symmetries and conservation laws obtained by lie group analysis for ...

DIPLOMA THESIS
SYMMETRIES AND
CONSERVATION LAWS OBTAINED
BY LIE GROUP ANALYSIS FOR
CERTAIN PHYSICAL SYSTEMS
MARCUS ERIKSSON
Uppsala School of Engineering
and
Department of Astronomy and Space Physics, Uppsala University, Sweden
APRIL 22, 2008

Abstract
The formal model of physical systems is typically made in terms of
differential equations. Conservation laws can be found for any system of
linear or non-linear differential equations, based on the concept of adjoint
equations. Provided that the number of equations is equal to the number
of variables, the adjoint equations for a system is used for construction of
Lagrangians. Lie group analysis provides us with a way of finding all the
symmetries of the original mathematical model. The symmetries are inherited
by the adjoint equations. By considering the total system of differential
equations together with its adjoint equations, we construct the Lagrangian
and once the symmetries are found, Noether’s theorem is used for determining
conservation laws. Consequently, for any symmetry found by Lie group
analysis a conservation law can be found.

The mass of mathematical truth is obvious and imposing; its practical
applications, the bridges and steamengines and dynamos, obtrude
themselves on the dullest imagination. The public does not need to
be convinced that there is something in mathematics.
C. G. Hardy

CONTENTS
Contents
Acknowledments
v
vii
I Group Theory 1
1 Introduction 3
2 Notation and the total differential operator 5
2.1 Functions of one independent variable x and one dependent variable
y 5
2.2 Functions of several variables 9
2.3 Important equations 11
3 Basic Lie group theory 13
3.1 Transformation groups 13
3.2 One-parameter group 15
3.2.1 Canonical parameter 18
3.3 The infinitesimal transformation and Lie’s equations 18
3.4 Invariant functions and equations 22
3.4.1 Invariant function 22
3.4.2 Invariant equations 23
3.5 Multi-parameter groups and Lie algebras 25
3.5.1 Multi–parameter groups 25
3.5.2 Short outline of Lie algebra 25
v

CONTENTS
4 Differential Algebra 27
4.1 The space A 27
4.2 Extended point transformation group 28
4.2.1 Extended pointtransformation in the plane 28
4.2.2 Point transformation involving many variables 32
4.2.3 Extended generators for differential equations with two
independent and one dependent variable 34
5 Operators 37
5.1 Euler-Lagrange operator 37
5.2 Lie-Bäcklund operators 39
5.3 Operators N i associated with Lie-Bäcklund operators and the
fundamental identity 40
II Symmetries and Conservation Laws 43
6 Hamilton’s Variational Principle 45
6.1 Hamilton´s variational principle in the plane 45
6.1.1 Euler-Lagrange equation with several variables 50
7 Noether’s Theorem and Conservation Laws 53
7.1 Noether´s Theorem 53
7.2 Many symmetries 60
8 Adjoint equations 63
8.1 Linear adjoint equations 63
8.2 Adjoint equations of arbitrary differential equations 66
8.3 Constructing Lagrangians by the use of adjoint equation 68
9 Symmetries and The Main Conservation Theorem 71
9.1 Determining equations 74
9.2 The main conservation theorem 79
9.3 Summary 85
Bibliography 87
vi

ACKNOWLEDMENTS
First, I would like to thank my supervisor Professor Bo Thidè for giving me the
opportunity to discover the bordercountry between mathematics and physics. I
would also like to thank him and everyone else how has supported me and given
me ideas as a result of many profound discussions, especially Dr Jan Bergman,
Siavoush Mohammadi, Jonny Jansson, Lars Daldorff and professor Nail Ibragimov.
You have all been of great importance for my work.
vii

Part I
Group Theory
1

1
INTRODUCTION
To confirm physical theories we need to make observations and measurements of
the physical system in interest. To be able to make a measurement of a physical
quantity, we need to know if that quantity is conserved in time and space. For
example;
We have a ruler of lenght l at one side of a room. During a time ∆t we move
it ( by throwing it ) to to the other side of the room. Now, if the ruler, after the
transport, would have a length diffrent from l we cannot measure its length since
it is changing in time and space, i.e the physical quantity l is not conserved and
therefore not very useful as a measurable. Of course, in reality l is conserved, and
this was just an example to illustrate the meaning of a conserved quantity.
In radio comunications we transmit information encoded in radiowaves. This
transmission of information is only useful if the information we transmit will be
the same at the reciever point. Thus, the information in radiowaves is a conserved
quantity.
The motion of the ruler and the transmission of radiowaves are examples of physical
systems. In mathematics we describe physical systems in terms of differential
equations. The motion of the ruler is described by the equation of motion for a
projectile [1] and the behavior of radiowaves, i.e electromagnetic waves, is described
by Maxwell’s equations [2].
In 2006 the mathematican Prof. Nail Ibragimov proved a main conservation theorem
for any well-posed system of differential equations [3]. The purpose of this
3

CHAPTER 1. INTRODUCTION
work is to present the mathematical metod and the underlying mathematics to find
conservation laws for physical systems described by differential equations. The
mathematical tools we need orginate from Lie group theory, the variational principle,
and the concept of adjoint equations. These three areas will be presented
and tied togheter to formulate the main conservation theorem.
This work is divided into Part I and Part II. Part I will focus on the notations
that will be used ( chapter 2 ) and serve as an introduction to Lie group theory
(chapter 3 and 4). In chapter 5 some useful definitions, lemmas and theorems will
be presented. In Part II, chapter 6-9, we will develop the method to find conservation
laws based on the Lie group theory from Part I. In chapter 6 we present
the variational principle, which togheter with Part I leed us to a formulation of
Noethers theorem in chapter 7. In chapter 8 we present the method for construction
of Lagrangians, based on the concept of adjoint equations. In chapter 9 we
show how symetries can be found by Lie group theory and how this togheter with
the concept of adjoint equations results in a main conservation theorem.
4

2
NOTATION AND THE TOTAL
DIFFERENTIAL OPERATOR
In this chapter we introduce some of the basic mathematical notations and the
total differentiation operator that will be used throughout the thesis. When dealing
with differential equations these notations comprise a convinient language
adopted from differential algebra. It will be explaind how the notations are defined
and how they simplify large equations.
2.1 Functions of one independent variable x and one dependent
variable y
In order to obtain information about a system we analyse the differential equations
which describe it. In most cases it is very complicated or impossible to find
a solution. Howerever, there are cases in which we know the general solutions to
a differential equation. This solutions is a family of curves depending on a number
of parameters (unknown constants), see [4] and [5]. We will start to find a
differential equation for a family of curves, and see how this procedure motivates
the introduction of a total differentiation operator [4].
Consider a family of curves in the xy-plane,
y = f (x,C 1 ,...,C n ), (2.1)
5

CHAPTER 2. NOTATION AND THE TOTAL DIFFERENTIAL OPERATOR
where the C k are parameters and k = 1,...,n. Equation (2.1) can be formulated
implicitly by
φ(x,y,C 1 ,...,C n ) = 0. (2.2)
Differentiating equation (2.2) with respect to x yields,
∂φ
∂x + ∂φ ∂y
∂y ∂x ≡ ∂φ
∂x
∂φ
+ y′ = 0. (2.3)
∂y
A differentiation of equation (2.3) with respect to x yields
∂ 2 φ
∂x 2 + ∂2 φ
∂y∂x
∂y
∂x + ∂2 φ ∂y
∂x∂y ∂x + ∂φ ∂ 2 y
∂y ∂x 2 + ∂2 φ ∂y ∂y
∂y 2 ∂x ∂x
≡ ∂2 φ
∂x 2 + 2 ∂2 φ
∂y∂x y′ + ∂2 φ
∂y 2 (y′ ) 2 + ∂φ
∂y y′′ = 0. (2.4)
Proceeding the differentiation in this way, we will eliminate one of the parameters
C k k = 1,...,n from φ(x,y,C 1 ,...,C n ). After n differentiations all C k are
eliminated and we arrive at an nth-order ordinary differential equation of the form
F(x,y,y ′ ,y ′′ ,...,y (n) ) = 0. (2.5)
We say, according to [4], that the function (2.1), y = f (x,C 1 ,...,C n ) provides a
solution to the differential equation (2.5). The equation F(x,y,y ′ ,y ′′ ,...,y (n) ) = 0
is termed the differential equation of a family of curves to y = f (x,C 1 ,...,C n ) or
φ(x,y,C 1 ,...,C n ) = 0.
Example 2.1
The solution set for a simple harmonic oscillator without damping in one dimension
is given by the family of curves [1]
y = A 1 cost + A 2 sint, (2.6)
where y is the position coordinate and A 1,2 are arbitrary constants (which may be
found by using boundary and initial conditions for a specific problem). Eq.(2.6)
is given implicit by
φ(t,y, A 1 , A 2 ) = 0, (2.7)
6

2.1. FUNCTIONS OF ONE INDEPENDENT VARIABLE X AND ONE DEPENDENT
VARIABLE Y
i.e, a function of two constants. Differentiation of (2.6) once with respect to time
yields
ẏ = −A 1 sint + A 2 cost. (2.8)
From (2.6) we get
A 1 = y − A 2 sint
. (2.9)
cost
Inserting (2.9) in (2.8) yields
ẏ = A 2 (sint tant + cost) − ytant, (2.10)
i.e, implicitly we have
φ(t,y,ẏ, A 2 ) = 0. (2.11)
Thus, one constant is eliminated. Differentiate (2.10) one more time yields
sint
ÿ = A 2
cos 2 t − y 1
cos 2 − ẏtant. (2.12)
t
From (2.10) we get
A 2 =
ẏ + ytant
sint tant + cost . (2.13)
Inserting (2.13) in (2.12) we get
( )
ẏ + ytant sint 1
ÿ =
sint tant + cost cos 2 − y
t cos 2 − ẏtant. (2.14)
t
Thus, we get an implicit function of the form
φ(t,y,ẏ,ÿ) = 0, (2.15)
where now all constants are eliminated.
(2.14) is reduced to
Using some trigonometric relations,
ÿ = −y (2.16)
7

CHAPTER 2. NOTATION AND THE TOTAL DIFFERENTIAL OPERATOR
or, we get a differential function of the form
F(t,y,ẏ,ÿ) = ÿ + y = 0. (2.17)
Equation (2.17) is termed the differential equation to the family of curves (2.6).
Note that in general the equation of motion for a simple harmonic oscillator in one
dimension is given by ÿ + ω 2 0 y = 0, where ω2 0 is a constant which in this example
is chosen to 1.
End of Example 2.1
To simplify the notation, we introduce the total differentiation operator D x . It acts
on functions depending on a finite number of variables x,y,y ′ ,y ′′ ,.... The total
differential operator is defined as [4]:
D x = ∂ ∂x + ∂ y′ ∂y + ∂
∂
y′′ + ··· + y(s+1)
∂y ′ ∂y (s) (2.18)
Using the total differentiation operator D x , we can write equation (2.3) and (2.4)
in a compact form as D x φ = 0 and D x D x φ ≡ D 2 xφ = 0, respectivly, up to D (n)
x φ = 0.
The set of all finite differential functions will be denoted by A, as in [6] and
[4]. The total derivate D x acts on differential functions, F(x,y,y ′ ,y ′′ ,...,y (n) ), of
any finite number of the variables x,y,y ′ ,y ′′ .... To understand the difference between
the total differentiation D x and the partial differentiation ∂/∂x, consider the
following example.
Example 2.2
Let f and g be two functions from A, defined as
f = xy (2.19)
g = y ′ (2.20)
To find the total derivitive of f and g we let (2.18) operate on them.
D x ( f ) = D x (xy) ≡ ( ∂ ∂x + y′ ∂ ∂y )(xy)
= ∂ ∂x (xy) + y′ ∂ ∂y (xy) = y + y′ x. (2.21)
8

2.2. FUNCTIONS OF SEVERAL VARIABLES
If we would only take the partial derivative, we would instead get
∂
(xy) = y. (2.22)
∂x
And for g we have
while
D x (g) = D x (y ′ ) = ( ∂ ∂x + y′ ∂ ∂y + y′′ ∂
∂y ′ )(xy) =
= ∂ ∂x y′ + y ′ ∂ ∂y y′ + y ′′ ∂
∂y ′ y′ = y ′′ , (2.23)
∂
∂x (y′ ) = 0. (2.24)
End of Example 2.2
2.2 Functions of several variables
So far we have considered functions of one independent variable i.e, x,y,y ′ ,y ′′ ...
we will now consider functions of several variables. The notation used here is
adopted from the calculus of differential algebra and has turned out to be very
convenient. The notations are as follows [4],
x = {x i } ≡ (x 1 ... x n ) (2.25)
u = {u α } ≡ (u 1 ...u m ) (2.26)
where u α = u α (x i ). The partial derivatives will be denoted
u (1) ≡ {u α i } ≡ ∂uα
∂x i (2.27)
u (2) ≡ {u α i j} ≡ ∂2 u α
∂x i ∂x j (2.28)
etc. for higher order derivatives.
The total differential operator D i denotes total differentiation and is defined as [4]
D i = ∂
∂x i + uα i
∂
∂u α + uα i j
∂
∂u α j
+ u α i jk
∂
∂u α jk
+ ··· (2.29)
9

CHAPTER 2. NOTATION AND THE TOTAL DIFFERENTIAL OPERATOR
Recall that we assume summation over repeteded indicies,i.e., Einsteins summation
convention. Differential algebra suggests that we treat (2.26)–(2.28) as independent
variables even if they are connected by the relations
u α i = D i (u α )
u α i j = D j (u α i ) = D j D i (u α ). (2.30)
The variables (2.26)–(2.28) are also referred to as the differential variables (we
differentiate them with respect to x = (x 1 ,... x n )).
Example 2.3
Consider a function f in A, such that f = f (x 1 , x 2 ,y,z) where y = y(x 1 , x 2 ) and
z = z(x 1 , x 2 ). According to the notation and terminology presented above, we
have that
α = 1,2, i = 1,2, u 1 = y, u 2 = z
Now, let f = y + z. Then
D 1 ( f ) = ( ∂
∂x 1 + ∂
u1 1
∂u 1 + ∂
u2 1
} {{
∂u 2 )(y + z) =
}
u α ∂
i ∂u α
= ∂ ∂y ∂ ∂z ∂
(y + z) +
∂x1 ∂x 1 (y + z) +
∂y ∂x 1 (y + z) =
∂z
= 0 + ∂y
∂x 1 · 1 + ∂z
∂x 1 · 1 = ∂y
∂x 1 + ∂z
∂x 1 (2.31)
For D 2 ( f ) we get, in a similar way,
D 2 ( f ) = ( ∂
∂x 2 + ∂
u1 2
∂u 1 + ∂
u2 2
} {{
∂u 2 )(y + z) =
}
u α ∂
i ∂u α
= ∂ ∂y ∂ ∂z ∂
(y + z) +
∂x2 ∂x 2 (y + z) +
∂y ∂x 2 (y + z) =
∂z
= 0 + ∂y
∂x 2 · 1 + ∂z
∂x 2 · 1 = ∂y
∂x 2 + ∂z
∂x 2 . (2.32)
End of Example 2.3
10

2.3. IMPORTANT EQUATIONS
2.3 Important equations
The main notations presented and needed for the subsequent chapters are equations
(2.25)–(2.29).
x = {x i } ≡ (x 1 ... x n ), u = u α ≡ (u 1 ...u m )
u (1) ≡ {u α i } ≡ ∂uα
∂x i ,
D i = ∂
∂x i + uα i
u (2) ≡ u α i j ≡ ∂2 u α
∂x i ∂x j
∂
∂u α + ∂
uα i j + u α ∂
i jk + ···
∂u α j
∂u α jk
11

3
BASIC LIE GROUP THEORY
In applied group analysis we will consider continuous transformation groups. A
transformation of the variables in a differential equation form a symmetry group
if it leaves the differential equation invariant [4]. These continuous symmetry
groups are represented by Lie group theory e.g [6, 4, 7]. We will employ Lie’s
theory (Lie, Marius Sophus, 1842–1899) which reduces the analysis from large
groups to a determination of infinitesimal transformations. In this chapter we introduce
the basic theory of Lie group analysis needed for finding symmetries of
differential equations. Lie’s theory will be presented and the meaning of invariance
will be clarified. We will also show how Lie groups lead to the concept of
Lie algebras [6, 7].
3.1 Transformation groups
Transformations of variables can be explained in the following general way [8];
Let the vector f be f = ( f 1 ,..., f m ) of m functions, each depending on n variables
x = (x 1 ,... x n ) in R n . Then f(x) defines a transformation from R n to R m . If y =
(y 1 ,...y m ) is the point in R m that corresponds to x under the transformation f, we
have the relation that y = f(x).
Consider the following transformations in R n [6]:
¯x i = f i (x) (3.1)
13

CHAPTER 3. BASIC LIE GROUP THEORY
where x = (x 1 ,..., x n ), ¯x = ( ¯x 1 ,..., ¯x n ) denotes the transformation. The group action
f = ( f 1 ,..., f n ) and its derivatives are continuous. Also let det‖ ∂ f i
‖≠ 0. This
∂x i
means that we can solve x i as a function of f i according to the implicit function
theorem [8, 9], i.e the transformations are invertible,
x i = ( f −1 ) i ( ¯x). (3.2)
Let us denote transformation (3.1) by T and the inverse transformation (3.2) by
T −1 .
Geomtrically T is carring a point x ∈ R n to a point ¯x ∈ R n , and T −1 carries ¯x back
to x. The identical transform I is just ¯x i = x i . Let T 1 and T 2 be two transformations
in R n , T 1 : f i 1 (x), T 2 : f i 2 (x) respectively. Then the successive transformation, T 1T 2
is defined as follow [6]:
¯x = f i 1(x)
¯x = f i 2( ¯x) = f i 2( f i 1(x)) (3.3)
This means that T 1 T 2 (x) carries the point x directly to ¯x. It follows that TT −1 =
T −1 T = I.
Definition 3.1
A group of transformations in R n is a set G of transformations (3.1), such that
it contains the identity I and all transformations T s and Ts
−1 within G and all
products T i T j , compare [6].
I ∈ G
T −1
s ∈ G T i T j ∈ G whenever T s ,T i ,T j ∈ G (3.4)
Exempel 3.1
Consider the translation group G of transformations T a (x) defined as
¯x = f (x) = x + a (3.5)
The successive transformation T a T b (x) is
¯x = T a T b (x) = T b ( ¯x) = T b (x + a) = x + a + b (3.6)
14

3.2. ONE-PARAMETER GROUP
The identity transform exists, ¯x = x for a = 0, i.e T 0 ∈ G. Since both
a,b ∈ R are arbitrary T a T b ∈ G. For the inverse transform Ts
−1 we have
x = f −1 ( ¯x) = ¯x − a. (3.7)
That is Ta
−1 = T −a and Ta
−1
group of transformations.
∈ G. According to Definition 1, equation (3.5) forms a
End of Example 3.1
3.2 One-parameter group
Now we consider the one-parameter groups. They consist of transformations of
the form [6]
¯x = f (x,a). (3.8)
In component form, transformation (3.8) is written as
¯x i = f i (x,a) (3.9)
where x ∈ R n and a is a real parameter. Let transformation (3.8) be the transform
T a and let the identity transform T a0 be
f (x,a 0 ) = x (3.10)
where a 0 (in general not zero) is the only value of a reducing T a to the identity
transform.
Definition 3.2
A set G of transformations T a in R n given by (3.9) is called a one-parameter local
group if there exist a subinterval U ′ ⊂ U containing a 0 , such that the functions
f i (x,a) satisfy [6],
f i ( f (x,a),b) = f i (x,c) i = 1,...,n (3.11)
∀a, b ∈ U ′ , c ∈ U and c = φ(a,b) such that
φ(a,b) = a 0 (3.12)
15

CHAPTER 3. BASIC LIE GROUP THEORY
has a unique solution b for any a ∈ U ′ . Equation (3.11) is called the composition
rule.
For a given a, the solution b of equation (3.12) is denoted a −1 (in general not 1/a).
Then Ta
−1 is given by
f i ( ¯x,a −1 ) = x i (3.13)
Definition 3.2 may seem quite complicated at first glance. Let us make a little
explanation:
Let T a and T b be two transformations of the type (3.8). Then, for the successive
transform T a T b = T c we must have a and b chosen from a smaller interval than c to
assure that the transform T c is still a transformation of the transformations (3.8).
In this smaller interval (a subinterval) we must have the value a 0 that reduces the
transformations T a and T b to the identity transform, otherwise they alone would
not form a group of transformations according to Definition 3.1. This means that
for any parameter a in the subinterval there is only one value of b that reduces
c to a 0 . I.e, there is only one value of b that reduces the successive transform
T a T b = T c to the identity transform. This value of b is denoted by a −1 and is,
consequently, the inverse transform.
The function
c = φ(a,b) (3.14)
is termed as the group composition law [6]. From the equations (3.9) and (3.11)
it follows that
φ(a,a 0 ) = a and φ(a 0 ,b) = b (3.15)
Indeed, if we, in equation (3.11), set a = a and b = a 0 , then we get
f i ( f (x,a),a 0 ) = f (x,a) ⇒ c = φ(a,a 0 ) = a
where equation (3.10) was used at the first step, and the result followed by identification
with the composition rule. Similarly for the second equation in (3.15).
Exampel 3.2
Consider the two transformations
a) ¯x = ax
b) ¯x = x + ax. (3.16)
16

3.2. ONE-PARAMETER GROUP
We wish to find their composition law. For a) we have:
¯x = f (x,a) = ax (3.17)
Using equation (3.11) we get for the LHS
f ( f (x,a),b) = f (ax,b) = abx (3.18)
and for the RHS
f (x,c) = f (ax,b) = cx. (3.19)
Thus, we get the group composition law
c = φ(a,b) = ab (3.20)
To find a −1 , we use equation (3.13) f ( ¯x,a −1 ) = x. Then we get
a −1 ¯x = x
⇒ a −1 (ax) = x
⇒ a −1 = 1 a
(3.21)
For b) we have:
¯x = f (x,a) = x + ax (3.22)
Using equation (3.11) we get for the LHS
f ( f (x,a),b) = f (x + ax,b) = x + (a + b + ab)x (3.23)
and for the RHS
f (x,c) = f (x + ax,b) = x + cx. (3.24)
Thus, we get the group composition law
c = φ(a,b) = a + b + ab (3.25)
To find a −1 we use (3.13), f ( ¯x,a −1 ) = x as before. Then we get
¯x + a −1 ¯x = x(1 + a + a −1 (1 + a))
⇒ x = x(1 + a + a −1 (1 + a))
⇒ a −1 = −
a
(3.26)
1 + a
Here we see that a −1 is not necessarily 1/a.
End of Example 3.2
17

CHAPTER 3. BASIC LIE GROUP THEORY
3.2.1 Canonical parameter
If we choose the group parameter a to be canonical, the composition law c =
φ(a,b) reduces by definition to, see [6] and [4],
c = φ(a,b) = a + b. (3.27)
Consider the group transformation (3.8) with a canonical parameter a.
¯x i = f i (x,a) (3.28)
it is assumed that f i (x,a) are defined in a neighborhood of a = 0, and that
f i (x,0) = x i (3.29)
Thus, for a canonical parameter a we get the composition rule
f i ( f (x,a),b) = f i (x,c) = f i (x,a + b) i = 1,...,n (3.30)
where a, b are any numerical values in a neighborhood of a = 0. The inverse
transform (3.13), when using a canonical parameter, is simply
Indeed,
x i = f i ( ¯x,−a) (3.31)
x i = f i ( ¯x,a −1 ) = f i ( f (x,a),a −1 ) = f i (x,a + a −1 ) = f i (x,0) ⇒ a −1 = −a
where in the second step we used (3.30) and in the last step we used (3.29). Thus,
local groups with a canonical parameter are well defined by the initial condition
(3.29), and the composition rule (3.30).
3.3 The infinitesimal transformation and Lie’s equations
We will now present the basic theory which reduces the analysis of a symmetry
group of transformations to analysis of its infintesimal transform. This is one of
the main ideas in Lie group theory, due to the fact that an element in a Lie group
is a continous differentiable transformation with an inverse with the same properties;
i.e., a diffeomorphism. See [7, 9].
18

3.3. THE INFINITESIMAL TRANSFORMATION AND LIE’S EQUATIONS
The infintesimal transformation of a group G, where G is a one-parameter group
of transformations (3.28), is obtained by Taylor expansion of (3.28) with respect
to the parameter a in a neigborhood a = 0 [6]. Only keeping the linear part in a
yields,
¯x i = f i (x,a) ≈ f i (x,0) + ∂ f i (x,a)
| a=0 (a − 0) + ··· (3.32)
∂a
thus, we get the infinitesimal transform
¯x i ≈ x i + aξ i (x) (3.33)
where ξ i (x) is defined as
ξ i (x) = ∂ ∂a [ f i (x,a)] a=0 (3.34)
Geometrically (3.34) defines the tangent vector ξ(x) = (ξ 1 (x),...ξ n (x)) at the point
x. Therefore, ξ is called the tangent vector field of group G.
The tangent vector field is associated with the first order differential operator
X = ξ i ∂
∂x i (3.35)
called the generator of group G.
Theorem 3.1
Let G be a local group defined by (3.8), where the functions f i obey (3.29) and
(3.30). Let (3.33) be the infintesimal transformation of group G. Then ¯x i = f i (x,a)
solves the system of first order ordinary differential equation, called the Lie equations
[4, 6, 7],
d ¯x i
da = ξi ( ¯x) with initial condition ¯x i | a=0 = x i (3.36)
i.e the solution to (3.36) provides a one-parameter local group with a given infintesimal
transformation.
19

CHAPTER 3. BASIC LIE GROUP THEORY
For a given infinitesimal transformation (3.33) or generator (3.35), the transformations
(3.8) are defined by integrating the Lie equations (3.36).
Proof of theorem 3.1
Set b = △a in (3.30). Then f i ( f (x,a),△a) = f i (x,a + △a). Now, expanding RHS,
f i (x,a + △a) in a Taylor series in a neighborhood where a + △a ≈ a, yields
f i (x,a+△a) ≈ f i (x,a)+ ∂ ∂a [ f i (x,a)]·(a+△a−a) = f i (x,a)+ ∂ ∂a [ f i (x,a)]·△a.
Now, expanding LHS, f i ( f (x,a),△a) in a Taylor series in a neighborhood where
△a ≈ 0, yields
f i ( f (x,a),△a) ≈ f i ( f (x,a),0) +
∂
∂△a [ f i ( f (x,a),△a)]| △a=0 · (△a − 0)
= f i (x,a) + ∂
∂△a [ f i ( f (x,a),△a)]| △a=0 · (△a).
Setting LHS = RHS and using (3.34), we get
∂
∂△a [ f i ( f (x,a),△a)]| △a=0 = ξ i ( f (x,a))
This leads to the result that
∂
∂a [ f i (x,a)] = ξ i ( f (x,a)).
We also need to show that the solution ¯x i = f i (x,a) of (3.36) satisfies the group
property (3.30). Let
u(b) = f ( ¯x,b) ≡ f ( f (x,a),b), (3.37)
v(b) = f (x,a + b) (3.38)
considered for a fixt a. We have that u(0) = v(0) = f (x,a), which solves (3.36).The
property (3.30) is staisfied if we can show that u(b) = v(b) in a neighborhood of
b = 0. Consider th Lie equations for u and v,
du
db ≡ f ( ¯x,b) = ξ(u)
db
(3.39)
dv f (x,a + b)
≡ = ξ(v)
db db
(3.40)
(3.41)
20

3.3. THE INFINITESIMAL TRANSFORMATION AND LIE’S EQUATIONS
with the initial conditions u(0) = f (x,a), v(0) = f (x,a) respectively. Let z = u(b)
and z = v(b), then z solves
dz
= ξ(z) (3.42)
db
with the initial condition z| b=0 = f (x,a). According to the uniqueness theorem
for initial value problems [10] we must have that u(b) = v(b). Thus, the group
property (3.30) is satisfied [6].
□
Exampel 3.3
Consider the infinitesimal transformation group [4],
¯x ≈ x + ax 2 ȳ ≈ y + axy. (3.43)
It has generator
X = ξ 1 ∂ ∂x + ξ2 ∂ ∂y , (3.44)
where ξ 1 (x,y) = x 2 and ξ 2 (x,y) = xy. The Lie equations are
d ¯x
da = ¯x2 with initial condition ¯x| a=0 = x
dȳ
da = ¯xȳ with initial condition ȳ| a=0 = y. (3.45)
Solving this system of differential equations by seperation of variables yields,
¯x −1 d ¯x = da ⇒ − 1¯x = a +C 1
⇒ ¯x = − 1
a +C 1
dȳ
ȳ = ¯xda = − 1 da
a +C 1
⇒ ȳ = C 2
a +C 1
⇒
¯x = − 1
a +C 1
,ȳ = C 2
a +C 1
(3.46)
21

CHAPTER 3. BASIC LIE GROUP THEORY
Using the initial conditions yields
C 1 = − 1 x
C 2 = − y x . (3.47)
We get the one-parameter local group
¯x =
x
1 − ax
ȳ =
y
1 − ax
(3.48)
End of Example 3.3
3.4 Invariant functions and equations
The meaning of invariant functions and invariant equations will now be clarified.
3.4.1 Invariant function
Definition 3.3
A function F(x) is called invariant of a group G of transformations (3.28) if
F( f (x,a)) = F(x) (3.49)
identically in x and a in a neighborhood a = 0 [6].
Theorem 3.2
A function F(x) is an invariant of a group G with generator X, (3.35), if and only
if it solves the homogeneous linear partial differential equation [6]
X(F) ≡ ξ i ∂F(x)
∂x i = 0. (3.50)
Proof of Theorem 3.2
Let F(x) be invariant. Consider F( f (x,a)) expanded in a Taylor series in a neighborhood
around a = 0, i.e
F( f (x,a)) ≈ F(x + aξ(x)) ≈ F(x) + aξ i ∂F(x)
∂x i
22

3.4. INVARIANT FUNCTIONS AND EQUATIONS
The invariance condition F(x) = F( f (x,a)) yields (3.50).
Conversely, let F(x) be any solution to the differential equation (3.50). Since
(3.50) is valid at any point, consider it at ¯x = f (x,a), i.e,
ξ i ∂F( ¯x)
( ¯x)
∂ ¯x i = 0
Hence, invoking the Lie equations:
dF( f (x,a))
da
=
∂F( ¯x) d f i (x,a)
∂ ¯x i = ξ i ∂F( ¯x) ¯x
da ∂ ¯x i = 0
Since F( f (x,0)) = F(x) we can conclude that u(a) = F( f (x,a)) solves the initial
value problem
du
da = 0
u| a=0 = F(x)
The solution to the initial value problem is, by direct calculation, given by u =
F(x) and this solution is unique according to the uniqueness theorem [10]. Hence,
the two solutions u(a) = F( f (x,a)) and u = F(x) are identical for any x, i.e F( f (x,a)) =
F(x) which is the invariance condition.
Remark:
From equation (3.50) we can see that ξ i really is a tangent vector field. Rewrite
X(F) ≡ ξ i ∂F(x) = 0, using vector notation, i.e ξ i ≡ ξ and ∂ ≡ ∇, we write (3.50)
∂x i
∂x i
as ξ • ∇F = 0. Recall from basic calculus that ∇F defines a normal vector to
the surface described by F. If ξ • ∇F = 0 we must have that ξ is orthogonal to the
normal vector to F, i.e ξ must be the tangent vector to F at the point x = (x 1 ... x n ).
See e.g. [8].
□
3.4.2 Invariant equations
Consider a system of equations [6]
F σ (x) = 0 σ = 1... s, (3.51)
23

CHAPTER 3. BASIC LIE GROUP THEORY
where x = (x 1 ... x n ) ∈ R n and s < n. Impose that for all x satisfying (3.51) we have
that the Jacobian matrix [8] is of rank s, i.e
rank‖ ∂F σ(x)
∂x i ‖ = s. (3.52)
Recall that for a system of s equations with n variables where s < n, we get a
solution involving (n − s) parameters if the rank of the system is equal to the
number of equation. These parameters will span the set of solutions. Thus, the
solution set x satisfying (3.51), F σ (x) = 0 is an (n − s)-dimensional surface in R n
or, more precise, an (n − s)-dimensional manifold M ⊂ R n [6],[7], [9].
Definition 3.4
The system (3.51) is said to be invariat, or admitted by a group G of transformations
¯x = f (x,a) if
F σ ( ¯x) = 0 σ = 1... s (3.53)
whenever x solves F σ (x) = 0. This means that any point in the manifold M is
carried to another point in M. Thus, any path curve of G passing through a point
x ∈ M lies in M. M is an invariant manifold [6].
Theorem 3.3
The system (3.51) is invariant under the group G with infinitesimal generator X if
and only if
XF σ | (3.51) = 0 σ = 1... s (3.54)
where | (3.51) means evaluated on its manifold [6].
Provided that F σ and XF σ are analytic functions in a neigborhood of M, the
invariance test (3.54) can be written as
XF σ (x) = λ ν σ(x)F ν (x) σ = 1... s, (3.55)
where λ ν σ are coefficients bounded in M [6]. For a more rigorous treatment of the
connection between Lie groups and Manifolds see [7].
24

3.5. MULTI-PARAMETER GROUPS AND LIE ALGEBRAS
3.5 Multi-parameter groups and Lie algebras
In previous sections we considered one–parameter groups. We will now extend
the theory to multi–parameter groups, i.e instead of one parameter a, we have r–
parameters, a = (a 1 ,...,a r ). This will lead to the concept of Lie algebras, which is
frequently used in theoretical physics, especially in quantum mechanics. See e.g.
[11].
3.5.1 Multi–parameter groups
Multi–parameter groups can be understood as one–parameter groups with the difference
that the parameter a is now a vector–parameter, a = (a 1 ,...,a r ) [6]. An
invertible transformation T a : R n → R n , or more precics T a r ≡ T a 1T a 2 ···T a r, in a
neighborhood of a = 0 ≡ (0,...,0) is
¯x = f (x,a) (3.56)
We have the same defintion for a local r–parameter group as for a local one–
parameter group, Definition 3.2. I.e, the transformation (3.56) is said to form a
local r–parameter group if
f ( f (x,a),b) = f (x,c) (3.57)
where b = (b 1 ,...,b r ) and the composition function c = c(a,b) is now a vector
function with components
c α = φ α (a,b), α = 1,...,r. (3.58)
with the same properties as in Defintion 3.2, see [6],[7].
3.5.2 Short outline of Lie algebra
A Lie algebra is a vector space spanned by infinitesimal generators X. Generators
of an r-parameter local groups, i.e multi-parameter groups leads to the concept of
Lie algebras. Almost all information in a group is contained in its Lie algebra [7].
We define a Lie algebras of operators as follow [6]:
25

CHAPTER 3. BASIC LIE GROUP THEORY
Definition 5.2
A Lie algebra is a vector space L of spanned by operators X α = ξα(x) i ∂
following property: If the operators
X 1 = ξ i 1(x) ∂
∂x i ,
are elements of L, then their commutator
∂x i
with the
X 2 = ξ i 2(x) ∂
∂x i (3.59)
[X 1 , X 2 ] ≡ X 1 X 2 − X 2 X 1 = (X 1 (ξ i 2) − X 2 (ξ i 1)) ∂
∂x i (3.60)
is also an element of L. We denote the dimension of the Lie algebra by L r [6].
The commutator is bilinear, skew-symetric and satisfies the Jacoby Identity [6, 7].
Consider a basis of a Lie algebra L r given by
X α = ξ i α(x) ∂
∂x i α = 1,...,r. (3.61)
Then, any operator X ∈ L r can be written as a linear combination of this basis,
i.e X = c α X α . Let X α and X β be basis vectors of L r then [ X α , X β
]
∈ Lr . Thus,
due to the bilinearity property of the commutator (which is a consequence of its
definition), we have that [6]
[X α , X β = c γ αβ X γ. (3.62)
This provides a simple test for the linear span of a Lie algebra. Thus, given r
independent operators, if (3.62) hold they span an r-dimensional Lie algebra. The
constants c γ αβ is called the structure constants [6],[7] of that Lie algebra.
Consider the composition of r one-parameter groups T a = T a 1 ···T a r, generated by
the base operators X α , via the Lie equations
d ¯x i
da α = ξi α( ¯x), ¯x i | a α =0 = x i (3.63)
where i = 1,...,n and α = 1,...,r. T a is an r-parameter local group G r if the operators
X α form an r-dimensional Lie algebra L r .
The invariance conditions stated in previous sections is the same for multiparameter
groups. This is due to the fact that we may consider a multi-parameter
group as a composition a several one-parameter groups [6].
26

4
DIFFERENTIAL ALGEBRA
Differential algebra furnishes us with a convenient language to use in modern
group analysis. Some of the notations that will be presented here have already
been introduced in chapter 2. Here, we will extend and generlaize the theory
presented earlier.
4.1 The space A
Let us denote an arbitrary sequence by z [6],
z = (x,u,u (1) ,u (2) ,...) (4.1)
with elements z ν , ν 1. Here
z i = x i
i = 1...n
z n+α = u α α = 1...m (4.2)
and the remaning elements are derivatives of u.
A finite subsequence of z will be denoted by [z].
Definition 4.1
A differential function f ([z]) is a locally analytic function. The order of f ([z]) is
the highest order of derivatives in the differential function f ([z]). We denote the
set of all differential functions of finite order by A [6].
27

CHAPTER 4. DIFFERENTIAL ALGEBRA
Properties in the space A:
If f ([z]) ∈ A and g([z]) ∈ A then,
1. a f + bg ∈ A for any constants a,b.
2. f g ∈ A
3. D i ( f ) ∈ A
The first and second property follow from the fact that both f and g are analytic
functions. For the third property we know that the derivative of an analytic
function is analytic. Furthere more, from the expression for the infinite operator
D i , (2.29), we see that it truncates when acting on a differential function, and
increasing the order of the differential function by one. i.e, if ord( f ) = s then
ord(D i ( f )) = s + 1 is still finite. According to Definition 4.1, statement three follows
[6].
4.2 Extended point transformation group
Extended point transformation means that we in the usual point transformation
(3.8) take the derivities into account. In that way we arrive at a transformation
including n + s variables, where n is the number of variables and s is its derivities
up to order s. We will consider extended point transformations in the plane and
then generilize it to many variables [6].
Definition 4.2
An extended point transformation up to the sth-derivative is called the sth-prolongation.
4.2.1 Extended pointtransformation in the plane
We start by looking at the point transformations in the plane [6], [4].
¯x = f (x,y,a) ≈ x + aξ(x,y)
ȳ = ϕ(x,y,a) ≈ y + aη(x,y) (4.3)
28

4.2. EXTENDED POINT TRANSFORMATION GROUP
where x is the independent variable and y the differential variable. we have that
ȳ ′ ≡ dȳ
d ¯x = ϕ xdx + ϕ y dy
f x dx + f y dy = ϕ x + y ′ ϕ y
f x + y ′ = D xϕ
f y D x f
(4.4)
By adding (4.4) to the system (4.3) of point transformation, we arrive at a transformation
group for three variables x,y,y ′ , where x is independent and y is the
differential variable. We treat them as if they where all independent of each other,
as before. If, for a canonical parameter a, (4.3) forms a one-parameter group with
the group property (3.30), i.e
¯x = f ( ¯x,ȳ,b) = f (x,y,a + b)
ȳ = ϕ( ¯x,ȳ,b) = ϕ(x,y,a + b) (4.5)
and
ȳ ′ = ψ(x,y,y ′ ,a) ≡ D x(ϕ(x,y,a))
D x ( f (x,y,a))
(4.6)
then ȳ ′ ≡ ψ( ¯x,ȳ,ȳ ′ ,a) must be equal to ψ(x,y,y ′ ,a + b) for (4.3) and (4.4) to form
a one-parameter group. i.e we need to show that
ȳ ′ ≡ ψ( ¯x,ȳ,ȳ ′ ,a) = ψ(x,y,y ′ ,a + b) (4.7)
To do that, consider the following expression for ȳ ′ :
ȳ ′ D
= ¯ x (ϕ( ¯x,ȳ,a))
D¯
x ( f ( ¯x,ȳ,a))
(4.8)
multiplying the numerator and the denominator of equation (4.8) by D x ( f (x,y,a)),
we get
ȳ ′ = D x( f (x,y,a)) D¯
x (ϕ( ¯x,ȳ,a))
D x ( f (x,y,a)) D¯
x ( f ( ¯x,ȳ,a)) = D x(ϕ( ¯x,ȳ,a))
D x ( f ( ¯x,ȳ,a))
= D x(ϕ(x,y,a + b))
D x ( f (x,y,a + b)) = ψ(x,y,y′ ,a + b) (4.9)
where we in the first step used the chain rule [4],
D x = D x ( f (x,y,a)) ¯ D x (4.10)
29

CHAPTER 4. DIFFERENTIAL ALGEBRA
and in the second step we used the group property (4.5).
Consider the one-parameter group (4.3) of infinitesimal transformations. The generator
to (4.3) is
X = ξ(x,y) ∂ ∂x + η(x,y) ∂ ∂y
(4.11)
The first order extended generator has the form
where
X (1) = ξ(x,y) ∂ ∂x + η(x,y) ∂ ∂y + ζ ∂
(1)
∂y ′ (4.12)
ζ (1) = D x (η) − y ′ D x (ξ) (4.13)
indeed, consider (4.3) in (4.4)and expand for the parameter a in a neighborhood
of 0, and only keeping linear terms in a, we get
ȳ ′ = D xϕ
D x f ≈ D x(y + aη(x,y))
D x (x + aξ(x,y)) = y′ + aD x (η)
1 + aD x (ξ)
≈ [y ′ + aD x (η)] · [1 − aD x (ξ)] ≈ y ′ + aD x (η) − y ′ aD x (ξ)
= y ′ + a[D x (η) − y ′ D x (ξ)] = y ′ + aζ (1) (4.14)
The (s + 1)-times extended infintesimal transformations, and their generators are
obtained recursively:
ȳ (s+1) ≈ [y (s+1) +aD x (ζ s )]·[1−aD x (ξ)] ≈ y (s+1) +a[D x (ζ s )−y (s+1) D x (ξ)] (4.15)
By setting ȳ (s+1) ≈ y (s+1) + aζ s+1 , we get the (s + 1)-times prolongation formula
ζ s+1 = D x (ζ s ) − y (s+1) D x (ξ) s = 1,2,.... (4.16)
Exampel 4.1
The spiral transform has the generator [4]
X = (x − y) ∂ ∂x + (x + y) ∂ ∂y . (4.17)
30

4.2. EXTENDED POINT TRANSFORMATION GROUP
Its twice extended generator has the form
X (2) = ξ(x,y) ∂ ∂x + η(x,y) ∂ ∂y + ζ (1)(x,y,y ′ ) ∂
∂y ′
+ ζ (2) (x,y,y ′ ,y ′′ ) ∂ , (4.18)
∂y ′′
where ξ(x,y) = x − y, η(x,y) = x + y and ζ (1) ,ζ (2) is given by:
ζ (1) = D x (η) − y ′ D x (ξ)
= D x (x + y) − y ′ D x (x − y)
= 1 + y ′ − y ′ (1 − y ′ ) = 1 + (y ′ ) 2 . (4.19)
Thus, the first extended generator is
X (1) = (x − y) ∂ ∂x + (x + y) ∂ ∂y + (1 + (y′ ) 2 ) ∂
∂y ′ . (4.20)
To get the second one we need to find ζ (2) .
ζ (2) = D x (ζ (1) ) − y ′′ D x (ξ)
= D x (1 + (y ′ ) 2 ) − y ′′ D x (x − y)
= 0 + 2y ′ y ′′ − y ′′ (1 − (y ′ ) 2 ) = y ′′ ((y ′ ) 2 + 2y ′ − 1). (4.21)
Hence, the second order extended generator is
X (2) = (x − y) ∂ ∂x + (x + y) ∂ ∂y + (1 + (y′ ) 2 ) ∂
∂y ′
+ (y ′′ ((y ′ ) 2 + 2y ′ − 1)) ∂ . (4.22)
∂y ′′
End of Example 4.1
In physical applications we often consider functions involving many differential
variables and one independent variable, e.g , three spatial coordinates depending
on time is a common case. More general, consider point transformations
¯x = f (x,u,a) ≈ x + aξ(x,u)
¯ u α = ϕ(x,u,a) ≈ u α + aη α (x,y) (4.23)
31

CHAPTER 4. DIFFERENTIAL ALGEBRA
with one independent variable x and m differential variables u = (u 1 ,,u m ). The
transformation of u¯
α 1
is obtained in precisely the same way as (4.4) where obtained.
We arrive at the prolongation formula [6],
ζ1 α = D x (η α ) − u α 1 D x (ξ)
ζs+1 α = D x (ζs α ) − u α s+1D x (ξ) s = 1,2,.... (4.24)
4.2.2 Point transformation involving many variables
In the previous section we considered point transforamtions in the plane and point
transformations involving several differential variables and one independent variable,
and found their prolongation formulas. In this section we generlize these
results into functions depending on several differential and several independent
variables [6]. This case is very common in physical applications in the study
of vector fields. For instance, physical systems described by Maxwell’s equations
[2]. In that case we have the components of the electromagentic fields as
the differential variables, and the spatial coordiantes and time as the independent
variables.
Let G be a one-parameter group of transformations, with independent variables
x = (x 1 ,..., x n ) and differential variables u = (u 1 ,...,u m ):
¯x i = f i (x,u,a), f i | a=0 = x i (4.25)
ū α = ϕ α (x,u,a), ϕ α | a=0 = u α (4.26)
with generator
where
X = ξ i (x,u) ∂
∂x i + ηα (x,u) ∂
∂u α (4.27)
ξ i = ∂ f i
∂a | a=0
η α = ∂ϕα
∂a | a=0 (4.28)
Let ¯D i denote the total derivatives in the new variables ¯x i . The chain rule yields:
D i = ( ∂ f j
∂x i + ∂ f j
uβ i
∂u β ) D¯
j ≡ D i ( f j ) D¯
j (4.29)
32

4.2. EXTENDED POINT TRANSFORMATION GROUP
The connection between the new differential variables and their derivatives are
ū α i = ¯D i (ū α ),
ū α i j = ¯ D j ¯D i (ū α ) ≡ ¯ D j (ū α i ) (4.30)
To find the first prolongation, consider the following relation,
D i (ϕ α ) ≡ D i (ū α ) = D i ( f j ) ¯ D j (ū α ) = (ū α j )D i ( f j )
⇒ ū α j = D i(ϕ α )
D i ( f j )
togheter with the following infinitesimal transformations
(4.31)
f i = x i + aξ i ,ϕ α = u α + aη α and set ū α i = u α i + aζ α i (4.32)
then, substituting the first two equations of (4.32) into (4.31) one obtains the first
prolongation formula [6],
ζ α i = D i (η α ) − u α j D i (ξ j ). (4.33)
The first extended generator X (1) to (4.27) is written
X (1) = ξ i (x,u) ∂
∂x i + ηα (x,u) ∂
∂u α + ζα i
∂
∂u α i
The second prolongation formula is given by
. (4.34)
ζ α i 1 i 2
= D i2 (ζ α i 1
) − u α ji 1
D i2 (ξ j ), (4.35)
and the twice extended generator is written
X (2) = ξ i (x,u) ∂
∂x i + ηα (x,u) ∂
∂u α + ζα i
∂
∂u α i
The higher order prolongations are defined recursively as
+ ζ α i 1 i 2
∂
∂u α i 1 i 2
. (4.36)
ζ α i 1 ...i s
= D is (ζ α i 1 ...i s−1
) − u α ji 1 ...i s−1
D is (ξ j ), (4.37)
where we have summation in repeteded indicies for every s.
The extension of generators is invariant under any change of variables [6]. For
a Lie algebra L r with r linearly independent operators of the form (4.27), their extension
up to the sth-prolongation also span a Lie algebra with the same structure
constants as L r . This means that the relations between the elements in the two
algebras are the same, i.e they are isomorphic [7].
33

CHAPTER 4. DIFFERENTIAL ALGEBRA
4.2.3 Extended generators for differential equations with two independent
and one dependent variable
In this section we will find the general extended generators for the special case of
two independent variables x and y, and one dependent variable u = u(x,y) as done
in [6]. We will consider partial differential equations up to second order. This
kind of partial differential equations arises alot in physical applications, for example
in gas dynamics where motion of gases are studied, particulary when they
interact with moving objects.
We consider a generator of a point transformation group of the form
X = ξ 1 (x,y,u) ∂ ∂x + ξ2 (x,y,u) ∂ ∂y + η(x,y,u) ∂ ∂u
(4.38)
The twice extension of (4.38) has the form
X = ξ 1 ∂ ∂ ∂x +ξ2 ∂y +η ∂ ∂u +ζ ∂ ∂ ∂ ∂ ∂
1 +ζ 2 +ζ 11 +ζ 12 +ζ 22 . (4.39)
∂u x ∂u y ∂u xx ∂u xy ∂u yy
The extendent tangent vector field components ζ i and ζ i j are given by (4.33) and
(4.36) respectively.
and
ζ 1 = D x (η) − u x D x (ξ 1 ) − u y D x (ξ 2 )
ζ 2 = D y (η) − u x D y (ξ 1 ) − u y D y (ξ 2 ) (4.40)
ζ 11 = D x (ζ 1 ) − u xx D x (ξ 1 ) − u xy D x (ξ 2 )
ζ 12 = D y (ζ 1 ) − u xx D y (ξ 1 ) − u xy D y (ξ 2 )
ζ 22 = D y (ζ 2 ) − u xy D y (ξ 1 ) − u yy D y (ξ 2 ). (4.41)
Using the definitions of D x and D y one obtains:
ζ 1 = η x + u x η u − u x ξ 1 x − (u x ) 2 ξ 1 u − u y ξ 2 x − u x u y ξ 2 u, (4.42)
ζ 2 = η y + u y η u − u x ξ 1 y − u x u y ξ 1 u − u y ξ 2 y − (u y ) 2 ξ 2 u. (4.43)
34

4.2. EXTENDED POINT TRANSFORMATION GROUP
For the second order prolongations we get,
ζ 11 = η xx + 2u x η xu + u xx η u + (u x ) 2 η uu − 2u xx ξ 1 x − u x ξ 1 xx
− 2(u x ) 2 ξ 1 xu − 3u x u xx ξ 1 u − (u x ) 3 ξ 1 uu − 2u xy ξ 2 x − u y ξ 2 xx
− 2u x u y ξ 2 xu − (u y u xx + 2u x u xy )ξ 2 u − (u x ) 2 u y ξ 2 uu, (4.44)
ζ 12 = η xy + u y η xu + u x η yu + u xy η u + u x u y η uu − u xy (ξ 1 x + ξy)
2
− u x ξ 1 xy − u xx ξy 1 − u x u y (ξ 1 xu + ξuy) 2 − (u x ) 2 ξyu
1
− (2u x u xy + u y u xx )ξu 1 − (u x ) 2 u y ξuu 1 − u y ξ 2 xy − u yy ξ 2 x
− (u y ) 2 ξ 2 xu − (2u y u xy + u x u yy )ξu 2 − u x (u y ) 2 ξuu, 2 (4.45)
ζ 22 = η yy + 2u y η yu + u yy η u + (u y ) 2 η uu − 2u yy ξ 2 y − u y ξ 2 yy
− 2(u y ) 2 ξ 2 yu − 3u y u yy ξ 2 u − (u y ) 3 ξ 2 uu − 2u xy ξ 1 y − u x ξ 1 yy
− 2u x u y ξ 1 yu − (u x u yy + 2u y u xy )ξ 1 u − u x (u y ) 2 ξ 1 uu. (4.46)
This is the general prolongation formulas for a second order partial differential
equation with one dependent variable and two independent variables. Examples
on extended generators and the use of prolongation formulas can be seen in chapter
9.
35

5
OPERATORS
In this chapter some differential operators that play a central role in the development
of conservation laws will be presented. We will mainly state their definitions,
and some useful Theorems and Lemmas. This is the last tools we need
before we procced to Part II. However, one could move on directly to Part II and
just look up the definitions, theorems and lemmas when needed.
5.1 Euler-Lagrange operator
We start by defining the Euler-Lagrange operator in the space A.
Definition 5.1
The Euler-Lagrange operator is defined by the formal sum [6],
δ
δu α =
∂ ∞
∂u α + ∑ (−1) s ∂
D i1 ··· D is
s=0
∂u α α = 1...m (5.1)
i 1 ...i s
where we have summation over repeted indicies i 1 ...i s for every s.
Exampel 5.1
Consider the Euler-Lagrange operator acting on a function f = f (x,u,u (1) ,u (2) )
where x = (x,y,z) are the independent variables, u is the only differential variable
37

CHAPTER 5. OPERATORS
and ( f ) = 2. I.e in (5.1) we have α = 1,i = 3 and s = 2. The Euler-Lagrange
operator takes the form
δ
δu = ∂ [
]
∂u + (−1) ∂ ∂ ∂
D x + D y + D z
∂u x ∂u y ∂u z
+ (−1) 2[ D 2 ∂
x + D 2 ∂
y + D 2 ∂
z
∂u xx ∂u yy ∂u zz
∂ ∂ ∂
]
+ D x D y + D x D z + D y D z . (5.2)
∂u xy ∂u xz ∂u yz
End of Example 4.1
Lemma 5.1
The total derivative and ∂/∂u α commutes [6], i.e
∂
[D i ,
∂u α ] ≡ D ∂
i
∂u α − ∂
∂u α D i = 0 (5.3)
This is veryfied by direct calculation,
D i
∂
∂u α ≡ ( ∂
∂x i + ∂
uα i
∂u α + uα i j
= ∂2
∂x i ∂u α + uα i
∂
∂u α D i ≡
∂
∂u α ( ∂
∂x i + uα i
= ∂2
∂ 2
∂
∂u α j
∂ 2 u α + uα i j
∂
∂u α + uα i j
+ ···) ∂
∂u α
∂u α j
∂
∂ 2
∂u α j
∂ 2
∂uα
+ ···
+ ···)
∂u α ∂x i + ∂uα i ∂
}{{}
∂u α ∂u α + uα i
∂ 2 u α + ∂uα i j
}{{}
∂u α
0
0
= ∂2
∂x i ∂u α + uα i
∂ 2
∂ 2 u α + uα i j
∂u α j
∂ 2
∂uα
+ ···
∂
∂u α j
∂ 2
+ u α i j
∂u α ∂u α j
+ ···
and Lemma 5.1 follows.
38

5.2. LIE-BÄCKLUND OPERATORS
Lemma 5.2
δ
δu α D i = 0 ∀α (5.4)
Lemma 5.1 is verified by direct calculation. The following theorem is a direct
consequense of Lemma 5.2 [6].
Theorem 5.1
A function f (x,u,u (1) ...u (s) ∈ A with independent variables x = (x 1 ... x n ) an differential
variables u = (u 1 ...u m is the divergence of a vector field H = (h 1 ...h n ),
where h i = h i (x,u,u (1) ...u (s−1) ) ∈ A i.e.
if and only if
f = div(H) ≡ D i (h i ) (5.5)
δ f
= 0. (5.6)
δuα
5.2 Lie-Bäcklund operators
Definition 5.3
A Lie-Bäcklund operator is defined by the formal sum [6],
where
with
X = ξ i ∂
∂x i + ∂
ηα
∂u α + ζα i
ξ i ([z]),η α ([z]) ∈ A
∂
∂u α i
+ ζ α i 1 i 2
∂
∂u α i 1 i 2
+ ··· , (5.7)
ζ α i = D i (W α ) + ξ j u α i j (5.8)
ζ α i 1 i 2
= D i1 D i2 (W α ) + ξ j u α ji 1 i 2
,...,
W α = η α − ξ j u α j . (5.9)
39

CHAPTER 5. OPERATORS
Note that in the Lie-Bäcklund operator the functions ξ and η also depend on the
derivatives of the differential variables, which is not the case when dealing with
point transformations.
The Lie-Bäcklund operator is the infinite order extension of X = ξ i ∂ + η α ∂
∂x i
The Lie-Bäcklund operator (5.7), is also written as [6]
∂u α .
X = ξ i D i + W α ∂
∂u α + D i(W α ) ∂
∂u α i
+ D i1 D i2 (W α ∂
)
∂u α + ... (5.10)
i 1 i 2
The Lie Bäcklund operator truncates when acting on any differential function, i.e
it belongs to A.
Indeed, let
X ν = ξν
i ∂
∂x i + ∂
ηα ν
∂u α ν = 1,2 (5.11)
be two Lie-Bäcklund operators (where the extension to infinite order is understood).
Their commutator is then given by
[X 1 , X 2 ] = X 1 X 2 − X 2 X 1 = (X 1 (ξ i 2)− X 2 (ξ i 1)) ∂
∂x i +(X 1(η α 2)− X 2 (η α 1)) ∂
∂u α (5.12)
Wich also is a Lie-Bäcklund operator. This means that all Lie-Bäcklund operators
form an infinte–dimensional Lie algebra, which will be denoted by L B [6].
5.3 Operators N i associated with Lie-Bäcklund operators and
the fundamental identity
Here we present the two last operators needed for our analysis to find conservation
laws.
40

5.3. OPERATORS N I ASSOCIATED WITH LIE-BÄCKLUND OPERATORS AND
THE FUNDAMENTAL IDENTITY
Definition 5.4
Given a Lie-Bäcklund operator X, (5.10), we define n operators N i ,i = (1...n) by
the formal sum [6]:
N i = ξ i + W α
δ
δu α i
+
∞
∑
s=1
D i1 ··· D is (W α δ
)
δu α . (5.13)
ii 1 ...i s
The fundamental identity is given in the following theorem, which is verified by
direct calculation.
Theorem 5.3
The Euler-Lagrange (5.1), Lie- Bäcklund (5.10) and the associated operators (5.13)
are connected by the identity [6],
X + D i (ξ i ) = W α δ
δu α + D iN i . (5.14)
41

Part II
Symmetries and Conservation
Laws
43

6
HAMILTON’S VARIATIONAL PRINCIPLE
6.1 Hamilton´s variational principle in the plane
We start with the basic problem of variational calculus, and treat it purly mathematically
in two dimensions only. The problem is to determine the functions y(x)
such that the integral
∫ x2
S = f (y(x),y ′ (x), x)dx (6.1)
x 1
is an extremum [1].
The quantity f depends on the dependent variable y(x) and is considered as given
for the particular problem. We will consider the limits of integration as fixed (note
that is not necessary, if we allow the limits to vary we need to find, as said above,
y(x) and x 1 , x 2 such that S is an extremum).
To find an extremum when y(x) is varied we may define a neigbouring function
as follow: We represent all neighbouring functions y in parametric form as y(α, x)
such that y(0, x) = y(x), we can then write
y(α, x) = y(0, x) + αη(x) (6.2)
where η(x) is an arbitrary function with a continuous first derivative and where
η(x 1 ),η(x 2 ) vanish. Equation (6.1) will now take the form
∫ x2
S (α) = f (y(α, x),y ′ (α, x), x)dx (6.3)
x 1
45

CHAPTER 6. HAMILTON’S VARIATIONAL PRINCIPLE
where y ′ (α, x) = ∂y ′ /∂x. A necessary condition for S (α) to have an extremum is
∂S
∂α | α=0 = 0, (6.4)
i.e that S is independent of α in first order along the path.
Performing the implied differentiation and assuming fixed boundaries yields
∂S
∂α = ∂ ∫ x2
∫ x2
( ∂ f
f (y(α, x),y ′ ∂y
(α, x), x)dx =
∂α x 1 ∂y ∂α + ∂ f ∂y ′ )
∂y ′ dx. (6.5)
∂α
Using (6.2) we see that
∂y
∂α = η(x)
∂y ′
∂α = ∂
∂α (y′ (x) + α dη(x)
dx ) = dη(x)
dx . (6.6)
Thus, (6.5) becomes
∂S
∂α = ∫ x2
x 1
( ∂ f
∂y η(x) + ∂ f dη(x)
∂y ′ dx
x 1
)
dx. (6.7)
Using integration by parts of the second term [8] in (6.7), i.e ∫ udv = uv − ∫ vdu
(with ∂ f
∂y
= u and dη(x)
′ dx
= dv), yields
∫ x2
x 1
∂ f dη(x)
∂y ′ dx dx = ∂ f
∂y ′ η(x)|x 2
x 1
−
∫ x2
x 1
η(x) d ∂ f
dx. (6.8)
dx ∂y ′
Using the fact that η(x) vanishes on the boundaries, we can write (6.7) as
∫
∂S x2
( ∂ f
∂α = x 1 ∂y η(x) − η(x) d ) ∫
∂ f
x2
( ∂ f
dx ∂y ′ dx =
x 1 ∂y − d )
∂ f
dx ∂y ′ η(x)dx. (6.9)
The condition that ∂S
∂α | α=0 = 0 and the fact that η(x) is an arbitrary function, we
must have that
( ∂ f
∂y − d )
∂ f
dx ∂y ′ | α=0 = 0 (6.10)
which is a necessary condition for S to have an extremum. Equation (6.10) is the
Euler-Lagrange equation in the plane.
46

6.1. HAMILTON´S VARIATIONAL PRINCIPLE IN THE PLANE
Example 6.1
We begin with a quite mathematical example. Suppose that we want to find the
shortest distance between two points in a plane, x 1 and x 2 . An arc lenght ds in a
plane is [8]
ds = √ √
( ) dy 2
dx 2 + dy 2 = 1 + dx. (6.11)
dx
The total lenght of a curve between x 1 and x 2 is then given by
√
∫ x 2 ∫ x 2 ( ) dy 2
I = ds = 1 + dx. (6.12)
x 1 dx
x 1
We have an extremum problem where we want to find the curve which gives the
minimum path between x 1 and x 2 , that is, the minimum of I. The function in (6.1)
is here given by
√
( ) dy 2
f = 1 + . (6.13)
dx
Using equation (6.10) we have
∂ f
∂y = 0,
∂ f
∂y ′ = y ′
√ . (6.14)
1 + y ′2
Thus, we arrive at the equation
( )
d y ′
√ = 0, (6.15)
dx 1 + y ′2
or
y ′
√
1 + y ′2 = c
⇒ y ′ =
c
√ = a, (6.16)
1 − c 2
where a is some other constant related to the constant c according to equation
(6.16). It is easy to see, by direct integration, that this is simply the equation of a
straight line
y = ax + b. (6.17)
Thus, the shortest distance between two points in a plane is a straight line!
47

CHAPTER 6. HAMILTON’S VARIATIONAL PRINCIPLE
Example 6.2: The brachistochrone problem
End of Example 6.1
Consider a particle falling from rest from a higher point to a lower point along a
curve, under the influence of gravity. The problem is to find the curve for which
the fall takes the shortest time [1].
Let the speed of the particle along the curve be v and an arc lenght of the curve
is ds as in example 6.1. Since the only external force acting on the particle i the
gravity, we get from conservation of energy that
mgy = 1 2 mv2 . (6.18)
I.e, gravitational potential energy transforms to kinetic energy. Solving this equation
for v, we get
v = √ 2gy. (6.19)
Now, the time it takes for a particle to travel from a point P 1 to P 2 is give by the
integral
√
∫ P2
∫
ds P2
t 12 =
P 1 v = 1 + y ′2
dx. (6.20)
P 1 2gy
Thus, equation (6.20) is a extremum problem and the function to be varied is
f (x,y,y ′ ) =
√
1 + y ′2
2gy . (6.21)
Now we would normaly directly use the Euler-Lagrange equation, (6.10) which is
the condition for (6.20) to have an extremum. However, f (x,y,y ′ ) does not have
any explicit dependence on x, therefore ∂ f /∂x = 0, which we can take advantage
of to simplify the calculations in the following way:
Take the total derivative of f with respect to x yields,
D x f ≡ d f
dx = ∂ f
∂x + y′ ∂ f
∂y + y′′ ∂ f
∂y ′ . (6.22)
48

6.1. HAMILTON´S VARIATIONAL PRINCIPLE IN THE PLANE
Solving this for the ∂ f /∂y term gives
y ′ ∂ f
∂y = d f
dx − ∂ f
∂x − y′′ ∂ f
∂y ′ . (6.23)
Now, multiplying (6.10) with y ′ gives
y ′ ∂ f
∂y − d ∂ f
y′ = 0. (6.24)
dx ∂y ′
Substitute (6.23) into (6.24) yields,
d f
dx − ∂ f
∂x − y′′ ∂ f
− ∂ f
∂x + d dx
In the case ∂ f /∂x = 0 we get
∂y ′ − d ∂ f
y′
dx ∂y
(
′
f − y ′ ∂ f )
∂y ′ = 0 (6.25)
f − y ′ ∂ f = C, (6.26)
∂y ′
where C is a constant of integration. Equation (6.26) was dicovered 1868 by
Beltrami and is known as the Beltrami identity in the plane, [12] Using this in our
case we get
1
√ √ = C. (6.27)
2gy 1 + y ′2
Squaring both sides and rearranging, we get
(1 + y ′2 )y = 1
2gC 2 = k2 , (6.28)
where kr is a new constant. The solution to this equation is given by the parametric
equations
x = 1 2 k2 (t − sint)
y = 1 2 k2 (1 − cost). (6.29)
This is the equations of a cycloid. Thus, the particle will fall from point P 1 to P 2
in the shortest time if it follows the path of a cycloid.
End of Example 6.2
49

CHAPTER 6. HAMILTON’S VARIATIONAL PRINCIPLE
6.1.1 Euler-Lagrange equation with several variables
Now we consider the case of several independent variables x = (x 1 ... x n ) and differential
variables u = (u 1 ...u m ). Consider a Lagrangian L ∈ A that is a differential
function of first order an an arbitrary volume V ⊂ R n , spanned by x = (x 1 ... x n )
with boundary ∂V. An action is the integral [6]
∫
l[u] = L(x,u,u (1) )dx. (6.30)
V
Consider a small variation h(x) in u which leads to a Lagrangian L = L(x,u +
h,u (1) + h (1) ). Taylor expanding this for small variations up to the linear part in h
yields
L(x,u+h,u (1)+h(1) ) = L(x,u+0,u (1) +0)+ ∂L
∂u α (hα −0)+ ∂L
∂u α (h α i −0)+.... (6.31)
i
The variation δl[u] caused by the variation u+h(x), defined as the principal linear
part in h of the integral
∫
V
takes the form
[L(x,u + h,u (1) + h (1) ) − L(x,u,u (1) )]dx (6.32)
∫
δl[u] =
V
[ ∂L
∂u α (hα ) + ∂L ]
∂u α (h α i ) dx. (6.33)
i
Integration of second term by parts yields
∫
∫ [ ] ∫ [ ]
∂L
∂L
∂L
V ∂u α (h α i )dx = D i
i
V ∂u α (h α ) dx − D i
i
V ∂u α (h α )dx. (6.34)
i
Leading to
∫ ( ∂L
δl[u] =
V ∂u α − D ∂L
i
∂u α i
) ∫ [ ] ∂L
h α dx + D i
V ∂u α (h α ) dx. (6.35)
i
Remark: The first term on RHS of equation (6.34) might seem strange, but remeber
that in basic calculus of one variable we have ∫ d f (x)
dx
dx = f (x), same rule
apply here but in this case f = f (x,u,u (1) ) = ∂L
∂u α (h α ) and d i
dx is replaced by D i.
50

6.1. HAMILTON´S VARIATIONAL PRINCIPLE IN THE PLANE
Recall the divergence theorem [8]
∫
∫
D i f (x,u,u (1),... )dx = f (x,u,u (1),... )ν i dx i = 1...n (6.36)
V
∂V
where f is an arbitrary function within a volume V with boundary ∂V and unit
outer normal ν = ν 1 ...ν n .
Using the divergence theorem, (6.35) can be written as
∫
δl[u] =
V
( ∂L
∂u α − D ∂L
i
∂u α i
) ∫
h α dx +
∂V
∂L
(h α )ν i dx. (6.37)
Provided that the functions h α (x) vanish on the boundary ∂V, we get
∫ ( )
∂L
δl[u] =
V ∂u α − D ∂L
i h α dx. (6.38)
∂u α i
A function u = u(x) is an extremum of the variational integral (6.30) if δl[u] = 0.
For an arbitrary volume V and increment h(x), that vanish on ∂V, we must have
that
∂L
∂u α − D ∂L
i = 0. (6.39)
∂u α i
∂u α i
The variational derivate
δL
δu α ≡ ∂L
∂u α − D i
∂L
∂u α i
(6.40)
is called the Euler-Lagrange equations.
51

7
NOETHER’S THEOREM AND
CONSERVATION LAWS
In this chapter we will formulate Noether’s theorem using the mathematical tools
we have developed so far, see also [13]. In the next chapter we will also explain
how one can find a Lagrangian for any system of differential equations by using
the theory of adjoint equations, leading to a more general conservation theorem.
7.1 Noether´s Theorem
Definition 7.1
Let G be a one-parameter group of transformations [6]
¯x i = f i (x,u,a) ū α = ϕ(x,u,a) (7.1)
where i = 1...n and α = 1...m, with generator
X = ξ i (x,u) ∂
∂x i + ηα (x,u) ∂
∂u α . (7.2)
A variational integral (6.30), l[u] = ∫ V L(x,u,u (1))dx is said to be invariant under
G if
∫
∫
L( ¯x,ū,ū (1) )d ¯x = L(x,u,u (1) )dx (7.3)
¯V
V
where ¯V ⊂ R n is the volume obtained from V by transformation (7.1) [6].
53

CHAPTER 7. NOETHER’S THEOREM AND CONSERVATION LAWS
Lemma 7.1
The variational integral (6.30) is invariant under the group G if and only if ([6])
X(L) + LD i (ξ i ) = 0, (7.4)
where X is the extension of (7.2), i.e
X(L) = ξ i ∂L ∂L
+ ηα
∂xi ∂u α + ζα i
where
ζ α i = D i (η α ) − u α j D i (ξ j )
∂L
∂u α i
(7.5)
Proof of Lemma 7.1:
Recall from calculus of several variables that for a change of variables, which
have a one-to-one transformation from a volume ¯V to a volume V, the volume
elements given by
d ¯V = JdV (7.6)
where J is the Jacobian for the transformation. Using this we can rewrite LHS of
(7.3) in the form
∫
∫
L( ¯x,ū,ū (1) )d ¯x = L( ¯x,ū,ū (1) )Jdx, (7.7)
¯V
V
where J = det‖D j ( ¯x i )‖ is the Jacobian of the first transformation in (7.1). Since
both integrals now is taken over the same arbitrary volume V, the integral equation
(7.3) is eqvivalent to
L( ¯x,ū,ū (1) )J = L(x,u,u (1) ). (7.8)
Consider the infinitesimal transformations
¯x i = x i + aξ i ū α = u α + aη α ū α i = u α i + aζ α i . (7.9)
54

7.1. NOETHER´S THEOREM
Taylor expansion of L( ¯x,ū,ū (1) ) around a = 0 yields,
L( ¯x,ū,ū (1) ) ≈ L( ¯x,ū,ū (1) )| a=0 + ∂L
∂ ¯x i ∂ ¯x i
∂a | a=0 · (a − 0)
+ ∂L
∂ū α ∂ū α
∂a | a=0 · (a − 0) + ∂L
∂ū α i
∂ū α i
∂a | a=0 · (a − 0)
= L(x,u,u (1) ) + ∂L
∂x i ξi a + ∂L
∂u α ηα a + ∂L
∂u α ζi α a
i
= L(x,u,u (1) ) + a[X(L)]. (7.10)
Taylor expansion of J around a = 0 yields
J ≈ J| a=0 + ∂J
∂a | a=0 · a
= det‖D j (x i )‖ + det‖D j ( ∂ ¯xi
∂a )‖ a=0
= 1 + aD i (ξ i ). (7.11)
Thus, the LHS of equation (7.8) yields
L( ¯x,ū,ū (1) )J ≈ ( L(x,u,u (1) ) + a[X(L)] )( 1 + aD i (ξ i ) )
≈ L(x,u,u (1) ) + a[X(L)] + aL(x,u,u (1) )D i (ξ i )
= L(x,u,u (1) ) + a [ [X(L)] + LD i (ξ i ) ] . (7.12)
This should be equal to RHS of (7.8). It follows that
[X(L)] + LD i (ξ i ) = 0.
□
Definition 7.2
A vector T i = (T 1 ...T n ) is a conserved vector if ([6])
D i (T i ) = 0 (7.13)
55

CHAPTER 7. NOETHER’S THEOREM AND CONSERVATION LAWS
Theorem 7.1
Let the variational integral (6.30) be invariant under the group G, with generator
(7.2). Then the vector T i ∈ A defined by
T i = Lξ i + (η α − ξ j u α j ) ∂L
∂u α i
i = (i,...n) (7.14)
is a conserved vector of the Euler-Lagrange equation, i.e D i (T i ) = 0 on the solutions
of Euler-Lagrange equations [6].
Proof of Theorem 7.1:
If the variational integral (6.30) is invariant it fulfills [X(L)] − LD i (ξ i ) = 0 according
to Lemma 7.1. By rewriting the operator (7.2) according to (5.10), i.e
X = ξ i D i + W α ∂
∂u α + D i(W α ) ∂
Straightforward calculations yield
∂u α i
. (7.15)
X(L) + LD i (ξ i ) = ξ i D i (L) + W α ∂L
∂u α + D i(W α ) ∂L
∂u α + LD i (ξ i )
i
= ξ i D i (L) + LD i (ξ i ) + W α ∂L
∂u α + D i(W α ∂L
∂u α )
i
− W α D i ( ∂L
∂u α )
i
= D i (Lξ i ) + D i (W α ∂L
∂u α ) + W α ( ∂L
i ∂u α − D ∂L
i
∂u α )
i
= D i (Lξ i + W α ∂L
∂u α ) + W α δL
i δu α . (7.16)
We know that for the solutions to the Euler-Lagrange equations we have that δL
δu α =
0 then, since
Hence
X(L) + LD i (ξ i ) ≡ D i (Lξ i + W α ∂L
∂u α ) + W α δL
i δu α = 0.
D i (Lξ i + W α ∂L
∂u α ) = 0. (7.17)
i
56

7.1. NOETHER´S THEOREM
According to (7.13) in definition 7.2 it follows that T i = Lξ i + (η α − ξ j u α j ) ∂L
∂u α i
conserved vector.
is a
According to Theorem 5.1 one can add any divergence type function F to the
Lagragian. The invariance condition (7.4) is then replaced by the divergence relation,
[6],
X(L) + LD i (ξ i ) = D i (B i ). (7.18)
Example 7.1
The differential equation, taken from [4]
y ′′ + 1 x (y′ + y ′3 ) = 0 (7.19)
has the Lagrangian
L = x √ 1 + y ′2 (7.20)
and admits the two-parameter group with generators
X 1 = ∂ ∂y , X 2 = x ∂ ∂x + y ∂ ∂y . (7.21)
In order to apply Noether’s theorem to find the conservation laws, we need to
check that the invariance condition (7.4) in lemma 7.1 is fullfilled. I.e, that
XL − LD i (ξ i ) = 0, (7.22)
where the extension of X is understood. For X 1 we have the twice extended generator
∂
Y 1 = X 1 + ζ 1
∂y ′ + ζ ∂
2
∂y ′′ (7.23)
The prolongation formulas is given by equation (4.24) in chapter 4. In this case
the prolongations are
ζ 1 = D x (η) − y ′ D x (ξ) = D x (1) − y ′ D x (0) = 0
ζ 2 = D x (ζ 1 ) − y ′′ D x (ξ) = D x (0) − y ′ D x (0) = 0. (7.24)
□
57

CHAPTER 7. NOETHER’S THEOREM AND CONSERVATION LAWS
The invariance test yields,
∂
∂y (L) − LD x(0) = 0 (7.25)
For X 2 we have the twice extended generator
∂
Y 2 = X 2 + ζ 1
∂y ′ + ζ ∂
2 . (7.26)
∂y ′′
The prolongations are in this case
ζ 1 = D x (y) − y ′ D x (x) = y ′ − y ′ = 0
ζ 2 = D x (ζ 1 ) − y ′′ D x (x) = o − y ′′ = −y ′′ . (7.27)
Thus, the extended generator is
Y 2 = x ∂ ∂x + y ∂ ∂y − ∂
y′′ . (7.28)
∂y ′′
The invariance test yields
(
x ∂ ∂x + y ∂ )
∂y − ∂ ( y′′
∂y ′′ x √ )
1 + y ′2 −
(x √ )
1 + y ′2 = L − L = 0. (7.29)
Thus, Noether’s theorem is applicabible to the two-parameter group.
Example 7.2
The nonlinear equation of motion for a pendulum is given by [14]
End of Example 7.1
y ′′ + ω 2 siny = 0, (7.30)
where ω 2 is a constant and y = y(t). The Lagrangian to equation (7.30) is given by
L = y′2
2 + ω2 cosy. (7.31)
Equation (7.30) does not contain any explicit dependence on t, consecuently, it is
invariant under time translation, and therefore has the generator
X = ∂ ∂t
(7.32)
58

7.1. NOETHER´S THEOREM
The conserved quantity associated with this symmetry is given by equation (7.15)
in theorem 7.1, thus
T i = Lξ i + (η α − ξ j u α j ) ∂L
∂u α , (7.33)
i
where, in this case, ξ i = ξ = 1, η α = 0,∀α, u α = y and u α i = y ′ hence,
T = L − y ′ ∂L
∂y ′ . (7.34)
Substituting in the expression for the Lagrangian and the fact that ∂L/∂y ′ = y ′ , we
get the conserved quantity
T = ω 2 cosy − y′2
2 . (7.35)
Theorem 7.2
End of Example 7.2
Let L(x,u,u (1) ,...,u (s) ) ∈ A be a differential function of any order. The Euler-
Lagrange equations are, using equation (5.1),
δL
= 0 α = 1,...,m. (7.36)
δuα Let X be a symmetry such that
X(L) + LD i (ξ i ) = D i (B i ), B i ∈ A. (7.37)
Then equations (7.36) admit a conservation law, D i (C i ) = 0, defined by
C i = N i (L) − B i , i = 1,...,n, (7.38)
where the operators N i are defined by (5.13) [6].
Proof of Theorem 7.2:
This follows directly from the fundamental identity (5.14). Indeed,
D i (C i ) = D i N i (L) − D i B i = 0
⇒ D i N i (L) = D i B i = X(L) + LD i (ξ i )
where the fact that δL
δu α = 0 was used.
59

CHAPTER 7. NOETHER’S THEOREM AND CONSERVATION LAWS
□
7.2 Many symmetries
If a differential equation has r symmetries X 1 ,..., X r of the form (7.2),i.e
X µ = ξ i µ(x,u) ∂
∂x i + ηα µ(x,u) ∂
∂u α µ = 1,...,r (7.39)
then (7.14) provides r conserved vectors T 1 ,...,T r with components
T i µ = Lξ i µ + (η α µ − ξ i µu α j ) ∂L
∂u α i
µ = 1,...r. (7.40)
See [6].
Example 7.3
This example is taken from [14].
The motion of a free particle in special relativity is described by the relativistic
Lagragian
L = −mc 2√ 1 − β 2 , (7.41)
where β 2 = |v|2 , |v| 2 =
c
∑ 3 2 i=1 (vi ) 2 . The space coordinates of the particle is x =
(x 1 , x 2 , x 3 ) and the velocity is v = (v 1 ,v 2 ,v 3 ) such that v = dx
dt
, where t is time.
We have that x is our differential variables and t is our only independent variable.
The generators of the Lorentz group (see e.g Classical dynamics) is given by
X 0 = ∂ ∂t
X i = ∂
∂x i
X i j = x j ∂
∂x i − ∂
xi
∂x j
X 0i = t ∂
∂x i + 1 c 2 xi ∂ ∂t . (7.42)
60

7.2. MANY SYMMETRIES
By applying Noether´s theorem to each of these generators with the given Lagragian
we wish to find the corresponding conservation law. We have generators of
the form
X = ξ(t,x) ∂ ∂t + ηα (t,x) ∂
∂x α α = 1,2,3. (7.43)
We will get a conserved quantity of the form
where
T = ξL + (η α − ξv α ) ∂L
∂v α , (7.44)
∂L
∂v α = 1
mvα γ γ = √ . (7.45)
1 − β 2
For X 0 = ∂ ∂t we have ξ = 1,η1 = η 2 = η 3 = 0. Equation (7.44) yields
T = L − v α ∂L
∂v α
= −mc 2 γ −1 − m|v| 2 γ
= −mc 2 γ. (7.46)
Putting T = −E we get E = mc 2 γ, i.e conservation of energy.
For X i = ∂
∂x i we get a conserved vector T = T i = (T 1 ,T 2 ,T 3 ). Here we have ξ = 0
for all i,while η α = δ iα . Equation (7.44) yields
T 1 = ∂L
∂v 1 = mv1 γ
T 2 = ∂L
∂v 2 = mv2 γ
T 3 = ∂L
∂v 3 = mv3 γ (7.47)
Leading to the conserved vector T = mvγ, i.e conservation of relativistic momentum.
61

CHAPTER 7. NOETHER’S THEOREM AND CONSERVATION LAWS
For the generator X i j = x j ∂ − x i ∂ , we have a rotational symmetry. With ξ = 0
∂x i ∂x j
and η α = x j δ αi − x i δ α j . Equation (7.44) yields
T 12 = x 2 ∂L ∂L
− x1
∂v1 ∂v 2
= mγ(x 2 v 1 − x 1 v 2 )
T 23 = x 3 ∂L ∂L
− x2
∂v2 ∂v 3
= mγ(x 3 v 2 − x 2 v 3 )
T 31 = x 1 ∂L ∂L
− x3
∂v3 ∂v 1
= mγ(x 1 v 3 − x 3 v 1 ). (7.48)
We get the conserved vector T = (T 23 ,T 31 ,T 12 ). Putting T = −L we get the conserved
vector L = m(x×vγ) = x×p rel , i.e conservation of angular momentum.
For the generator X 0i = t ∂
∂x i + 1 c 2 x i ∂ ∂t , we get a vector T = (T 1 ,T 2 ,T 3 ). For the
first component we have ξ = 1 c 2 x 1 ,η 1 = t,η 2 = η 3 = 0. Equation (7.44) yields
T 01 = L 1 c 2 x1 + (t − 1 c 2 x1 v 1 ) ∂L
∂v 1
= L 1 c 2 x1 + tmv 1 γ − 1 c 2 x1 (v 1 ) 2 γ
= m γ (−x1 + tv 1 γ 2 − x 1 [βγ] 2 )
= mγ(tv 1 − x 1 ). (7.49)
A similar procedure for the remaining two components and putting T = Q, leads
to the conserved vector Q = mγ(tv−x), i.e conservation of relative center of mass.
End of Example 7.3
62

8
ADJOINT EQUATIONS
In this chapter the very basic theory of adjoint equations is discussed. We start
with linear adjoint equations and then we consider adjoint equations for arbitrary
differential equations, as made in [15]. The latter will be used to find a Lagrangian
for any arbitrary physical system described by arbitrary differential equations.
8.1 Linear adjoint equations
To get a feeling for the concept of adjoint equations we start by considering linear
adjoint equations before we move on to arbitrary differential equations.
Definition 8.1
Let L be a linear differential operator of any order. An adjoint operator L ∗ to L is
given by [4]
vL[u] − uL ∗ [v] = D i (P i ) (8.1)
∀u,v where P = (p 1 ,..., p n ) is a vector field. The equation L ∗ [v] = 0 is termed the
adjoint equation to L[u] = 0.
63

CHAPTER 8. ADJOINT EQUATIONS
Consider a general second-order operator L [15],
L = a i j (x)D i D j + b i (x)D i + c(x) (8.2)
where i, j = 1,...,n and the coefficients are symetric in i, j, i.e a i j = a ji .
Theorem 8.1
The adjoint operator L ∗ to L is uniquely determined and has the form, see [15],
L ∗ [v] = D i D j (a i j v) − D i (b i v) + cv (8.3)
Proof of Theorem 8.1:
Let u,v be two differential functions. Then
vL[u] = va i j (x)D i D j u+vb i (x)D i u+vc(x)u
} {{ } } {{ } } {{ }
1
2
3
= D i (va i j (x)D j u) − D i (va i j (x))D j u
} {{ }
1
+ D i (vb i (x)u) − uD i (vb i (x)) +vc(x)u.
} {{ } } {{ }
2
3
The second term in 1 can be rewritten as
−D i (va i j (x))D j u = −D j (uD i (va i j (x))) + uD i D j (a i j (x)v)
= −D i (uD j (va i j (x))) + uD i D j (a i j (x)v).
(8.4)
In the last step we changed the index of i, j and used the fact that the coefficients
are symmetric. Hence
vL[u] = D i (va i j (x)D j u) − D i (uD j (va i j (x))) + uD i D j (a i j (x)v)
+ D i (vb i (x)u) − uD i (vb i (x)) + vc(x)u
= u [ D i D j (a i j (x)v) − D i (vb i (x)) + c(x)v ]
+ D i
[
va
i j (x)u j − uD j (va i j (x)) + vb i (x)u ]
⇒
vL[u] − u [ D i D j (a i j (x)v) − D i (vb i (x)) + c(x)v ]
= D i
[
va
i j (x)u j − uD j (va i j (x)) + vb i (x)u ] . (8.5)
64

8.1. LINEAR ADJOINT EQUATIONS
Comparing this with (8.1), we see that the adjoint operator L ∗ has the form
L ∗ [v] = D i D j (a i j (x)v) − D i (vb i (x)) + c(x)v
and it follows that
P i = va i j (x)u j − uD j (va i j (x)) + vb i (x)u. (8.6)
The same result applies if u,v are m-dimensional vector functions, and the coefficients
are m × m-matrices according to [15].
Definition 8.2
An operator is said to be self adjoint if, for any function u,
L[u] = L ∗ [u], (8.7)
see [15] or [4].
Theorem 8.2
The operator L = a i j (x)D i D j + b i (x)D i + c(x)is self adjoint if and only if
b i (x) = D j (a i j ) (8.8)
Proof of Theorem 8.2:
From Definition 8.1 we have that if L is self-adjoint, i.e according to Definition
8.2 L[u] = L ∗ [v] when we put v = u, we must have that D i (P i ) vanish. Thus, for
v = u
D i (P i ) ≡ D i (ua i j (x)u j − uD j (ua i j (x)) + ub i (x)u)
= D i u [ a i j D j u − D j (ua i j ) + b i u ]
= D i u [ a i j D j u − a i j D j u − uD j a i j + b i u ]
= D i u [ u ( b i − D j a i j)] = 0 (8.9)
has to be true for any u, and equation (8.8) follows. For alternative proof see [4].
□
□
65

CHAPTER 8. ADJOINT EQUATIONS
8.2 Adjoint equations of arbitrary differential equations
We will now consider adjoint equations for arbitrary differential equations, to be
used in the construction of Lagrangians as made in [15] and [3].
Definition 8.3
Consider a system of sth-order partial differential equations,
F α (x,u,u (1) ,...,u (s) ) = 0 α = 1,...,m (8.10)
where F α (x,u,u (1) ,...,u (s) ) ∈ A are differential functions with n independent variables
x = (x 1 ,..., x n ) and m dependent variables u = (u 1 ,...,u m ),u = u(x). The
system of adjoint equations to (8.10) is defined by
F ∗ α(x,u,v,...,u (s) ,v (s) ) ≡ δ(vβ F β )
δu α = 0 α = 1,...,m (8.11)
where v = (v 1 ,...,v m ) are new dependent variables v = v(x) [15].
Definition 8.4
A system of equations (8.10) is said to be self adjoint if the system obtained from
the adjoint equations (8.11) by the substitution v = u:
F ∗ α(x,u,u,...,u (s) ,u (s) ) = 0 α = 1,...,m (8.12)
is identical with the original system (8.10) [15].
Exampel 8.1
The heat equation is
u t − c(x)u xx = 0, (8.13)
where c(x) is some arbitrary function with correct dimensions. To find the adjoint
equation we use Definition 8.3. Here we have that
F = F(x,t,u,u (1) ,u (2) ) = c(x)u xx − u t = 0, (8.14)
66

8.2. ADJOINT EQUATIONS OF ARBITRARY DIFFERENTIAL EQUATIONS
where x,t are independent variables, and u is the differential variable. By definition
we obtain the adjoint equation as
F ∗ = F ∗ (x,t,u,v,u (1) ,u (2) ,v (1) ,v (2) ) = δ
δu (v[c(x)u xx − u t ])
[ ]
∂
≡
∂u − D ∂ ∂
t − D x + D 2 ∂
x + D 2 ∂ ∂
t + D x D t
∂u t ∂u x ∂u xx ∂u tt ∂u xt
· (v[c(x)u xx − u t ])
[
]
∂
= −D t + D 2 ∂
x (v[c(x)u xx − u t ])
∂u t ∂u xx
= −D t (−v) + D 2 x(c(x)v) = v t + (cv) xx = 0, (8.15)
which is the adjoint equation to u t − c(x)u xx = 0 with new dependent variable v.
End of Example 8.1
Exampel 8.2
The Kortweg-de Vries equation is a mathematical model of waves on shallow
water surfaces. It has the form, see [6] and [15],
u t = u xxx + uu x . (8.16)
Here we have that
F = F(t, x,u,u (1) ,u (2) ,u (3) ) = u t − u xxx − uu x = 0. (8.17)
The adjoint equation is
F ∗ (t, x,u,v,u (1) ,u (2) ,u (3) ,v (1) ,v (2) ,v (3) ) = δ
δu (v[u t − u xxx − uu x ])
[ ]
∂
=
∂u − D ∂ ∂
t − D x − D 3 ∂
x (v[u t − u xxx − uu x ])
∂u t ∂u x ∂u xxx
= −vu x − D t v + D x (vu) + D 3 xv
= −vu x − v t + v x u + vu x + v xxx
= −v t + v x u + v xxx . (8.18)
67

CHAPTER 8. ADJOINT EQUATIONS
The adjoint equation to u t = u xxx +uu x is v t = v x u+v xxx . Furthermore, if we replace
v by u in the adjoint equation, we have that
F ∗ (t, x,u,u,u (1) ,u (2) ,u (3) ,u (1) ,u (2) ,u (3) )
= −(u t − u xxx − uu x ) = −F(t, x,u,u (1) ,u (2) ,u (3) ). (8.19)
Thus, according to Definition 8.4, Korteweg-de Vries equation is self adjoint.
End of Example 8.2
8.3 Constructing Lagrangians by the use of adjoint equation
The following theorem provides us with a powerful method to construct a Lagrangian
for a system of arbitrary differential equations, if they are considered
with its adjoint system. The only restriction is that the number of equations has to
be equal to the number of dependent variables. This is the main result in [15].
Theorem 8.4
Any system of sth-order differential equations (8.10),
F α (x,u,u (1) ,...,u (s) ) = 0
α = 1,...,m
considered together with its adjoint equation (8.11),
F ∗ α(x,u,v,...,u (s) ,v (s) ) ≡ δ(vβ F β )
δu α = 0 α = 1,...,m
has a Lagrangian. The simultaneous system (8.10) and (8.11) with 2m dependent
variables, u = (u 1 ,...,u m ) and v = (v 1 ,...,v m ), is the system of Euler-Lagrange
equations with a Lagrangian L defined by
L = v β F β . (8.20)
This is easy to verify, since we have
δL
δv α = F α(x,u,u (1) ,...,u (s) ) = 0, (8.21)
68

8.3. CONSTRUCTING LAGRANGIANS BY THE USE OF ADJOINT EQUATION
and
δL
δu α = F∗ α(x,u,v,...,u (s) ,v (s) ) = 0, (8.22)
i.e The Euler-Lagrange equation is satisfied.
Example 8.3
Consider example 8.1. The heat equation considered togheter with its adjoint
equation has a Lagrangian, namely
L = vu t − c(x)u x . (8.23)
Indeed, the Euler-Lagrange are
δL
δv = u t − c(x)u x = 0
δL
δu = δv[u t − c(x)u x ]
≡ F ∗ (t, x,u,v,u (1) ,u (2) ,v (1) ,v (2)
δu
= v t + (cv) xx = 0, (8.24)
i.e the Lagrangian (8.23) satifies the Euler-Lagrange equation.
End of Example 8.3
69

9
SYMMETRIES OF DIFFERENTIAL
EQUATIONS AND THE MAIN
CONSERVATION THEOREM
In this chapter we clarify the important step of finding symmetries from a physical
system described by differential equation. Some symmetries can easily be
found by using the invariance condition, that is for exampel the symmetry groups
that consists of translatation and scaling transformations. However, the main concern
is to find a method that determines all symmetries of differential equations.
This is done via the determining equation. First we consider some examples of
simple symmetries obtained by using the fact that the group in interest leaves our
differential equation invariant.
Example 9.1
Consider The Riccati equation [6]
y ′ + y 2 − 2 x 2 = 0 (9.1)
i.e, a function F(x,y,y ′ ) = 0, y ′ = dy
dx
. We set our symetry group G to be a dilatiation
group ¯x = kx and ȳ = ly. Then equation (9.1) takes the form
ȳ ′ + ȳ 2 −
2¯x 2
= 0 (9.2)
71

CHAPTER 9. SYMMETRIES AND THE MAIN CONSERVATION THEOREM
i.e, F( ¯x,ȳ,ȳ ′ ) = 0. The invariance condition states that
ȳ ′ + ȳ 2 − 2¯x 2 = λ(y′ + y 2 − 2 ). (9.3)
x2 Using our dilatation group in the expression above yields,
ȳ ′ + ȳ 2 −
2¯x 2
≡
dȳ
d ¯x + ȳ2 − 2¯x 2
= l k y′ + l 2 y 2 − 2
k 2 x
(9.4)
i.e, we must have l k = l2 = 1 . It follows that l = 1 k 2 k
, where k is an arbitrary constant.
Let k = e a , we then get the dilatation group
¯x = e a x, ȳ = e −a y. (9.5)
The Lie equations yields
d ¯x
da | a=0 = x
dȳ
da | a=0 = −y. (9.6)
It follows that the Riccati equation admitts the generator (symmetry)
X = x ∂ ∂x − y ∂ ∂y . (9.7)
Example 9.2
Consider the equation of motion for a simple harmonic oscillator [1]
End of Example 9.1
ẍ + ω 2 x = 0, (9.8)
where ω is a physical constant. It does not have an explicit dependence on t, and
is therefore invariant under the group of time translations, ¯t = t + a. It follows that
(9.8) admits the generator (symmetry)
X = ∂ ∂t . (9.9)
End of Example 9.2
72

We shall now present a more general method for finding all symmetries of differential
equations. The main requirement for finding symmetries is that the differential
equations are invariant under a group G of point transformations.
Consider a kth-order differential function
F σ (x,u,u (1) ...u (k) ) = 0 σ = 1... s, (9.10)
where F σ ∈ A, x = (x 1 ,..., x n ) and u = (u 1 ,...,u m ).
Definition 9.1
Given any differential function F ∈ A, ord(F) = s, the equation
F(x,u,u (1) ...u (s) ) = 0 (9.11)
defines a manifold in the space of variables x,u,u (1) ...u (s) . This manifold is called
the frame of the sth-order partial differential equation [6].
This can be seen as a space spanned by the variables x,u,u (1) ...u (s) , i.e a (s + 2)-
dimensional space, in which we have a (s + 1)-dimensional manifold [6].
Definition 9.2
A system of partial differential equations (9.10), is said to be locally solvable if,
for any generic point z 0 = (x 0 ,u 0 ,u 0 (1) ...,u0 (k) ) lying on its frame F σ(x 0 ,u 0 ,u 0 (1) ...,u0 (k) ) =
0, there exist a solution u = φ(x) of the system passing through z 0 such that φ(x)
and its derivities assumes the values u 0 ,u 0 (1) ...,u0 (k) at x = x0 [6].
The symmetry of the differential equation can be found by concerning its solution
or its frame [6]. In the first case we need to know the solution of the differential
equation. If the solutions are merely permuted among themselves under a group
G, we say that the differential equation is invariant under the group G of point
transformations. The solution set is described by a family of curves. [5]
In the other case, we say that the differential equation is invariant under group G
if its frame is an invariat manifold for the extension of the group G to the same
order as the differential equation. In the second case, where no solutions of the
differential equation is needed, we use determining equations to find symmetries.
73

CHAPTER 9. SYMMETRIES AND THE MAIN CONSERVATION THEOREM
9.1 Determining equations
The fact that a differential equation is invariant if its frame is an invariant manifold,
togheter with Theorem 3.3 gives us a criterion for finding a symmetry group
[6].
Theorem 9.1
The system (9.10) is invariant under the group with generator X, extended to all
its derivatives, if and only if
XF σ | (9.10) = 0 (9.12)
This equation determines all infinitesimal symmetries of the system (9.10) and is
therefore called the determining equation. As in equation (3.55) the determining
equation can be written as
XF σ = φ ν σF ν , (9.13)
where φ ν σ are differential functions bounded on the frame (9.10)[6].
The left-hand side of the determining equation is a function of all variables in
F, i.e x,u,u (1) ...u (k) togheter with the functions ξ i and η α which are functions of
x,u (they are the functions in our extended generator X). Since this should be
equal to zero we must have that the coefficients for the partial derivities u (i) must
equal zero for each i = 1...k. Thus, the determining equation breaks down to a
system of differential equation involving derivities of ξ i , η α with respect to x,u.
This system is often overdetermined and can therefore be solved analytically for
the functions ξ i and η α . When we know ξ i and η α , we also know the infintesimal
symmetry of our differential equation (9.10).
Theorem 9.2
The solutions of any determining equations form a Lie algebra [6].
74

9.1. DETERMINING EQUATIONS
Example 9.3
Consider the time independent partial differential equation from transonic gas
dynamics [6]
u x u xx + u yy = 0. (9.14)
To find the symmetries of equation (9.14) we use the determining equation (9.12),
where X is the twice extended generator given by (4.39) in chapter 4. Thus, from
the determining equation we get
u xx ζ 1 + u x ζ 11 + ζ 22 = 0. (9.15)
Inserting the expressions for ζ 1 ,ζ 11 ,ζ 22 from equation (4.42),(4.44) and (4.46) in
chapter 4, and set u yy = −u x u xx . Thus, the partial differential equation to solve for
ξ 1 ,ξ 2 ,η becomes
u xx η x + u xx u x η u − u xx u x ξ 1 x − u xx (u x ) 2 ξ 1 u − u xx u y ξ 2 x − u xx u x u y ξ 2 u
+ u x η xx + 2(u x ) 2 η xu + u x u xx η u + (u x ) 3 η uu − 2u x u xx ξ 1 x − (u x ) 2 ξ 1 xx
− 2(u x ) 3 ξ 1 xu − 3(u x ) 2 u xx ξ 1 u − (u x ) 4 ξ 1 uu − 2u x u xy ξ 2 x − u x u y ξ 2 xx
− 2(u x ) 2 u y ξ 2 xu − u x u y u xx ξ 2 u − 2(u x ) 2 u xy ξ 2 u − (u x ) 3 u y ξ 2 uu
+ η yy + 2u y η yu − u x u xx η u + (u y ) 2 η uu + 2u x u xx ξ 2 y − u y ξ 2 yy
− 2(u y ) 2 ξ 2 yu + 3u y u x u xx ξ 2 u − (u y ) 3 ξ 2 uu − 2u xy ξ 1 y − u x ξ 1 yy
− 2u x u y ξ 1 yu − (u x ) 2 u xx ξ 1 u − 2u y u xy ξ 1 u − u x (u y ) 2 ξ 1 uu = 0. (9.16)
This might seem quite complicated but this is a function of x,y,u,u x ,u y ,u xx ,u xy .
Since ξ 1 ,ξ 2 ,η only depends on x,y,u we may isolate the terms containing u x ,u y ,u xx ,u xy
and those free of these variables, and set each term equal to zero. For the terms
containing u xy we get,
Leading to
u xy
(
−2ux ξ 2 x − 2u 2 xξ 2 u − 2ξ 1 y − 2u y ξ 1 u)
= 0. (9.17)
ξ 1 y = 0, ξ 1 u = 0, ξ 2 x = 0, ξ 2 u = 0. (9.18)
For the terms containing u xx ,and insertion of (9.18), we get,
u xx
(
ηx + u x
(
ηu − 3ξ 1 x + 2ξ 2 y))
. (9.19)
75

CHAPTER 9. SYMMETRIES AND THE MAIN CONSERVATION THEOREM
Leading to
η x = 0, η u − 3ξ 1 x + 2ξ 2 y = 0. (9.20)
Now, using the results from equations (9.18) and (9.20), equation (9.16) reduces
to
Leading to
(u x ) 3 η uu − (u x ) 2 ξ 1 xx + η yy + u y (η yu − ξ 2 yy) + (u y ) 2 η uu = 0 (9.21)
η uu = 0, η yy = 0, ξ 1 xx = 0, η yu − ξ 2 yy = 0. (9.22)
Thus, the system (9.16) is now split into the overdetermined system of linear
partial differential equations (9.18),(9.20) and (9.22). This system is solved by
straight forward calculation. From equation (9.18) it follows that ξ 1 = ξ 1 (x) and
ξ 2 = ξ 2 (y). Using the fact that ξ 1 xx = 0 yields
ξ 1 = C 1 x +C 2 . (9.23)
From equation (9.20) we have that η = η(y,u) and
η u = 3ξ 1 x + 2ξ 2 y = 3C 1 + 2ξ 2 y, (9.24)
where equation (9.23) was used in the last step. From the first two equations in
(9.22) we also know that η is linear in y and u. Now, differentiate (9.24) with
respect to y and inserting it into the last equation in (9.22) yields,
thus,
ξ 2 yy = 0, (9.25)
ξ 2 = C 3 y +C 4 . (9.26)
Substituting (9.26) into (9.24) and integrate with respect to u, keeping in mind
that η is linear in y, yields,
η = (3C 1 + 2C 3 )u +C 5 y +C 6 . (9.27)
Thus, the general solution to (9.15) is
ξ 1 = C 1 x +C 2 , ξ 2 = C 3 y +C 4 , η = (3C 1 + 2C 3 )u +C 5 y +C 6 . (9.28)
76

9.1. DETERMINING EQUATIONS
This solution depends on six arbitrary constants (C 1 ,...,C 6 ). Thus, we arrive at a
six-dimensional Lie algebra spanned by the operators
X 1 = ∂ ∂x ,
X 3 = ∂ ∂u ,
X 2 = ∂ ∂y
X 4 = y ∂ ∂u
X 5 = x ∂ ∂x + 3u ∂ ∂u ,
X 6 = y ∂ ∂y − 2u ∂ ∂u
(9.29)
This is obtained by successively letting one of the constants C i be equal to one,
while the others are zero, for all i = 1,...,6.
Example 9.4
End of Example 9.3
The motion of a free particle (i.e no external forces are acting on it) in space is
described by the differential equation
∂ 2 x α
= 0 α = 1,2,3 (9.30)
∂t2 where x α = x α (t) is the dependent, or differential variables (x 1 , x 2 , x 3 ) and t is the
independent variable. To find the symmetries of the partial differential equation
(9.30) we look for a generator of the form
X = ξ(t, x 1 , x 2 , x 3 ) ∂ ∂t + ηα (t, x 1 , x 2 , x 3 ) ∂
∂x α (9.31)
and determining the tangent vector fields by solving equation (9.12) for ξ and η α .
F σ in equation (9.12) is equation (9.30) here, and the twice extended generator is
given by
X = ξ ∂ ∂t + ∂
ηα
∂x α + ζα 1
∂
∂x α 1
+ ζ α 2
∂
∂x α 2
, (9.32)
where (x1 α, xα 2
) denotes the first and second derivatives of component α respectively.
The prolongations ζ1 α and ζα 2
is given by (4.24),
ζ α 1 = D t (η α ) − x α 1 D t (ξ)
ζ α 2 = D t (ζ α 1 ) − x α 2 D t (ξ) (9.33)
77

CHAPTER 9. SYMMETRIES AND THE MAIN CONSERVATION THEOREM
Now, when (9.32) acting on (9.30), i.e,
(
ξ ∂ ∂t + ∂
) ηα
∂x α + ∂
ζα 1 + ζ2
α ∂ ∂ 2 x α
= 0, (9.34)
∂t2 ∂x α 1
∂x α 2
we arrive at the following system of equations to be solved:
ζ α 2 = 0. (9.35)
We use the prolongation formulas (9.33) to calculate ζ2 α and ζα 1
(note that the
prolongations are defined recursively, so in order to calculate ζ2 α we first need to
calculate ζ1 α).
ζ1 α = ∂ηα + x β ∂η α ( )
∂ξ
t
∂t ∂x β − xα t
∂t + ∂ξ
xγ t
∂x γ
= ∂ηα + x β ∂η α
t
∂t ∂x β − ∂ξ
xα t
∂t − xα t xt
γ ∂ξ
∂x γ , (9.36)
where the lowercase letter t on x denotes the derivatives, i.e x t ≡ ∂x/∂t. α is a
component index, while β and γ are dummy summation indicies. When calculating
ζ α 2 we use the fact that x tt ≡ ∂ 2 x/∂t 2 = 0. Note that we directly invoking the
chain rule.
ζ α 2 = D t (ζ α 1 )
( ∂η
α
= D t
∂t
) ( )
+ D t xt
β ∂η
α
(
( )
− D t x
α ∂ξ ∂ξ
t
∂t − xα t D t
∂t
( ) ∂η
α
∂x β + xβ t D t
∂x
)
β ( )
− D t x
α γ ∂ξ
t xt
∂x γ
− x α t D t
(
x
γ
t
) ∂ξ
∂x γ − xα t x γ t D t
( ∂ξ
∂x γ )
= ∂2 η α
∂t 2 + ∂ 2 η α
xβ t
∂t∂x β + ∂x xγ t
β ∂η α
tt
∂xt
γ ∂x β
+ xt
β ∂ 2 η α
∂t∂x β + xβ t xt
γ ∂ 2 η α
∂x γ ∂x β − ∂ξ
xα tt
∂t
− xt
α ∂ 2 ξ
∂t 2 − xα t xt
β ∂ 2 ξ
∂t∂x β − xα ttxt
γ ∂ξ
∂x γ
− xt α xtt
γ ∂ξ
∂x γ − xα t xt
γ ∂ 2 ξ
∂t∂x γ − xα t xt γ xt
β ∂ 2 ξ
∂x γ ∂x β . (9.37)
78

9.2. THE MAIN CONSERVATION THEOREM
Putting all x tt = 0 yields,
ζ α 2 = ∂2 η α
∂t 2
+2xβ t
∂ 2 η α
∂t∂x β + xβ t x γ t
∂ 2 η α
∂x γ ∂x β − ∂ 2 ξ
xα t
∂t 2 −2xα t xt
β
∂ 2 ξ
∂t∂x β − xα t x γ t x β t
∂ 2 ξ
∂x γ ∂x β
(9.38)
This system is split into several systems by equating to zero the terms free of first
order derivatives x t , the terms linear, quadratic and cubic in x t . Thus, we arrive at
the following system of equations:
∂ 2 η α
∂t 2 = 0
∂
∂t
(2 ∂ηα
∂x β − δα β
(
∂ ∂η
α
∂x γ ∂x β − 2δα β
)
∂ξ
∂t
∂ξ
∂t
= 0
)
= 0
∂ 2 ξ
∂x β ∂x γ = 0 (9.39)
The solution to the system (9.39) depends on 24 arbitrary constants and therefor
possess 24 infinitesimal symmetries. They form a Lie algebra spanned by the
following operators (see [6]):
X 0 = ∂ ∂t , X α = ∂
∂x α , S = t ∂ ∂t , P α = x α ∂ ∂t , Q ∂
α = t
∂x α
Y αβ = x α ∂
∂x β , Z 0 = t 2 ∂ ∂t + ∂ txβ ∂x β , Z α = x α t ∂ ∂t + xα x β ∂
∂x β (9.40)
End of Example 9.4
9.2 The main conservation theorem
In chapter 4 we introduced the concept of extended operators due to the fact that
we treat all varibles x i ,u α and all derivities as independent even if they are connected
by the relations given in (2.30) in chapter 2. To find a Lagrangian for an
arbitrary system of differential equation we used the concept of adjoint equations
and the lagrangian were constructed accordingly to theorem 8.4 in chapter 8, by
79

CHAPTER 9. SYMMETRIES AND THE MAIN CONSERVATION THEOREM
considering the simultanious system. By introducing the adjoint equations we
arrived at a system of 2m equations with a total of 2m dependent variables, m dependent
variables from the original system and m new dependent variables from
the system of adjoint equations. Now we need to show that the system of adjoint
equations inherit the symmetries of the original system.
Consider a system of m differential equations of order s,
F α (x,u,u (1) ,...,u (s) ) = 0 α = 1,...,m, (9.41)
with n independent variables x = (x 1 ,..., x n ) and m dependent variables
u = (u 1 ,...,u m ). The adjoint system is
F ∗ α(x,u,v,...,u (s) ,v (s) ) ≡ δ(vβ F β )
δu α = 0 α = 1,...,m (9.42)
where v = (v 1 ,...,v m ) are new dependent variables v = v(x). Consider a point
symmetry for the original system (9.42) with a generator of the form
X = ξ i ∂
∂x i + ∂
ηα
∂u α , (9.43)
where the prolongation to all derivities involved in equation (9.42) is understood.
The condition for equation (9.42) to be invariant is given by (3.55) in chapter 3,
XF α = λ β αF β (9.44)
The Lagrangian for the simultanious system 9.41 and 9.42 is given by [15]
L = v α F α . (9.45)
The extension of the generator (9.43) to the adjoint variables and their derivities
(which is understood) is the generator [3]
Y = ξ i ∂
∂x i + ∂
ηα
∂u α + ∂
ηα ∗
∂v α . (9.46)
The Lagrangian should remain invariant under a certian point transformation, we
can find a condition on the infinitesimal tangent vector field associated with the
adjoint variables, η α ∗ , by require the the invariance condition for the variational
integral, according to lemma 7.1, should be satisfied, i.e
Y(L) + LD i (ξ i ) = 0 (9.47)
80

9.2. THE MAIN CONSERVATION THEOREM
Direct calculations of equation (9.47), yields,
Y(L) + LD i (ξ i ) = Y(v α F α ) + v α F α D i (ξ i )
= Y(v α )F α + v α Y(F α ) + v α F α D i (ξ i )
= Y(v α )F α + v α X(F α ) + v α F α D i (ξ i )
= η α ∗ F α + v α λ β α(F β ) + v α F α D i (ξ i )
= [η α ∗ + v β λ α β + v α D i (ξ i )]F α = 0, (9.48)
where λ α β is defined by equation (9.44). To guarantee invariance of the simultanious
system we arrive at the following condition for tangent vectorfield associated
with the adjoint variables:
η α ∗ = −[v β λ α β + v α D i (ξ i )] (9.49)
Thus, if the original system admitts the operator (9.43), the adjoint system admitts
the operator
Y = ξ i ∂
∂x i + ∂
ηα
∂u α − [vβ λ α β + v α D i (ξ i )] ∂
∂v α . (9.50)
Since a operator of this form (9.50) provids a solution for the invariance condition
(9.47), which is a determining equation, we have accordingly to theorem 9.2 that
this operator form a Lie algebra. Thus, it follows that the adjoint system inherit
the symmetries of the original system. This can be formulated in a theorem.
Theorem 9.3
For a system described by equation (9.41) that admits a generator (9.43), its adjoint
system (9.42) inherits the symmetries of the original system, i.e the adjoint
system admitts the generator (9.43) extended to the adjoint variables according to
(9.50) [3].
Theorem 9.4: The main conservation theorem
Every Lie point, Lie-Bäcklund and non-local symmetry
X = ξ i (x,u,u (1) ,...) ∂
∂x i + ηα (x,u,u (1) ,...) ∂
∂u α (9.51)
81

CHAPTER 9. SYMMETRIES AND THE MAIN CONSERVATION THEOREM
of differential equations
F α (x,u,u (1) ,...,u (s) ) = 0, α = 1,...,m, (9.52)
provides a conservation law for the system of differential equation (9.52) considered
togheter with its adjoint system of equations [3].
By using the determinig equation we can find the symmetries for any system of m
differential equations. If the system of differential equations is well-posed, i.e the
number of equations is equal to the number of dependent variables, we construct a
Lagrangian by considering the original system togheter with the adjoint system of
equations. Since the adjoint system inherit the symmetries of the original system,
according to theorem 9.3, we now have a total system of 2m equations with known
symmetries and a Lagrangian. Thus, we can now use Noethers theorem to find a
conservation law. We illustrate the proccedure with an example.
Example 9.5
The mathematical formulation for the motion of a free particle in space is given
by the differential equations
mẍ = 0, (9.53)
where x = (x 1 , x 2 , x 3 ) is the spatial coordinates, and m is the mass of the particle.
Thus, we have a system of differential equations of the form
F α (t, x α , ẋ α , ẍ α ) = 0 α = 1,2,3. (9.54)
Here we have that t is the independent variable and x = x(t) is the dependent
variable. The differential variables are x α , ẋ α , ẍ α and we treat the as if they were
independent according to the definitions given in chapter 2.
Equation (9.54) is invariant under rotational symmetry with generators of the form
X i j = x j ∂
∂x i − ∂
xi
∂x j . (9.55)
82

9.2. THE MAIN CONSERVATION THEOREM
Note that this generators is the linear combination Y αβ − Y βα from example 9.3.
Thus, we have the following three generators:
X 12 = x 2 ∂
∂x 1 − ∂
x1
∂x 2
X 13 = x 3 ∂
∂x 1 − ∂
x1
∂x 3
X 23 = x 3 ∂
∂x 2 − ∂
x2
∂x 3 . (9.56)
Consider rotation around the x 3 -axis, that is generator X 12 . For that one we have
a tangent vector field with components ξ = 0, η 1 = x 2 , η 2 = −x 1 and η 3 = 0. The
twice extended generator, i.e the prolongation to the differential variables involved
in (9.53) is given by
X 12,(2) = x 2 ∂
∂x 1 − ∂
x1
∂x 2 + ∂
ζα 2
∂ẍ α α = 1,2,3, (9.57)
where ζ α 1 and ζα 2
is given by
ζ α 1 = D t
(
ζ
α
1
)
− ẋ α D t (ξ)
ζ α 2 = D t
(
ζ
α
2
)
− ẍ α D t (ξ). (9.58)
First we calculate ζ α 1
and use the fact that ξ = 0. For α = 1,2,3 we get
ζ 1 1 = D t (η 1 ) = D t (x 2 ) = ẋ 2
ζ 2 1 = D t (η 2 ) = −D t (x 1 ) = −ẋ 1
ζ 3 1 = 0. (9.59)
For ζ α 2
we now get in a similar way
ζ2 1 = D t (ζ1) 1 = D t (ẋ 2 ) = ẍ 2
ζ2 2 = D t (ζ1) 2 = −D t (ẋ 1 ) = −ẍ 1
ζ2 3 = 0. (9.60)
Thus, the twice extended generator is given by
X 12,(2) = x 2 ∂
∂x 1 − ∂
x1
∂x 2 + ∂
ẍ2
∂ẍ 1 − ∂
ẍ1
∂ẍ 2 . (9.61)
83

CHAPTER 9. SYMMETRIES AND THE MAIN CONSERVATION THEOREM
The simultanious system (9.53) considered togheter with its adjoint system
has the Lagrangian, according to (9.45)
L = y β F β = my 1 ẍ 1 + my 2 ẍ 2 + my 3 ẍ 3 . (9.62)
The extension of X 12 to the new adjoint differential variables y α is
Y = x 2 ∂
∂x 1 − ∂
x1
∂x 2 + ∂
ηα ∗
∂y α , (9.63)
where η α ∗ is given by equation (9.49). In this case, since ξ = 0 it is simply
λ α β
η α ∗ = −λ α βy β . (9.64)
is obtaind from the invariance condition (9.44). In our case we have
X 12,(2) F α = λ β αF β . (9.65)
Performing this calculation yields
ẍ 2 = λ 1 1ẍ 1 + λ 2 1ẍ 2 + λ 3 1ẍ 3
−ẍ 1 = λ 1 2ẍ 1 + λ 2 2ẍ 2 + λ 2 1ẍ 3
0 = λ 1 3ẍ 1 + λ 2 3ẍ 2 + λ 3 3ẍ 3 , (9.66)
for α = 1,2,3 respectivley. For these equations to hold we must have that λ 2 1 = 1
and λ 1 2 = −1 and the others must be zero. Using equation (9.64) we get
η 1 ∗ = −λ 1 1y 1 − λ 1 2y 2 − λ 1 3y 3 = y 2
η 2 ∗ = −λ 2 1y 1 − λ 2 2y 2 − λ 2 3y 3 = −y 1
η 3 ∗ = 0 (9.67)
We now have all components of the extended tangent vector field for our simultanious
system. To find the conserved quantity associated with the extended generator
(9.63) we apply Noether’s theorem for a second order Lagrangian. Thus,
we will get a conservd quantity C of the form
C = ξL + W α [ ∂L
∂u α t
]
∂L
− D t
∂u α + D t (η α ) ∂L
tt ∂u α , (9.68)
tt
84

9.3. SUMMARY
where u α = (x 1 , x 2 , x 3 ,y 1 ,y 2 ,y 3 ),u α t = (ẋ 1 , ẋ 2 , ẋ 3 ,ẏ 1 ,ẏ 2 ,ẏ 3 ),u α tt = (ẍ 1 , ẍ 2 , ẍ 3 ,ÿ 1 ,ÿ 2 ,ÿ 3 ).
For the tangent vector field components we have ξ = 0 and η α = (x 2 ,−x 1 ,0,y 2 ,−y 1 ,0).
Since ξ = 0 we have that W α = η α , recall definition from chapter 5.3. In this case
we also have that ∂L/∂u α t = 0,∀α. The conserved quantity (9.68) reduces to
C = −η α ∂L
D t
∂u α + D t (η α ) ∂L
tt ∂u α tt
= −x 2 D t (my 1 ) − (−x 1 )D t (my 2 ) − 0 − y 2 D t (0) − (−y 1 )D t (0) − 0
+ D t (x 2 )my 1 + D t (−x 1 )my 2 + D t (0)my 3 + D t (y 2 ) · 0 + D t (−y 1 ) · 0 + 0
= −mx 2 ẏ 1 + mx 1 ẏ 2 + my 1 ẋ 2 − my 2 ẋ 1 . (9.69)
Using the fact that the system (9.53) is self-adjoint, i.e we can replace the adjoint
variable y with x and also that ẋ ≡ v, the velocity, we get
C = −mx 2 v 1 + mx 1 v 2 + mx 1 v 2 − mx 2 v 1
= 2m ( x 1 v 2 − x 2 v 1) . (9.70)
Now, putting M = C/2, we recognice this as the third component of angular momentum.
Doing the same calculations for X 13 and X 23 we arrive at the conserved
quantity
M = m(x×v). (9.71)
I.e, conservation of angular momentum!
End of Example 9.5
This example illustrates the mathematical method to find a conservation law from
a known symmetry.
9.3 Summary
For an arbitrary system of differential equations with a known symmetry and a
known Lagrangian, we may use Noethers theorem to determinig a conservation
law, i.e Noethers theorem gives a connection between symmetries and conservation
laws and provides a simple method for calculating them for variational
85

CHAPTER 9. SYMMETRIES AND THE MAIN CONSERVATION THEOREM
problems.
The symmetries of differential equations may be found by using Lie group theory,
presented in Part I. By solving the determining equation we have the symmetries.
At this point we have two cases to consider. First, if we have a Lagrangian, then
we simply use Noethers theorem to calculate the conservation law for the symmetries
obtained by Lie group theory. On the other hand, if we don’t have a
Lagrangian but the number of equations is equal to the number of dependent variables,
the Main conservation theorem states that we have a conservation law if we
consider our original system with its adjoint system. From the adjoint system we
can construct a Lagrangian and hence, Noethers theorem can be used to calculate
the associated conservation law. For more examples on this I recommend Refs.
16 and 3.
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88