrix}1\\\ -1/2\end{pmatrix}\lambda\ ,$ (38)

where $\lambda$ is arbitrary. The general solution is the sum of particular
and homogeneous solutions:

$\left.\begin{pmatrix}\lambda_{1}\\\ \lambda_{2}\end{pmatrix}\right|_{\rm
general}=\begin{pmatrix}\lambda\\\ -(\lambda+p_{4})/2\end{pmatrix}\ .$ (39)

Thus, the restriction (31) on the Lagrange multipliers yields
$\lambda_{1}=\lambda$ and $\lambda_{2}=-(\lambda+p_{4})/2$, where $\lambda$ is
an arbitrary function of time.

##  IX Total Hamiltonian

The total Hamiltonian $H_{T}$ is obtained from the primary Hamiltonian $H_{P}$
by inserting the general solution for the Lagrange multipliers:

$\displaystyle H_{T}$
$\displaystyle=H_{P}\Bigr{|}_{\lambda_{1}=\lambda,\,\lambda_{2}=-(\lambda+p_{4})/2}=H_{C}+\lambda\phi_{1}-(\lambda+p_{4})\phi_{2}/2$
$\displaystyle=\frac{1}{2}\left[p_{3}^{2}-p_{2}p_{4}+p_{3}p_{4}-2p_{3}q_{1}-(q_{1}+2q_{2})(q_{1}+2q_{4})\right]$
$\displaystyle\quad+\lambda(p_{1}-p_{2}/2+p_{3}/2-p_{4}/2)\ .$ (40)

Physical phase space trajectories are defined by the total Hamiltonian as the
weak equality $\dot{F}\approx[F,H_{T}]$, with initial data that satisfy the
complete set of constraints, ${\cal C}_{A}=0$.

Hamilton’s equations for the total Hamiltonian $H_{T}$ are

$\displaystyle\dot{q}_{1}$ $\displaystyle\approx\lambda\ ,$ (41a)
$\displaystyle\dot{q}_{2}$ $\displaystyle\approx-p_{4}/2-\lambda/2\ ,$ (41b)
$\displaystyle\dot{q}_{3}$ $\displaystyle\approx
p_{3}+p_{4}/2-q_{1}+\lambda/2\ ,$ (41c) $\displaystyle\dot{q}_{4}$
$\displaystyle\approx p_{3}/2-p_{2}/2-\lambda/2\ ,$ (41d) and
$\displaystyle\dot{p}_{1}$ $\displaystyle\approx p_{3}+q_{1}+q_{2}+q_{4}\ ,$
(41e) $\displaystyle\dot{p}_{2}$ $\displaystyle\approx q_{1}+2q_{4}\ ,$ (41f)
$\displaystyle\dot{p}_{3}$ $\displaystyle\approx 0\ ,$ (41g)
$\displaystyle\dot{p}_{4}$ $\displaystyle\approx q_{1}+2q_{2}\ .$ (41h)

Since these are weak equalities, we can use the constraints to simplify the
results. Observe that the constraints ${\cal C}_{A}=0$ imply $p_{1}=p_{3}=0$,
$p_{2}+p_{4}=0$ and $q_{1}+q_{2}+q_{4}=0$. Therefore, we can set $p_{1}$ and
$p_{3}$ to zero, replace $p_{4}$ with $-p_{2}$, and replace $q_{4}$ with
$-q_{1}-q_{2}$. Then Equations (41d,e,g,h) are either redundant or vacuous,
and the remaining equations are

$\displaystyle\dot{q}_{1}$ $\displaystyle\approx\lambda\ ,$ (42a)
$\displaystyle\dot{q}_{2}$ $\displaystyle\approx p_{2}/2-\lambda/2\ ,$ (42b)
$\displaystyle\dot{q}_{3}$ $\displaystyle\approx-p_{2}/2-q_{1}+\lambda/2\ ,$
(42c) $\displaystyle\dot{p}_{2}$ $\displaystyle\approx-q_{1}-2q_{2}\ .$ (42d)

These equations, along with the constraints ${\cal C}_{A}=0$, give a complete
description of the physical system.

Let us check the results. The Lagrange multiplier $\lambda$ can be eliminated
from Equations (42a,b) to give $\dot{q}_{1}+2\dot{q}_{2}\approx p_{2}$. Now
differentiate this equation and eliminate $\dot{p}_{2}$ with Equation (42d) to
obtain $\ddot{q}_{1}+2\ddot{q}_{2}\approx-q_{1}-2q_{2}$. The constraints allow
us to set $q_{1}\approx-q_{2}-q_{4}$, which gives

$\ddot{q}_{4}-\ddot{q}_{2}\approx q_{2}-q_{4}\ .$ (43)

This is Equation (5c), which follows directly from Lagrange’s equations. The
result (5a) from Lagrange’s equations is simply the secondary constraint
${\cal C}_{4}\equiv 2(q_{1}+q_{2}+q_{4})=0$. Finally, the result (5b) is
obtained by summing Equations (42b) and (42c).

Recall that Equations (5a–c) are equivalent to Lagrange’s equations. Thus, we
have verified that Hamilton’s Equations (41), along with the primary and
secondary constraints ${\cal C}_{A}=0$, are equivalent to Lagrange’s
equations.

##  X First and Second Class Constraints

A first class function $F$ is a phase space function that has weakly vanishing
Poisson brackets with all primary and secondary constraints:

$[F,{\cal C}_{A}]\approx 0\Longleftrightarrow F{\rm\ is\ first\ class.}$ (44)

It can be shown that the Poisson bracket of any two first class functions is
itself a first class function [10, 11].

The constraints themselves can be first class; constraints that are not first
class are called second class. The constraints are separated into first and
second class by examining the matrix of Poisson brackets:

$[{\cal C}_{A},{\cal C}_{B}]=\begin{pmatrix}0&0&-1&-2\\\ 0&0&-2&-4\\\
1&2&0&0\\\ 2&4&0&0\end{pmatrix}\ .$ (45)

The rank of this $4\times 4$ matrix is $2$, and its nullity is $4-2=2$. It
follows that there are $2$ independent eigenvectors with eigenvalues equal to
zero; for example, $u^{A}=(1,-1/2,0,0)$ and $v^{A}=(0,0,1,-1/2)$. Then there
are two independent combinations of constraints that are first class, namely
$u^{A}{\cal C}_{A}$ and $v^{A}{\cal C}_{A}$. (A sum over the repeated index
$A$ is implied.) The first class constraints are

$\displaystyle{\cal C}_{1}^{(fc)}$
$\displaystyle\equiv\phi_{1}-\phi_{2}/2=p_{1}-p_{2}/2+p_{3}/2-p_{4}/2\
,\qquad$ (46a) $\displaystyle{\cal C}_{2}^{(fc)}$
$\displaystyle\equiv\psi_{1}-\psi_{2}/2=p_{3}\ .$ (46b)

One can check that the first class conditions $[{\cal C}_{1}^{(fc)},C_{B}]=0$
and $[{\cal C}_{2}^{(fc)},C_{B}]=0$ hold. The most general first class
constraint is a linear combination of ${\cal C}_{1}^{(fc)}$ and ${\cal
C}_{2}^{(fc)}$.

There are two remaining linear combinations of constraints, which we take to
be

$\displaystyle{\cal C}_{1}^{(sc)}$
$\displaystyle\equiv(\phi_{1}+\phi_{2})/3=(p_{1}+p_{2}-p_{3}+p_{4})/3\
,\qquad$ (47a) $\displaystyle{\cal C}_{2}^{(sc)}$
$\displaystyle\equiv(\psi_{1}+\psi_{2})/3=p_{3}/3+q_{1}+q_{2}+q_{4}\ .$ (47b)

These are the second class constraints. They have nonvanishing Poisson
brackets with each other,

$[{\cal C}_{2}^{(sc)},{\cal C}_{1}^{(sc)}]=1\ .$ (48)

The most general second class constraint is a linear combination of ${\cal
C}_{1}^{(sc)}$, ${\cal C}_{2}^{(sc)}$, ${\cal C}_{1}^{(fc)}$ and ${\cal
C}_{2}^{(fc)}$, with nonzero coefficients on one or both of ${\cal
C}_{1}^{(sc)}$ and ${\cal C}_{2}^{(sc)}$.

The splitting of constraints into first and second class is independent of the
splitting into primary and secondary. In this example, the first class
constraints are mixtures of primary and secondary constraints. Likewise, the
second class constraints are mixtures of primary and secondary constraints.

##  XI First Class Hamiltonian, Gauge Generators and the Dirac Conjecture

The total Hamiltonian (IX) includes the product of an arbitrary Lagrange
multiplier $\lambda$ with the first class constraint ${\cal
C}_{1}^{(fc)}\equiv\phi_{1}-\phi_{2}/2$. We refer to ${\cal C}_{1}^{(fc)}$ as
a primary first class constraint, since it is constructed entirely from
primary constraints.

If we remove the primary first class constraint from the total Hamiltonian,
what remains is the first class Hamiltonian $H_{fc}$. That is, the total
Hamiltonian can be written as

$H_{T}=H_{fc}+\lambda{\cal C}_{1}^{(fc)}\ ,$ (49)

where

$H_{fc}=\frac{1}{2}\Bigl{[}p_{3}^{2}-p_{2}p_{4}+p_{3}p_{4}-2p_{3}q_{1}-(q_{1}+2q_{2})(q_{1}+2q_{4})\Bigr{]}$
(50)

is the first class Hamiltonian. A common notation for $H_{fc}$, the notation
used by Dirac [10], is $H^{\prime}$.

We can check directly that the first class Hamiltonian (50) is a first class
function. However, this is not necessary, because we know that the constraints
are preserved under the time evolution defined by $H_{T}$. That is, $\dot{\cal
C}_{A}=[{\cal C}_{A},H_{T}]\approx 0$. Thus, the total Hamiltonian must be
first class, $[H_{T},{\cal C}_{A}]\approx 0$. Of course, the primary first
class constraint ${\cal C}_{1}^{(fc)}$ is first class. It then follows from
the definition (49) that $H_{fc}$ must also be a first class function.

The splitting (49) of the total Hamiltonian into the first class Hamiltonian
and the primary first class constraint is not special to our example. This
splitting will occur for any constrained Hamiltonian system [10, 11]. In
general, $H_{T}$ will include the products of every primary first class
constraint with an arbitrary multiplier.

Primary first class constraints generate gauge transformations. Consider the
change in a phase space function $F$ generated by the primary first class
constraint ${\cal C}_{1}^{(fc)}$,

$\delta F=\delta\epsilon[F,{\cal C}_{1}^{(fc)}]\ .$ (51)

This transformation does not change the physical state of the system. We can
see this by considering $F$ to be evaluated as a function of the $q$’s and
$p$’s at some particular time $t$. At an infinitesimally later time $t+\delta
t$, this function becomes $F(t+\delta t)=F(t)+[F,H_{T}]\delta t$. In terms of
the first class Hamiltonian, we have

$F(t+\delta t)=F(t)+\left\\{[F,H_{fc}]+\lambda[F,C_{1}^{(fc)}]\right\\}\delta
t\ .$ (52)

The Lagrange multiplier is arbitrary, so we can make a different choice during
the time interval from $t$ to $t+\delta t$, say, $\tilde{\lambda}$. Then, the
function $F$ at time $t+\delta t$ will be

$\tilde{F}(t+\delta
t)=F(t)+\left\\{[F,H_{fc}]+\tilde{\lambda}[F,C_{1}^{(fc)}]\right\\}\delta t\
.$ (53)

The physical state of the system at $t+\delta t$ should not depend on our
choice of Lagrange multiplier, so $F(t+\delta t)$ and $\tilde{F}(t+\delta t)$
must represent the same physical state. The result (51) is obtained by
subtracting Equation (52) from Equation (53) and defining $\delta
F\equiv\tilde{F}-F$ and $\delta\epsilon\equiv(\tilde{\lambda}-\lambda)\delta
t$.

For the phase space coordinates, the gauge transformation generated by the
primary first class constraint ${\cal C}_{1}^{(fc)}$ is

$\displaystyle\delta q_{1}$ $\displaystyle=\delta\epsilon\ ,$ (54a)
$\displaystyle\delta q_{2}$ $\displaystyle=-\delta\epsilon/2\ ,$ (54b)
$\displaystyle\delta q_{3}$ $\displaystyle=\delta\epsilon/2\ ,$ (54c)
$\displaystyle\delta q_{4}$ $\displaystyle=-\delta\epsilon/2\ .$ (54d)

The transformations of the $p$’s all vanish. This result agrees with the gauge
transformation from Equation (14), with the change of notation
$\epsilon\leftrightarrow\delta\epsilon$. Here, we denote the gauge parameter
by $\delta\epsilon$ because the transformation is infinitesimal; in Equation
(14), we used $\epsilon$ because the transformation was finite. It is clear
that the infinitesimal transformation (54) can be iterated to obtain the
finite transformation (14).

In general, a gauge transformation is defined as a transformation $\delta
F=\delta\epsilon[F,{\mathbb{G}}]$ that does not alter the physical state of
the system. The function ${\mathbb{G}}$ is the gauge generator. We have seen
that the primary first class constraints generate gauge transformations.
However, not all gauge transformations are generated by primary first class
constraints. In fact, it can be shown [10, 11] that the Poisson bracket
between any primary first class constraint and the first class Hamiltonian is
itself a first class constraint that generates a gauge transformation.444For
systems with more than one primary first class constraint, the Poisson
brackets of any two primary first class constraints is also a first class
constraint that generates a gauge transformation [10, 11].

For our example problem, the Poisson bracket of the primary first class
constraint ${\cal C}_{1}^{(fc)}$ and the first class Hamiltonian $H_{fc}$ is

$[{\cal C}^{(fc)}_{1},H_{fc}]=p_{3}\ .$ (55)

This is the secondary first class constraint, ${\cal
C}_{2}^{(fc)}\equiv\psi_{1}-\psi_{2}/2=p_{3}$. Thus, we see that in this
example both the primary and secondary first class constraints are generators
of gauge transformations. Explicitly, the transformation $\delta
F=\delta\epsilon[F,{\cal C}_{2}^{(fc)}]$ is

$\displaystyle\delta q_{1}$ $\displaystyle=0\ ,$ (56a) $\displaystyle\delta
q_{2}$ $\displaystyle=0\ ,$ (56b) $\displaystyle\delta q_{3}$
$\displaystyle=\delta\epsilon\ ,$ (56c) $\displaystyle\delta q_{4}$
$\displaystyle=0\ ,$ (56d)

with the transformations of the $p$’s all vanishing. We can iterate this
infinitesimal gauge transformation (56) to obtain the finite transformation
(16).

The “doubling” of the gauge freedom identified in Section III appears quite
naturally in the Hamiltonian formalism. The two types of gauge transformation
are generated by the two first class constraints, ${\cal C}_{1}^{(fc)}$ and
${\cal C}_{2}^{(fc)}$.

The Dirac conjecture [10] says that all first class constraints (whether they
are primary, secondary, etc., or a combination of primary, secondary, etc.)
generate gauge transformations. This conjecture does not hold as a general
theorem—there are known examples in which the transformation generated by a
secondary first class constraint does not coincide with any invariance of the
original Lagrangian system.555Counterexamples to the Dirac conjecture are
discussed in Refs. [20, 11, 21, 22, 23, 24, 25, 26] and elsewhere. Proofs of
the conjecture have been constructed by adopting various simplifying
assumptions [27, 28, 11]. The status of the conjecture is a subtle issue; see
for example Refs. [29, 18, 30]. Nevertheless, the Dirac conjecture is usually
taken as an assumption. It appears that in practice, for systems of physical
interest, all first class constraints generate gauge transformations.

##  XII Extended Hamiltonian

The Dirac conjecture tells us that all first class constraints generate gauge
transformations and should be treated on an equal footing. The extended
Hamiltonian $H_{E}$ is defined by adding all first class constraints ${\cal
C}_{a}^{(fc)}$ with Lagrange multipliers $\gamma_{a}$ to the first class
Hamiltonian:

$H_{E}=H_{fc}+\gamma_{a}{\cal C}_{a}^{(fc)}\ .$ (57)

(A sum over the index $a$ is implied.)

For our example, the extended Hamiltonian is

$\displaystyle H_{E}=$
$\displaystyle\frac{1}{2}\Bigl{[}p_{3}^{2}-p_{2}p_{4}+p_{3}p_{4}-2p_{3}q_{1}-(q_{1}+2q_{2})(q_{1}+2q_{4})\Bigr{]}$
$\displaystyle+\gamma_{1}(p_{1}-p_{2}/2+p_{3}/2-p_{4}/2)+\gamma_{2}p_{3}\ .$
(58)

The equations of motion $\dot{F}\approx[F,H_{E}]$ are

$\displaystyle\dot{q}_{1}$ $\displaystyle\approx\gamma_{1}\ ,$ (59a)
$\displaystyle\dot{q}_{2}$ $\displaystyle\approx-p_{4}/2-\gamma_{1}/2\ ,$
(59b) $\displaystyle\dot{q}_{3}$ $\displaystyle\approx
p_{3}+p_{4}/2-q_{1}+\gamma_{1}/2+\gamma_{2}\ ,$ (59c)
$\displaystyle\dot{q}_{4}$ $\displaystyle\approx p_{3}/2-p_{2}/2-\gamma_{1}/2\
,$ (59d) and $\displaystyle\dot{p}_{1}$ $\displaystyle\approx
p_{3}+q_{1}+q_{2}+q_{4}\ ,$ (59e) $\displaystyle\dot{p}_{2}$
$\displaystyle\approx q_{1}+2q_{4}\ ,$ (59f) $\displaystyle\dot{p}_{3}$
$\displaystyle\approx 0\ ,$ (59g) $\displaystyle\dot{p}_{4}$
$\displaystyle\approx q_{1}+2q_{2}\ .$ (59h)

Let us compare these results to the equations of motion (41) obtained from the
total Hamiltonian $H_{T}$. There are just two differences. The first is
trivial: the Lagrange multiplier $\lambda$ in Equations (41) has changed names
to $\gamma_{1}$ in Equations (59). The second difference is significant: the
equation for $\dot{q}_{3}$ has an extra term $\gamma_{2}$ on the right-hand
side. This is a new feature of the extended Hamiltonian. It makes explicit the
fact that the gauge freedom allows $q_{3}$ to be changed arbitrarily, and
independently, from the other variables.

We can check the equations of motion for $H_{E}$ following the same reasoning
that was applied to the equations of motion for $H_{T}$. First, recall that
the (first and second class) constraints imply $p_{1}=p_{3}=0$, $p_{4}=-p_{2}$
and $q_{4}=-q_{1}-q_{2}$. Then Equations (59e) and (59g) are vacuous, and
Equations (59f) and (59h) are redundant. It also follows that with the
constraints imposed, Equation (59d) is a consequence of Equations (59a) and
(59b). The remaining equations are

$\displaystyle\dot{q}_{1}$ $\displaystyle\approx\gamma_{1}\ ,$ (60a)
$\displaystyle\dot{q}_{2}$ $\displaystyle\approx p_{2}/2-\gamma_{1}/2\ ,$
(60b) $\displaystyle\dot{q}_{3}$ $\displaystyle\approx-
p_{2}/2-q_{1}+\gamma_{1}/2+\gamma_{2}\ ,$ (60c) $\displaystyle\dot{p}_{2}$
$\displaystyle\approx-q_{1}-2q_{2}\ .$ (60d)

These agree with Equations (42), apart from the change of notation
$\lambda\to\gamma_{1}$ and the extra term $\gamma_{2}$ on the right-hand side
of the $\dot{q}_{3}$ equation.

By eliminating $\gamma_{1}$, the equations of motion generated by the extended
Hamiltonian $H_{E}$ become

$\displaystyle\dot{q}_{1}+2\dot{q}_{2}$ $\displaystyle\approx p_{2}\ ,$ (61a)
$\displaystyle\dot{q}_{2}+\dot{q}_{3}$ $\displaystyle\approx-q_{1}+\gamma_{2}\
,$ (61b) $\displaystyle\dot{p}_{2}$ $\displaystyle\approx-q_{1}-2q_{2}\ .$
(61c)

If we differentiate the first equation, combine with the second, and use the
constraint $q_{1}+q_{2}+q_{4}=0$, we obtain
$(\ddot{q}_{4}-\ddot{q}_{2})\approx-(q_{4}-q_{2})$. This is the expected
result (5c). In fact, the only difference between Hamilton’s equations
$\dot{F}\approx[F,H_{E}]$ and the results (5) (which are equivalent to
Lagrange’s equations) is the extra term $\gamma_{2}$ in Equation (61b) above.
That term does not appear in the corresponding Lagrangian Equation (5b).

Note that we can use the first class constraints to simplify the extended
Hamiltonian $H_{E}$. For example, using $C_{2}^{(fc)}=p_{3}$, we can set
$p_{3}=0$ everywhere in Equation (XII), except of course in the term
$\gamma_{2}p_{3}$. The extended Hamiltonian becomes

$H_{E}=\frac{1}{2}\Bigl{[}-p_{2}p_{4}-(q_{1}+2q_{2})(q_{1}+2q_{4})\Bigr{]}+\gamma_{1}(p_{1}-p_{2}/2-p_{4}/2)+\gamma_{2}p_{3}\
.$ (62)

This amounts to replacing the Lagrange multiplier $\gamma_{2}$ in Equation
(XII) by

$\gamma_{2}\to\gamma_{2}-p_{3}/2-p_{4}/2+q_{1}-\gamma_{1}/2\ .$ (63)

This replacement does not change the physical content of the theory, since the
Lagrange multiplier $\gamma_{2}$ is arbitrary.

##  XIII Extended Action

The equations of motion for the extended theory can be derived from the action
[11]

$S_{E}[q,p,\gamma,\sigma]=\int_{0}^{T}dt\Bigl{\\{}p_{i}\dot{q}_{i}-H_{E}-\sigma_{1}\,{\cal
C}_{1}^{(sc)}-\sigma_{2}\,{\cal C}_{2}^{(sc)}\Bigr{\\}}\ ,$ (64)

which includes the second class constraints with Lagrange multipliers
$\sigma_{a}$. Recall that the first class constraints ${\cal C}_{a}^{(fc)}$
are included in the extended Hamiltonian $H_{E}$ with multipliers
$\gamma_{a}$, so $S_{E}$ includes all four constraints.

We can use either form of the extended Hamiltonian, Equation (XII) or (62), in
the extended action. Let us use Equation (XII). Then, the equations of motion
that follow from extremizing $S_{E}$ with respect to the momenta $p_{i}$ are

$\displaystyle\dot{q}_{1}$ $\displaystyle=\gamma_{1}+\sigma_{1}/3\ ,$ (65a)
$\displaystyle\dot{q}_{2}$ $\displaystyle=-p_{4}/2-\gamma_{1}/2+\sigma_{1}/3\
,\qquad$ (65b) $\displaystyle\dot{q}_{3}$
$\displaystyle=p_{3}+p_{4}/2-q_{1}+\gamma_{1}/2+\gamma_{2}-\sigma_{1}/3+\sigma_{2}/3\
,$ (65c) $\displaystyle\dot{q}_{4}$
$\displaystyle=p_{3}/2-p_{2}/2-\gamma_{1}/2+\sigma_{1}/3\ .$ (65d) Extremizing
$S_{E}$ with respect to the coordinates yields $\displaystyle\dot{p}_{1}$
$\displaystyle=p_{3}+q_{1}+q_{2}+q_{4}-\sigma_{2}\ ,$ (65e)
$\displaystyle\dot{p}_{2}$ $\displaystyle=q_{1}+2q_{4}-\sigma_{2}\ ,$ (65f)
$\displaystyle\dot{p}_{3}$ $\displaystyle=0\ ,$ (65g)
$\displaystyle\dot{p}_{4}$ $\displaystyle=q_{1}+2q_{2}-\sigma_{2}\ ,$ (65h)
and the constraints $\displaystyle{\cal C}_{1}^{(fc)}$ $\displaystyle\equiv
p_{1}-p_{2}/2+p_{3}/2-p_{4}/2=0\ ,$ (65i) $\displaystyle{\cal C}_{2}^{(fc)}$
$\displaystyle\equiv p_{3}=0\ ,$ (65j) $\displaystyle{\cal C}_{1}^{(sc)}$
$\displaystyle\equiv(p_{1}+p_{2}-p_{3}+p_{4})/3=0\ ,$ (65k)
$\displaystyle{\cal C}_{2}^{(sc)}$ $\displaystyle\equiv
p_{3}/3+q_{1}+q_{2}+q_{4}=0\ ,$ (65l)

follow from extremizing $S_{E}$ with respect to the Lagrange multipliers
$\gamma_{a}$ and $\sigma_{a}$.

Let us check these equations of motion. The constraints imply $p_{1}=p_{3}=0$,
$p_{4}=-p_{2}$ and $q_{1}+q_{2}+q_{4}=0$. Then the equation of motion (65e)
gives $\sigma_{2}\approx 0$. The constraints also imply
$\dot{q}_{1}+\dot{q}_{2}+\dot{q}_{4}=0$. The sum of Equations (65a), (65b) and
(65d) then yields $\sigma_{1}\approx 0$. Now, if we set $\sigma_{1}$ and
$\sigma_{2}$ to zero, the equations of motion (65a–h) agree precisely with
Hamilton’s Equations (59) for the extended Hamiltonian $H_{E}$. In Section
XII, we showed that the equations generated by $H_{E}$ agree with Lagrange’s
equations apart from the extra term $\gamma_{2}$ in the equation for
$\dot{q}_{3}$. This term extends the original Lagrangian theory by making
explicit the fact that the gauge freedom allows for independent
transformations of $q_{3}$.

Finally, we note that the extended action is invariant under the
transformation defined by

$\delta F=\epsilon_{1}[F,C^{(fc)}_{1}]+\epsilon_{2}[F,C^{(fc)}_{2}]$ (66a) for
the phase space variables and $\displaystyle\delta\gamma_{1}$
$\displaystyle=\dot{\epsilon}_{1}\ ,$ (66b) $\displaystyle\delta\gamma_{2}$
$\displaystyle=\dot{\epsilon}_{2}+\epsilon_{1}\ ,$ (66c)
$\displaystyle\delta\sigma_{1}$ $\displaystyle=0\ ,$ (66d)
$\displaystyle\delta\sigma_{2}$ $\displaystyle=0\ ,$ (66e)

for the Lagrange multipliers. Here, the gauge parameters $\epsilon_{1}$ and
$\epsilon_{2}$ are functions of time. These equations express the gauge
invariance at the level of the action $S_{E}$.

##  XIV Dirac Brackets and the Partially Reduced Hamiltonian

We now return to the evolution defined by the extended Hamiltonian $H_{E}$ of
Equation (XII), and Hamilton’s Equations (59). To obtain a physically allowed
trajectory, we must choose initial data that satisfy the four constraints
${\cal C}_{a}^{(fc)}=0$ and ${\cal C}_{a}^{(sc)}=0$. Apart from restricting
the initial data, the second class constraints play no role in the formalism.
It would be convenient if we could restrict the variables from the outset such
that the second class constraints are automatically satisfied. For example, we
could use ${\cal C}_{1}^{(sc)}=0$ and ${\cal C}_{2}^{(sc)}=0$ from Equations
(65k,l) to replace $q_{1}$ with $-q_{2}-q_{4}-p_{3}/3$ and replace $p_{2}$
with $-p_{1}+p_{3}-p_{4}$.

We are not allowed to apply the second class constraints in this way. For
example, consider the Poisson brackets $[q_{1},p_{1}]=1$. If we were to
replace $q_{1}$ with $-q_{2}-q_{4}-p_{3}/3$, we would find a different answer:
$[-q_{2}-q_{4}-p_{3}/3,p_{1}]=0$. The second class constraints cannot be
imposed before Poisson brackets are computed.

Dirac devised a way to allow the second class constraints to be imposed from
the outset by modifying the Poisson brackets [10]. The result is the Dirac
brackets.

To construct Dirac brackets, we first compute the matrix of Poisson brackets
among the second class constraints:

$M_{ab}\equiv[{\cal C}^{(sc)}_{a},{\cal C}^{(sc)}_{b}]=\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}\ .$ (67)

Let

$M^{ab}=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$ (68)

denote the inverse of $M_{ab}$. Then, the Dirac brackets $[F,G]^{*}$ of two
phase space functions $F$ and $G$ are defined by

$[F,G]^{*}\equiv[F,G]-[F,{\cal C}^{(sc)}_{a}]M^{ab}[{\cal C}^{(sc)}_{b},G]\ .$
(69)

Dirac brackets, like Poisson brackets, are antisymmetric and satisfy the
Jacobi identity [10, 11].

Explicitly, the Dirac brackets among the coordinates are

$[q_{i},q_{j}]^{*}=\frac{1}{9}\begin{pmatrix}0&0&1&0\\\ 0&0&1&0\\\
-1&-1&0&-1\\\ 0&0&1&0\end{pmatrix}\ ,$ (70)

and the Dirac brackets between the $q$’s and $p$’s are

$[q_{i},p_{j}]^{*}=\frac{1}{3}\begin{pmatrix}2&-1&0&-1\\\ -1&2&0&-1\\\
1&1&3&1\\\ -1&-1&0&2\end{pmatrix}$ (71)

For our example, the Dirac brackets among the momenta all vanish:
$[p_{i},p_{j}]^{*}=0$.

There are two key properties that make Dirac brackets relevant. First, the
Dirac brackets agree weakly with Poisson brackets if one of the two functions
is first class. Since the extended Hamiltonian is first class, we have
$[F,H_{E}]^{*}\approx[F,H_{E}]$ for any $F$. It follows that we can write the
equations of motion as

$\dot{F}\approx[F,H_{E}]^{*}\ ,$ (72)

using Dirac brackets.

The second key property of the Dirac brackets is that they weakly vanish if
one of the functions is a second class constraint: $[F,{\cal
C}_{a}^{(sc)}]^{*}\approx 0$. This allows us to apply the second class
constraints before computing brackets. For example, we can use either $q_{1}$
or $-q_{2}-q_{4}-p_{3}/3$ to compute Dirac brackets with $p_{1}$:

$[q_{1},p_{1}]^{*}=[-q_{2}-q_{4}-p_{3}/3,p_{1}]^{*}=2/3\ .$ (73)

With Dirac brackets, the second class constraints can be treated as strong
equations and imposed before computing the equations of motion.

Let us use the second class constraints (65k,l) to eliminate $q_{1}$ and
$p_{2}$ and write the extended Hamiltonian in terms of the smaller set of
variables $q_{2}$, $q_{3}$, $q_{4}$, $p_{1}$, $p_{3}$ and $p_{4}$. Setting

$\displaystyle q_{1}$ $\displaystyle=-q_{2}-q_{4}-p_{3}/3\ ,$ (74a)
$\displaystyle p_{2}$ $\displaystyle=-p_{1}+p_{3}-p_{4}\ ,$ (74b)

we have

$H_{PR}=7p_{3}^{2}/9+\frac{1}{2}\left[p_{4}^{2}+p_{1}p_{4}+2p_{3}(q_{2}+q_{4})+(q_{2}-q_{4})^{2}\right]+\gamma_{1}(3p_{1}/2)+\gamma_{2}p_{3}\
.\\\ $ (75)

This is the partially reduced Hamiltonian, obtained from the extended
Hamiltonian by applying the second class constraints.

Of course, the partially reduced Hamiltonian is not unique. We could use the
second class constraints to eliminate some other pair of variables instead of
$q_{1}$ and $p_{2}$.

The equations of motion generated by the partially reduced Hamiltonian,
$\dot{F}\approx[F,H_{R}]^{*}$, are

$\displaystyle\dot{q}_{2}$ $\displaystyle\approx-p_{1}/6-p_{4}/2-\gamma_{1}/2\
,$ (76a) $\displaystyle\dot{q}_{3}$ $\displaystyle\approx
4p_{3}/3+p_{1}/6+p_{4}/2+q_{2}+q_{4}+\gamma_{1}/2+\gamma_{2}\ ,$ (76b)
$\displaystyle\dot{q}_{4}$ $\displaystyle\approx p_{1}/3+p_{4}/2-\gamma_{1}/2\
,$ (76c) $\displaystyle\dot{p}_{1}$ $\displaystyle\approx 2p_{3}/3\ ,$ (76d)
$\displaystyle\dot{p}_{3}$ $\displaystyle\approx 0\ ,$ (76e)
$\displaystyle\dot{p}_{4}$ $\displaystyle\approx-p_{3}/3+q_{2}-q_{4}\ .$ (76f)

We can also use $H_{PR}$ and the Dirac brackets to compute $\dot{q}_{1}$ and
$\dot{p}_{2}$. The results are equivalent to those obtained by differentiating
the right-hand sides of Equations (74) and using the equations of motion (76).

Let us check the equations of motion. With the second class constraints
applied, the first class constraints imply $p_{1}=p_{3}=0$ and $p_{4}=-p_{2}$.
Thus, Equations (76d) and (76e) are vacuous and the remaining equations become

$\displaystyle\dot{q}_{2}$ $\displaystyle\approx p_{2}/2-\gamma_{1}/2\ ,$
(77a) $\displaystyle\dot{q}_{3}$ $\displaystyle\approx-
p_{2}/2+q_{2}+q_{4}+\gamma_{1}/2+\gamma_{2}\ ,$ (77b)
$\displaystyle\dot{q}_{4}$ $\displaystyle\approx-p_{2}/2-\gamma_{1}/2\ ,$
(77c) $\displaystyle\dot{p}_{2}$ $\displaystyle\approx-q_{2}+q_{4}\ ,$ (77d)

Compare these to the independent Equations (60) that follow from the extended
Hamiltonian. Equations (60b,c,d) agree with Equations (77a,b,d) once we use
$q_{1}=-q_{2}-q_{4}$. The final Equation (60a) is obtained by differentiating
$q_{1}=-q_{2}-q_{3}$ in time and using Equations (77a) and (77b).

##  XV Partially Reduced Action

The partially reduced equations of motion (76) can be obtained from the
extended action $S_{E}$ by eliminating the superfluous variables. Note that
the equations of motion obtained by varying $S_{E}$ with respect to $p_{2}$,
$q_{1}$, $\sigma_{1}$ and $\sigma_{2}$ are Equations (65b,e,k,l),
respectively. We can eliminate these variables by solving these equations and
substituting the results into the action.666Any action can be reduced by using
the equations of motion obtained by varying with respect to a subset of
variables, solving those equations for the same subset of variables, then
substituting the results into the action. In general, it is not permissible to
reduce an action by using the equations obtained by varying with respect to
one subset of variables but solving those equations for a different subset of
variables. The results are:

$\displaystyle q_{1}$ $\displaystyle=-p_{3}/3-q_{2}-q_{4}\ ,$ (78a)
$\displaystyle p_{2}$ $\displaystyle=-p_{1}+p_{3}-p_{4}\ ,$ (78b)
$\displaystyle\sigma_{1}$ $\displaystyle=3\dot{q}_{2}+3p_{4}/2+3\gamma_{1}/2\
,$ (78c) $\displaystyle\sigma_{2}$ $\displaystyle=-\dot{p}_{1}+2p_{3}/3\ .$
(78d)

Inserting these into the extended action (64), we find

$\displaystyle
S_{PR}[q_{2},q_{3},q_{4},p_{1},p_{3},p_{4},\gamma_{1},\gamma_{2}]=\int_{0}^{T}dt\bigl{\\{}$
$\displaystyle-p_{1}\dot{p}_{3}/3+(p_{3}-p_{4}-2p_{1})\dot{q}_{2}$
$\displaystyle+p_{3}\dot{q}_{3}+(p_{4}-p_{1})\dot{q}_{4}-H_{PR}\bigr{\\}}\ .$
(79)

This is the partially reduced action.

The equations of motion obtained from varying $S_{PR}$ with respect to the
phase space variables are

$\displaystyle 2\dot{p}_{1}-\dot{p}_{3}+\dot{p}_{4}-p_{3}-q_{2}+q_{4}$
$\displaystyle=0\ ,$ (80a) $\displaystyle-\dot{p}_{3}$ $\displaystyle=0\ ,$
(80b) $\displaystyle\dot{p}_{1}-\dot{p}_{4}-p_{3}+q_{2}-q_{4}$
$\displaystyle=0\ ,$ (80c)
$\displaystyle-2\dot{q}_{2}-\dot{q}_{4}-\dot{p}_{3}/3-p_{4}/2-3\gamma_{1}/2$
$\displaystyle=0\ ,$ (80d)
$\displaystyle\dot{q}_{2}+\dot{q}_{3}+\dot{p}_{1}/3-14p_{3}/9-q_{2}-q_{4}-\gamma_{2}$
$\displaystyle=0\ ,\qquad$ (80e)
$\displaystyle\dot{q}_{2}+\dot{q}_{4}-p_{1}/2-p_{4}$ $\displaystyle=0\ ,$
(80f)

and the equations obtained by varying with respect to the Lagrange multipliers
$\gamma_{1}$ and $\gamma_{2}$ are

$\displaystyle-3p_{1}/2$ $\displaystyle=0\ ,$ (81a) $\displaystyle-p_{3}$
$\displaystyle=0\ .$ (81b)

These are, of course, the first class constraints, reduced by using the second
class constraints to eliminate $q_{1}$ and $p_{2}$.

We can now solve Equation (80) for the time derivatives of $q_{2}$, $q_{3}$,
$q_{4}$, $p_{1}$, $p_{3}$ and $p_{4}$. The result coincides with the equations
of motion (76) obtained from the partially reduced Hamiltonian $H_{PR}$ and
the Dirac brackets.

The partially reduced action $S_{PR}$ is invariant under the transformation
defined by

$\delta F=\epsilon_{1}[F,C^{(fc)}_{1}]^{*}+\epsilon_{2}[F,C^{(fc)}_{2}]^{*}$
(82a) for the phase space variables and $\displaystyle\delta\gamma_{1}$
$\displaystyle=\dot{\epsilon}_{1}\ ,$ (82b) $\displaystyle\delta\gamma_{2}$
$\displaystyle=\dot{\epsilon}_{2}+\epsilon_{1}\ ,$ (82c)

for the Lagrange multipliers. These equations express the gauge invariance of
the theory at the level of the action principle with the second class
constraints eliminated.

##  XVI Partially Reduced Lagrangian

It is not too difficult to find a change of variables that will bring $S_{PR}$
into “canonical form”. For example, let

$\displaystyle q_{2}$ $\displaystyle=Q_{1}-P_{2}/9\ ,$ (83a) $\displaystyle
q_{3}$ $\displaystyle=Q_{2}+P_{2}/9\ ,$ (83b) $\displaystyle q_{4}$
$\displaystyle=Q_{3}-P_{2}/9\ ,$ (83c) $\displaystyle p_{1}$
$\displaystyle=(-P_{1}+P_{2}-P_{3})/3\ ,$ (83d) $\displaystyle p_{3}$
$\displaystyle=P_{2}\ ,$ (83e) $\displaystyle p_{4}$
$\displaystyle=(-P_{1}+P_{2}+2P_{3})/3\ ,$ (83f)

define a new set of variables $Q_{\alpha}$, $P_{\alpha}$ for the secondary
constraint surface. (The index $\alpha$ ranges over $1$, $2$ and $3$.) The
partially reduced action becomes

$S_{PR}[Q,P,\gamma]=\int_{0}^{T}dt\left\\{P_{\alpha}\dot{Q}_{\alpha}-H_{PR}\right\\}$
(84)

with

$\displaystyle H_{PR}=$
$\displaystyle\frac{1}{18}\left[2P_{1}^{2}+12P_{2}^{2}+2P_{3}^{2}-4P_{1}P_{2}+5(P_{2}-P_{1})P_{3}+18P_{2}(Q_{1}+Q_{3})+9(Q_{1}-Q_{3})^{2}\right]$
$\displaystyle+\gamma_{1}(-P_{1}+P_{2}-P_{3})/2+\gamma_{2}P_{2}\ .$ (85)

The equations of motion $\delta S_{PR}=0$ include
$\dot{Q}_{\alpha}=[Q_{\alpha},H_{PR}]$ and
$\dot{P}_{\alpha}=[P_{\alpha},H_{PR}]$, where $[\,\cdot\,,\,\cdot\,]$ are the
usual Poisson brackets. The partially reduced action (84) is invariant under
the transformation

$\displaystyle\delta F$
$\displaystyle=\epsilon_{1}[F,C^{(fc)}_{1}]+\epsilon_{2}[F,C^{(fc)}_{2}]\ ,$
(86a) $\displaystyle\delta\gamma_{1}$ $\displaystyle=\dot{\epsilon}_{1}\ ,$
(86b) $\displaystyle\delta\gamma_{2}$
$\displaystyle=\dot{\epsilon}_{2}+\epsilon_{1}\ ,$ (86c)

where $C_{1}^{(fc)}=(-P_{1}+P_{2}-P_{3})/2$ and $C_{2}^{(fc)}=P_{2}$ are the
first class constraints.

We can use the equations of motion $\delta S_{PR}/\delta P_{\alpha}=0$ to
eliminate the momenta $P_{\alpha}$ from the partially reduced action (84).
These equations are

$\displaystyle\dot{Q}_{1}$
$\displaystyle=(4P_{1}-4P_{2}-5P_{3})/18-\gamma_{1}/2\ ,$ (87a)
$\displaystyle\dot{Q}_{2}$
$\displaystyle=-(4P_{1}-24P_{2}-5P_{3})/18+Q_{1}+Q_{3}+\gamma_{1}/2+\gamma_{2}\
,$ (87b) $\displaystyle\dot{Q}_{3}$
$\displaystyle=-(5P_{1}-5P_{2}-4P_{3})/18-\gamma_{1}/2\ ,$ (87c)

with solutions

$\displaystyle P_{1}$
$\displaystyle=(-71\dot{Q}_{1}+9\dot{Q}_{2}-100\dot{Q}_{3})/10-9(Q_{1}+Q_{3})/10-9\gamma_{1}-9\gamma_{2}/10\
,$ (88a) $\displaystyle P_{2}$
$\displaystyle=9(\dot{Q}_{1}+\dot{Q}_{2}-Q_{1}-Q_{3}-\gamma_{2})/10\ ,$ (88b)
$\displaystyle P_{3}$ $\displaystyle=-10\dot{Q}_{1}-8\dot{Q}_{3}-9\gamma_{1}\
.$ (88c)

Inserting these results into the partially reduced action, we find

$\displaystyle S_{PR}[Q,\gamma]=\int_{0}^{T}dt\biggl{\\{}$
$\displaystyle(-71\dot{Q}_{1}^{2}+9\dot{Q}_{2}^{2}-80\dot{Q}_{3}^{2})/20+(9\dot{Q}_{2}-100\dot{Q}_{3})\dot{Q}_{1}/10$
$\displaystyle-9(Q_{1}+Q_{3})(\dot{Q}_{1}+\dot{Q}_{2})/10-(Q_{1}^{2}+Q_{3}^{2}-38Q_{1}Q_{3})/20$
$\displaystyle-9(\dot{Q}_{1}+\dot{Q}_{3})\gamma_{1}-9(\dot{Q}_{1}+\dot{Q}_{2}-Q_{1}-Q_{3})\gamma_{2}/10$
$\displaystyle-9(10\gamma_{1}^{2}-\gamma_{2}^{2})/20\biggr{\\}}\ .$ (89)

The partially reduced Lagrangian is the integrand of this action.

We can go one step further and eliminate the Lagrange multipliers using the
equations of motion $\delta S/\delta\gamma_{1}=0$ and $\delta
S/\delta\gamma_{2}=0$. These equations have solutions

$\displaystyle\gamma_{1}$ $\displaystyle=-\dot{Q}_{1}-\dot{Q}_{3}\ ,$ (90a)
$\displaystyle\gamma_{2}$ $\displaystyle=\dot{Q}_{1}+\dot{Q}_{2}-Q_{1}-Q_{3}\
.$ (90b)

Inserting these results back into the action yields

$S_{PR}[Q]=\int_{0}^{T}dt\left\\{\frac{1}{2}(\dot{Q}_{1}-\dot{Q}_{3})^{2}-\frac{1}{2}(Q_{1}-Q_{3})^{2}\right\\}\
.$ (91)

This is the action for a harmonic oscillator in the variable $Q_{1}-Q_{3}$.
Note that the coordinate transformation (83) implies
$(Q_{1}-Q_{3})=(q_{2}-q_{4})$, so once again, we find that the coordinate
combination $q_{2}-q_{4}$ describes a simple harmonic oscillator. In addition,
we observe that the action (91) leaves the variable $Q_{2}$ completely
unspecified. This expresses the gauge freedom generated by the first class
constraint $C_{2}^{(fc)}=P_{2}=p_{3}$.

##  XVII Gauge Conditions and the Fully Reduced Hamiltonian

Let us return to the theory described by the extended Hamiltonian, prior to
the elimination of the second class constraints.

For our example problem, phase space is eight-dimensional. The physical
trajectories fill the constraint “surface”, which is the four-dimensional
subspace where all first and second class constraints hold. Each point in the
constraint surface can be mapped into a physically equivalent state by the
gauge generators, namely, the first class constraints ${\cal C}_{a}^{(fc)}$.
Since there are two independent gauge generators, each physical state of the
system corresponds to a two-dimensional subspace of the constraint surface.
The constraint surface is foliated by these two-dimensional slices, referred
to as gauge “orbits”.

We can select a single phase space point on each gauge orbit to represent the
physical state. We do this by applying gauge conditions. In particular, we
will consider a canonical gauge777Canonical gauges restrict the phase space
variables. Noncanonical gauges [11] involve the Lagrange multipliers. which
takes the form ${\cal G}_{a}(q,p)\approx 0$ with $a=1,2$. A good canonical
gauge condition must not be gauge invariant, otherwise it would allow more
than one point on the gauge orbit to represent the physical state of the
system. To be precise, the matrix of Poisson brackets of gauge conditions and
gauge generators, $[{\cal G}_{a},C_{b}^{(fc)}]$, must be nonsingular [11].

As an example, let us choose

$\displaystyle{\cal G}_{1}$ $\displaystyle=q_{1}-q_{2}\ ,$ (92a)
$\displaystyle{\cal G}_{2}$ $\displaystyle=q_{3}+p_{4}$ (92b)

as our gauge conditions. This is a good gauge:

${\rm det}[{\cal G}_{a},C_{b}^{(fc)}]=\begin{vmatrix}3/2&0\\\
1/2&1\end{vmatrix}=3/2\ .$ (93)

The matrix $[{\cal G}_{a},C_{b}^{(fc)}]$ is nonsingular, as required.

The gauge conditions ${\cal G}_{a}=0$, like the first and second class
constraints, restrict the phase space variables. The full set of restrictions

${\cal C}^{(all)}_{A}=\\{{\cal G}_{1},{\cal G}_{2},{\cal C}^{(fc)}_{1},{\cal
C}^{(fc)}_{2},{\cal C}^{(sc)}_{1},{\cal C}^{(sc)}_{2}\\}$ (94)

reduces the available phase space from eight dimensions to two dimensions.
(Here, the index $A$ ranges from $1$ to $6$.) Taken as a whole, the six
conditions ${\cal C}^{(all)}_{A}$ are second class. We see this by computing
the Poisson brackets

$M_{AB}\equiv[{\cal C}^{(all)}_{A},{\cal
C}^{(all)}_{B}]=\frac{1}{6}\begin{pmatrix}0&0&9&0&0&0\\\ 0&0&3&6&-2&-4\\\
-9&-3&0&0&0&0\\\ 0&-6&0&0&0&0\\\ 0&2&0&0&0&-6\\\ 0&4&0&0&6&0\end{pmatrix}\ .$
(95)

This matrix has a nonzero determinant, $\det(M)=9/4$, which is the condition
for the set of constraints and gauge conditions to be second class.

We can eliminate the constraints and gauge conditions by constructing Dirac
brackets. The inverse of $M_{AB}$ is

$M^{AB}=\frac{1}{3}\begin{pmatrix}0&0&-2&1&0&0\\\ 0&0&0&-3&0&0\\\
2&0&0&0&0&0\\\ -1&3&0&0&-2&1\\\ 0&0&0&2&0&3\\\ 0&0&0&-1&-3&0\end{pmatrix}\ ,$
(96)

and the Dirac brackets are defined by

$[F,G]^{*}=[F,G]-[F,{\cal C}^{(all)}_{A}]M^{AB}[{\cal C}^{(all)}_{B},G]\ .$
(97)

The Dirac brackets among the phase space variables are

$\displaystyle\left[q_{1},q_{3}\right]^{*}=[q_{1},p_{2}]^{*}=-[q_{1},p_{4}]^{*}$
$\displaystyle=1/3\ ,\quad$ (98a)
$\displaystyle\left[q_{2},q_{3}\right]^{*}=[q_{2},p_{2}]^{*}=-[q_{2},p_{4}]^{*}$
$\displaystyle=1/3\ ,$ (98b)
$\displaystyle\left[q_{3},q_{4}\right]^{*}=-[q_{4},p_{2}]^{*}=[q_{4},p_{4}]^{*}$
$\displaystyle=2/3\ ,$ (98c)

with all other brackets vanishing.

The constraints can be solved in various ways and the results can be used
freely, either before or after computing Dirac brackets. For example, the
constraints imply

$\displaystyle q_{1}$ $\displaystyle=-q_{4}/2\ ,$ (99a) $\displaystyle q_{2}$
$\displaystyle=-q_{4}/2\ ,$ (99b) $\displaystyle q_{3}$ $\displaystyle=p_{2}\
,$ (99c) $\displaystyle p_{1}$ $\displaystyle=0\ ,$ (99d) $\displaystyle
p_{3}$ $\displaystyle=0\ ,$ (99e) $\displaystyle p_{4}$ $\displaystyle=-p_{2}\
.$ (99f)

We can use these to eliminate the variables $q_{1}$, $q_{2}$, $q_{3}$,
$p_{1}$, $p_{3}$ and $p_{4}$. Then the extended Hamiltonian $H_{E}$ becomes
the fully reduced Hamiltonian

$H_{FR}=\frac{1}{2}\left[p_{2}^{2}+\frac{9}{4}q_{4}^{2}\right]\ ,$ (100)

which depends only on $q_{4}$ and $p_{2}$.

The Dirac brackets of the variables that remain are $[q_{4},p_{2}]^{*}=-2/3$.
Thus, the equations of motion become

$\displaystyle\dot{q}_{4}=[q_{4},H_{FR}]^{*}$ $\displaystyle=-2p_{2}/3\ ,$
(101a) $\displaystyle\dot{p}_{2}=[p_{4},H_{FR}]^{*}$ $\displaystyle=3q_{4}/2\
.$ (101b)

These are the equations for a simple harmonic oscillator with solution

$\displaystyle q_{4}(t)$ $\displaystyle=\alpha\sin t+\beta\cos t\ ,$ (102a)
$\displaystyle p_{2}(t)$ $\displaystyle=-(3\alpha/2)\cos t+(3\beta/2)\sin t\
,\quad$ (102b)

where $\alpha$ and $\beta$ are arbitrary constants.

With the gauge fixed, the dynamics take place on the fully reduced phase
space, the two-dimensional surface defined by the constraints and gauge
conditions ${\cal C}^{(all)}_{A}=0$. There are many different choices of
coordinates for this surface. Instead of solving the constraints for $q_{4}$
and $p_{2}$, we could solve them for $q_{2}$ and $q_{3}$. In that case, the
fully reduced Hamiltonian is

$H_{FR}=\frac{9}{2}q_{2}^{2}+\frac{1}{2}q_{3}^{2}$ (103)

and Hamilton’s equations are

$\displaystyle\dot{q}_{2}$ $\displaystyle=[q_{2},H_{FR}]^{*}=q_{3}/3\ ,$
(104a) $\displaystyle\dot{q}_{3}$ $\displaystyle=[q_{3},H_{FR}]^{*}=-3q_{2}\ $
(104b)

Again, this describes the simple harmonic oscillator.

We can choose other coordinates on the fully reduced phase space. For example,
let $q_{4}=Q+q_{2}$ and $p_{2}=-P$, then use the constraints to eliminate
$q_{1}$, $q_{2}$, $q_{3}$, $p_{1}$, $p_{3}$ and $p_{4}$. The fully reduced
Hamiltonian becomes

$H_{FR}=\frac{1}{2}(Q^{2}+P^{2})\ .$ (105)

The nonzero Dirac brackets are $[Q,P]^{*}=1$, and the equations of motion are
simply $\dot{Q}=P$ and $\dot{P}=-Q$.

In each case, the fully reduced theory exhibits the single physical degree of
freedom that we expect.

##  XVIII Fully Reduced Action

The fully reduced equations of motion can be derived from the action that
includes all of the constraints and gauge conditions. For lack of a better
name, let us denote this action with the subscript “all”:

$S_{all}=\int_{0}^{T}dt\left\\{p_{i}\dot{q}_{i}-H_{fc}-\gamma_{a}{\cal
C}_{a}^{(fc)}-\sigma_{a}{\cal C}_{a}^{(sc)}-\rho_{a}{\cal G}_{a}\right\\}\ .$
(106)

Now extremize $S_{all}$ with respect to variations in $p_{1}$, $p_{3}$,
$p_{4}$, $q_{1}$, $q_{2}$, $q_{3}$ and $q_{4}$:

$\displaystyle\dot{q}_{1}$ $\displaystyle=\gamma_{1}+\sigma_{1}/3\ ,$ (107a)
$\displaystyle\dot{q}_{3}$
$\displaystyle=-q_{1}+p_{3}+p_{4}/2+\gamma_{1}/2+\gamma_{2}-\sigma_{1}/3+\sigma_{2}/3\
,$ (107b) $\displaystyle\dot{q}_{4}$
$\displaystyle=-p_{2}/2+p_{3}/2-\gamma_{1}/2+\sigma_{1}/3+\rho_{2}\ ,$ (107c)
$\displaystyle\dot{p}_{1}$
$\displaystyle=q_{1}+q_{2}+q_{4}+p_{3}-\sigma_{2}-\rho_{1}\ ,$ (107d)
$\displaystyle\dot{p}_{2}$ $\displaystyle=q_{1}+2q_{4}+\sigma_{2}-\rho_{1}\ ,$
(107e) $\displaystyle\dot{p}_{3}$ $\displaystyle=-\rho_{2}\ .$ (107f)

In addition, vary $S_{all}$ with respect to the Lagrange multipliers to obtain
the constraints ${\cal C}^{(all)}_{A}=0$. The solution of the full set of
equations, (107) and ${\cal C}^{(all)}_{A}=0$, is given by Equations (99)
along with

$\displaystyle\gamma_{1}$
$\displaystyle=-p_{2}/3-2\dot{p}_{3}/2+2\dot{q}_{1}/3-2\dot{q}_{4}/3\ ,\qquad$
(108a) $\displaystyle\gamma_{2}$
$\displaystyle=p_{2}-3q_{4}/4+\dot{p}_{1}/6+\dot{p}_{2}/6+\dot{p}_{3}+\dot{q}_{3}+\dot{q}_{4}\
,$ (108b) $\displaystyle\sigma_{1}$
$\displaystyle=p_{2}+2\dot{p}_{3}+\dot{q}_{1}+2\dot{q}_{4}\ ,$ (108c)
$\displaystyle\sigma_{2}$ $\displaystyle=3q_{4}/4-\dot{p}_{1}/2-\dot{p}_{2}/2\
,$ (108d) $\displaystyle\rho_{1}$
$\displaystyle=-3q_{4}/4-\dot{p}_{1}/2+\dot{p}_{2}/2\ ,$ (108e)
$\displaystyle\rho_{2}$ $\displaystyle=-\dot{p}_{3}\ .$ (108f)

We now insert these results into $S_{all}$ to obtain the fully reduced action

$S_{FR}[q_{4},p_{2}]=\int_{0}^{T}dt\left\\{-\frac{3}{2}p_{2}\dot{q}_{4}-\frac{1}{2}\left[p_{2}^{2}+\frac{9}{4}q_{4}^{2}\right]\right\\}\
,$ (109)

which is a functional of $q_{4}$ and $p_{2}$. The equations of motion $\delta
S_{FR}=0$ are

$\displaystyle 0$ $\displaystyle=\frac{\delta S_{FR}}{\delta
q_{4}}=\frac{3}{2}\dot{p}_{2}-\frac{9}{4}q_{4}\ ,$ (110a) $\displaystyle 0$
$\displaystyle=\frac{\delta S_{FR}}{\delta
p_{2}}=-\frac{3}{2}\dot{q}_{4}-p_{2}\ .$ (110b)

These are equivalent to Hamilton’s Equations (101) for the fully reduced
Hamiltonian.

We can place $S_{FR}$ into “canonical form” by defining new variables
$P=-p_{2}$ and $Q=3q_{4}/2$. Then

$S_{FR}[Q,P]=\int_{0}^{T}dt\left\\{P\dot{Q}-\frac{1}{2}[P^{2}+Q^{2}]\right\\}\
,$ (111)

which is the familiar action for the harmonic oscillator.

##  XIX Summary and Discussion

Here is the Dirac–Bergmann algorithm:

  * •

Compute the conjugate momenta $p_{i}=\partial L/\partial\dot{q}_{i}$ and
define the canonical Hamiltonian $H_{C}$ as $p_{i}\dot{q}^{i}-L(q,\dot{q})$,
written in terms of $p$’s and $q$’s.

  * •

Identify the primary constraints. The primary Hamiltonian $H_{P}$ is obtained
from $H_{C}$ by adding the primary constraints with Lagrange multipliers.

  * •

Apply Dirac’s consistency conditions to identify higher-order constraints and
restrictions on the Lagrange multipliers.

  * •

The total Hamiltonian $H_{T}$ is found from $H_{P}$ by incorporating the
restrictions on the Lagrange multipliers.

  * •

Separate the primary, secondary, and higher-order constraints into first and
second class.

  * •

The first class Hamiltonian $H_{fc}$ is the part of $H_{T}$ with the primary
first class constraints removed.

  * •

The extended Hamiltonian $H_{E}$ is obtained from the first class Hamiltonian
$H_{fc}$ by adding all of the first class constraints with Lagrange
multipliers.

  * •

The partially reduced Hamiltonian $H_{PR}$ is found from $H_{E}$ by using
Dirac brackets to eliminate the second class constraints.

  * •

Gauge freedom is removed by assigning gauge conditions. The fully reduced
Hamiltonian $H_{FR}$ is obtained from $H_{E}$ by using Dirac brackets to
impose all constraints and gauge conditions.

The theory defined by the singular Lagrangian (1) provides a relatively
complete example of each step in the algorithm.

One reason the Dirac–Bergmann algorithm can be confusing is that typical
examples are chosen for simplicity, allowing some of the logical steps to be
skipped. This causes the distinction between Hamiltonians to become blurred.
For example, if there are no restrictions on the Lagrange multipliers, then
the primary Hamiltonian $H_{P}$ and the total Hamiltonian $H_{T}$ coincide.
Likewise, if there are no secondary (or higher-order) first class constraints,
then the total Hamiltonian $H_{T}$ and the extended Hamiltonian $H_{E}$
coincide. Moreover, for theories with no second class constraints and no gauge
conditions imposed, Dirac brackets and the reduction process are not needed.

Another confusing aspect of the Dirac–Bergmann algorithm is that for many
important theories, the Lagrangian is given to us in a form that contains
Lagrange multipliers. For example, consider the Einstein–Hilbert action of
general relativity. The Lagrangian density is the spacetime curvature scalar.
A 3+1 splitting of the spacetime metric [31, 32] allows us to write the
Lagrangian density as

$L=R+K_{ij}(g^{ik}g^{j\ell}-g^{ij}g^{k\ell})K_{k\ell}\ ,$ (112)

apart from a total derivative term that integrates to the boundary. Here, $R$
and $g_{ij}$ are the spatial scalar curvature and spatial metric. In addition,
$K_{ij}$ is the extrinsic curvature of space, built from the lapse function
$N$, shift vector $N^{i}$ and spatial derivatives of $g_{ij}$. The Lagrangian
density depends on the Lagrange multipliers $N$ and $N^{i}$ as well as the
configuration space coordinates $g_{ij}$.

Because the Einstein–Hilbert Lagrangian (112) depends on Lagrange multipliers,
it is not analogous to the singular Lagrangian (1) of our example problem.
Rather, it is analogous to the partially reduced Lagrangian that appears in
the integrand of the action $S_{PR}[Q,\gamma]$ of Equation (XVI). Recall that
the partially reduced Hamiltonian $H_{PR}$ contains only first class
constraints and no restrictions on the Lagrange multipliers. Likewise, the
Hamiltonian for general relativity [32] is constructed from first class
constraints (the Hamiltonian and momentum constraints), and the Lagrange
multipliers (the lapse function and shift vector) are unrestricted.

We can attempt to eliminate the lapse and shift from the Einstein–Hilbert
action, just as we eliminated the $\gamma$’s from $S_{PR}[Q,\gamma]$ and
obtained the result $S_{PR}[Q]$ in Equation (91). It is straightforward to
eliminate the lapse $N$—the result is the Baierlein–Sharp–Wheeler action [33].
However, the shift vector cannot be eliminated algebraically because the
equations of motion obtained by varying the action with respect to the lapse
and shift depend on spatial derivatives of $N^{i}$.

So, for general relativity, we do not have a singular Lagrangian analogous to
Equation (1), and we cannot expect to apply the Dirac–Bergmann algorithm from
beginning to end as laid out by Dirac [10]. Nevertheless, the general
Dirac–Bergmann algorithm serves as the foundation for our understanding and
interpretation of the Hamiltonian form of the theory.

###### Acknowledgements.

I would like to thank Claudio Bunster and Marc Henneaux for sparking my
interest in this subject many years ago. I also thank Ashiqul Islam Dip for
helpful discussions.

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