Solution of Differential Equations by Means of One-Parameter Groups

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©JMH1II, 1982 AMS Subject Classifications: (main) 34-XX, 35-XX (subsidiary) 20-XX, 22-XX

British Library Cataloguing in Publication Data

Hill, J. M. Solution of differential equations by means of one-parameter groups. 1. Differential equations I. Title 515.3'S 0A371 ISBN 0-273-08506-9

Library of Congress Cataloging in Publication Data

Hill, J. M. Solution of differential equations by means of one-parameter groups (Research notes in mathematics; 63) Bibliography; p. 1. Differential equations—Numerical solutions. 2. Groups, Theory of. I. T. II. Series. 0A371.H56.

515.3'5 82-621 ISBN 0-273-08506-9 AACR2.

Australian Cataloguing in Publication Data

Hill, J. M. (James M.) Solution of differential equations by means of one-parameter groups. Includes bibliographical references.

ISBN 0858968932. 1. Differential equations. 2. Groups, Theory of. I. Title. (Series: Research notes in mathematics; 63).

515.3'S

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JMHi11 University of Wollongong

Solution of differential equations by means of one -parameter groups

Pitman Advanced Publishing Program BOSTON LONDON. MELBOURNE

Preface

The trouble with solving differential equations is that whenever we are

successful we seldom stop to ask why.

The concept of one—parameter

transformation groups which leave the differential equation invariant provides the only unified understanding of all known special solution techniques.

In these notes I have attempted to present a fairly concise and

self—contained account of the use of one—parameter groups to solve differential equations.

The presentation is formal and is intended to appeal

to Applied Mathematicians and Engineers whose principal concern is obtaining solutions of differential equations.

I have included only the essentials of

the subject, sufficient to etiable the reader to attempt the group approach

when solving differential equations.

I have purposely not included all known

results since this would inevitably lead to unnecessarily reproducing large portions of existing accounts.

For example, for ordinary differential

equations, the account of the subject by L.E. Dickson, "Differential equations from the group standpoint", is still extremely readable and is recommended to the reader interested in pursuing the subject further.

For partial

differential equations the books by G.W. Bluman and J.D. Cole, "Similarity methods for differential equations", and by L.V. Ovsjannikov "Group properties of differential equations", contain several applications and examples which I have not reproduced here.

The first two chapters are introductory.

Chapter 1 gives a general

introduction with simple examples involving both ordinary and partial differential equations.

In Chapter 2 the concepts of one—parameter groups

and Lie series are introduced.

Just as ordinary methods of solving

differential equations often require a certain ingenuity so does the group approach.

In order to establish some familiarity with the group method I

have attempted to exploit our experience with linear equations. are aware that linear differential equations for the transformation

x1 =

f(x),

y1 =

g(x)y

devoted to implications of this result.

y(x)

Most of us

remain linear under

and Chapter 3 of these notes is

In Chapters 4 and 5 I have tried to

relate the usual theory for the group method with the results obtained in the third chapter.

In this respect these notes differ from most accounts of the

subject and I believe that a number of results given, especially in Chapter 3 are new.

The remaining two chapters are devoted to partial differential equations. For the most part the theory is illustrated with reference to diffusion related partial differential equations.

The theory for linear partial

differential equations is introduced in Chapter 6 for the classical diffusion or heat conduction equation and the Fokker—Planck equation. equations are treated in Chapter 7.

Non—linear

For partial differential equations the

group approach is less satisfactory since for boundary value problems both the equation and boundary conditions must remain invariant.

In these notes

we principally consider only the invariance of the equation and view the group method as a means of systematically deducing solution types of a given partial differential equation.

Although these notes appear as a research monograph they actually represent advanced teaching material and in fact form the basis of a post—graduate course given at the University of Wollongong for the past six years. therefore included numerous examples and exercises.

I have

In addition to the

exercises I have used the problems at the end of each chapter to conveniently

locate standard results for differential equations.

On occasions I have also

used these problems to include summaries of theory which is already adequately described in the literature. The existing theory of the solution of differential equations by means of one—parameter groups is by no means complete. the subject are highlighted in the text.

Many of the inadequacies of

When it does work it is very easy

and it is therefore an area of knowledge which every Applied Mathematician ought to be aware of.

Whatever the limitations of the group method may be,

it will always represent a profoundly interesting idea towards solving differential equations.

I hope these notes prove to be useful and complement

the existing literature.

James M. Hill, The University of Wollongong, Australia.

Acknowledgement

The author wishes to thank all of his students over the

past six years who have discovered various errors, spelling mistakes and omissions in a preliminary draft of these notes.

He is also grateful to Mrs.

Kerrie Gamble for her careful and thoroughly professional typing of the manuscript.

For

Desley,

and

Ruth.

Contents

1.

Introduction 1.1 1.2

2.

3.

First 4.1 4.2

4.3 4.4 4.5 4.6

5.

5.4

of standard linear equation8

First order equation y' + p(x)y = q(x) Second order homogeneous equation y" + p(x)y = 0 Third order equation y"+ p(x)y' + q(x)y = 0 Fourth order self—adjoint equation y" + [p(x)y']' + q(x)y = 0 Problems

order differential equation8 Infinitesimal versions of y' and y' = F(x,y) and the fundamental problem Integrating factors and canonical coordinates for y' = F(x,y) The alternative problem The fundamental problem and singular solutions of y' = F(x,y) Invariance of the associated first order partial differential equation Lie's problem and area preserving groups Problems

Second 5.1 5.2 5.3

groups and Lie series

One—parameter transformation groups Lie series and the commutation theorem Problems

Invariance 3.1 3.2 3.3 3.4

4.

Ordinary differential equations Partial differential equations Problems

One-parameter 2.1 2.2

1

and higher order differential equation8

Infinitesimal versions of y" and y" = F(x,y,y') Examples of the determination of and Determination of the most general differential equation invariant under a given one—parameter group Applications Problems

1

6 9

12 12 18

21

26 26 30 33 36 39

48

49 51

55 58 59 62 66

77 77 79

83 87 92

6.

Linear partial differential 6.1 6.2

equations

Formulae for partial derivatives

Classical groups for the diffusion equation 6.3 Simple examples for the diffusion equation

6.4 Moving boundary problems 6.5 Fokker-Planck equation 6.6 Examples for the Fokker—Planck equation 6.7 Non—classical groups for the diffusion equation Problems

7.

7.4

99

ioi 103 105 109 116 120 123

partial differential equations

135

Formulae for partial derivatives Classical groups for non—linear diffusion

136 140

Non-linear

7.1 7.2 7.3

97

Non—classical groups for non—linear diffusion Transformations of the non—linear diffusion equation Problems

References

146 148 150

159

1 Introduction

Although

a good deal of research over the past two centuries has been devoted

to differential equations our present understanding of them is far from complete.

These notes are concerned with obtaining solutions of differential

equations by means of one—parameter transformation groups which leave the equation invariant.

This subject was initiated by Sophus Lie [1] over a

hundred years ago.

Such an approach is not always successful in deriving

solutions.

However it does provide a framework in which existing special

methods of solution can be properly understood and also it is applicable to linear and non—linear equations alike.

In formulating differential equations

the Applied Mathematician inevitably makes certain assumptions.

Using group

theory these assumptions can be seen to hold the key to obtaining solutions of their equations.

The purpose of this chapter is to present a simple introduction to the subject for both ordinary and partial differential equations by means of simple familiar examples.

For ordinary differential equations comprehensive

accounts of the subject are given by Cohen [2], Dickson [3], Page [4] and more recently Bluman and Cole [5] and Chester [6].

For partial differential

equations the reader may consult Bluman and Cole [5] and Ovsjannikov [7] where additional references may also be found.

1.1

ORDINARY DIFFERENTIAL EQUATIONS

In order to illustrate some of the ideas developed in these notes we consider a simple example.

It is well known

differential equation

that

the 'homogeneous' first order

2

2

dx

(11)

xy

can be made separable by the substitution

u(x,y) =

y/x

and the resulting

solution is given by

=C, where

(1.2)

denotes an arbitrary constant.

C

We might well ask the following

questions:

Why does the substitution

Question I

equation for Question 2

u(x,y) =

y/x

lead to a separable

u ?

How do we interpret the degree of freedom embodied in the

arbitrary constant

C

in the solution?

Answers to these questions can be provided within the framework of transformations which leave the differential equation unaltered.

Consider

the following transformation,

x1ex,

y1ey,

C

where

C

is

C

(1.3)

an arbitrary constant.

We notice that (1.1) remains invariant

under (1.3) in the sense that the differential equation in the new variables x1

and

is identical to the original equation, namely

y1

2

2

—= x1 +y1 dy1

x1y1

dx1

(1.4)

.

Moreover we see that (1.3) satisfies the following: gives the identity transformation

(1)

c = 0

(ii)

—c

characterizes the inverse transformation

(iii)

if

x2 =

e6x1,

y2 = e6y1

x1 =

x,

y1 =

y, y =

x =

eCy1,

then the product transformation is also a

member of the set of transformations (1.3) and moreover is characterized by the parameter 2

c+5,

that

is

x2 = e

c+5

x, y2 = e

c+cS

y.

A transformation satisfying these three properties is said to be a one—

parameter

group

of transformatione.

We

observe that the usual associativity

law for groups follows from the property (iii).

With this terminology

established we might answer the above questions as follows: Answer 1 u

because

u(x1,y1) =

The substitution

y/x

leads to a separable equation for

is an invariant of (1.3) in the sense that

u(x,y)

u(x,y)

since, yl

u(x1,y1) =

u(x,y) =

= u(x,y)

—=

(1.5)

,

and it is this property which results in a simplification of (1.1).

In

general we shall see that if a differential equation is invariant under a one—parameter group of transformations then use of an invariant of the group results in a simplification of the differential equation.

If the

differential equation is of first order then it becomes separable while if the equation is of higher order then use of an invariant of the group permits a reduction in the order of the equation by one. Answer 2

From (1.2) and (1.3) we see that we have 2

y1

— =C xl

log x1 —

+ c

(1.6)

,

so that the degree of freedom in the solution (1.2) resulting from the arbitrary constant

C

is related to the invariance of the differential

equation (1.1) under the group of transformations (1.3) which is characterized That is, the transformation (1.3) permutes

by the arbitrary parameter

c.

the solution curves (1.2).

In general we shall see that for every one—

parameter group in two variables there are functions

u(x,y)

and

v(x,y)

such that the group becomes u(x1,y1) =

u(x,y)

,

v(x1,y1)

=

v(x,y)

+ c

.

(1.7) 3

Moreover if a first order differential equation is invariant under this group then in terms of these new variables =

4(u)

u

and

v

it takes the form,

(1.8)

,

and consequently has a solution of the form

v +

=

C

(1.9)

,

for appropriate functions

and

In order to give the reader some indication of the usefulness of the above we consider the following non—trivial equation, (1.10)

This is an Abel equation of the second kind (Murphy [8], page 25) which we see is not readily amenable to any of the standard devices.

However the

equation is clearly invariant under the group

x1ex, C

y1=e—Cy,

and therefore we choose

(1.11)

u(x,y) = xy

as the new dependent variable and the

differential equation (1.10) becomes, (1.12)

which can be readily integrated.

It is worthwhile emphasizing that not all

equations can be solved in such a simple manner. —

Consider for example,

(1.13)

,

which arises in finite elasticity (see Hill [9]).

This equation is again an

Abel equation of the second kind but in this case there is apparently no simple group such as (1. 11) which leaves the equation invariant.

In this general introduction it may be appropriate to mention here possible research areas for which group theory has not yet been applied.

4

The reader

might well like to bear these problems in mind with a view to developing results in these areas. Research area 1

Differential—difference equations.

It is well known that formal solutions of linear differential—difference equations, for example

-y(x—x0)

=

where

(1.14)

,

is a constant, can be expressed as

x0

r

y(x)

C.e

=

—W4X

(1.15)

,

j

are arbitrary constants and

where

denote the roots of

w =

If the equation is non—linear then there are no such general methods of Consider for example Hutchinson's equation which can be written as

solution.

4x)

=

y(x)[l

—

y(x—x0)]

(1.16)

.

This equation arises in theory of populations (see Hutchinson [10]).

What are

the implications of group theory, if any, for equations of this type?

(See

problems 19 and 20 of Chapter 4). Research area 2

Differential equations invariant under transformations which

cannot be characterized as one—parameter groups. A differential equation occurring in fluid dynamics is Tuck's equation (see Tuck [11]),

— 2

dt2

dx + dt

(5+3x)

+ 3x(1—x)

4x(1+x)(dtJ

It can be verified that if the usual way we let y

3x(1-.x)

dx

(1+x)

y =

+ 2y

x(t)

dx/dt

(117)

(1+x)

is a solution then so is

x(t)'.

If in

then (1.17) becomes

+ (5+3x) 4x(1+x)

2

'

which is again an Abel equation of the second kind.

(1 18)

From the invariance

5

property of (1.17) we can deduce that (1.18) remains invariant under the transfo rmat ion

(1.19) which clearly cannot be characterized as a one—parameter group.

Can we use

such invariance properties to determine solutions of differential equations? Research area 3

Abel equation of the second kind;

As we have already Indicated one of the most frequently occurring differential equations which is not always amenable to standard devices is the Abel equation of the second kind.

The general equation can be expressed

in the form (see Murphy [8], page 26) y

Equation

dx

=

a(x)

(1.20)

+ b(x)y .

(1.20) with arbitrary functions

a(x)

and

b(x)

would appear to be

a problem worthwhile studying.

1.2

PARTIAL DIFFERENTIAL EQUATIONS

Unlike ordinary differential equations the success of the group approach for partial differential equations depends to a considerable extent on the accompanying boundary conditions.

That is, the group approach is only

effective in the solution of boundary value problems if both the equation and boundary conditions are left unchanged by the one—parameter group.

For the

most part we confine our attention to specific differential equations rather than boundary value problems.

For any particular boundary value problem we

should always first look for any simple invariance properties.

These may be

more apparent from the physical hypothesis of the problem rather than its mathematical formulation.

If no such invariance

can be found and if the

problem merits a numerical solution then the group approach might still be

6

relevant as a means of checking the numerical technique with artificially imposed boundary conditions which permit an exact analytic solution. As an illustration we consider a boundary value problem for which both the partial differential equation and the boundary conditions are invariant under a simple one—parameter group.

Consider the problem of determining the source

solution for the one—dimensional diffusion or heat conduction equation for c(x,t), namely 2 3c

=

(t

—°°

>

x




solution is of course well known.

—°° < x

< co)

.

(1.29)

For our purposes it firstly serves as

a specific non—trivial boundary value problem for which the differential equation and boundary conditions are both invariant under a one—parameter group.

Secondly it serves to illustrate that knowledge of a one—parameter

group leaving the equation invariant enables, at least in the case of two independent variables, the partial differential equation to be reduced to an ordinary differential equation.

For more independent variables knowledge of

a group leaving the equation unchanged reduces the number of independent variables by one.

In these notes we give the general procedure for determining the group such as (1.23) whIch leaves a specific equation invariant.

We also give the

general technique for establishing the functional form of the solution such as that

8

given

by (1.25).

PROBLEMS

1.

Determine in each case the constants

a

and

8

such that the one—

parameter group ac x1 = e x

y1 = e

,

y

leaves the following differential equations invariant.

Use an invariant

of the group to integrate the equation.

(a) E =

3,12

+ By3

and

(A

B

are constants)

(b)

+ By = 0

(c)

x(A + xytl)

2.

Verify that,

(A, B

and

n

are constants)

y1=e —2c y,

x1=x+C,

is a one—parameter group of transformations and hence integrate the

differential equation (1 — 2x —

3.

logy)

2y = 0

Integrate the differential equation (x—y)2

=

(A

is a constant)

by observing that the equation admits the group

x1x+C, 4.

Given that

p(x)

y1=y+C. is a solution of the linear differential—difference

equation (1.14) show that p(x—x0) y(x)

p(x)

=

is a solution of the non—linear differential—difference equation dy(x)

=

y(x)[y(x)

—

y(x—x0)] 9

5.

Show that the transformation e

y(x)=

x

f(eXX0)

reduces equation (1.16) to the differential equation

f(t)

df(t) —

f(At)

dt

where 6.

t=e x-xO and X=e -xO

Show that with

w =

3(1-x)

=

dx

(1+x)

y/x

the differential equation (1.18) becomes

+

+

(1-x) (1+x)

Show further that the substitution 2

4

s =

(1—x)/(1+x)

2

(s —1)

w

2

dw SW — = 35 + 2w + — ds 4

and observe that the transformation (1.19) becomes Si

7.

yields

w1 =

—w

and

= —S.

Observe that the partial differential equation (1.21) remains invariant under the transformation c

x1 = e x

t1

,

= e

2€ t

,

c1 = c

so that the equation admits solutions of the form

Deduce the ordinary differential equation for

c(x,t) where 8.

A

and

Continuation.

4

c(x,t) =

and hence show that

2

"4dy + B

= A,j'

B

denote arbitrary constants.

For the non—linear diffusion equation

D(c)j—) where the diffusivity

D

is a function of

c

only, use the one—

parameter group and functional f6rm of the solution in the previous problem to deduce the ordinary differential equation 10

+ dD(4)

9.

Continuation.

+

For the case

= 0

D(c) = c

show that

the

ordinary

differential equation of the previous problem remains invariant under the group

41e 2c

c

=

and that with

the equation reduces to the Abel equation of

the second kind

+ p2 + p =

p = 10.

+

+

and

log

y =

Show that the singular solution

E.

corresponds to the solution

Continuation.

4(e)

constant.

Show that the transformation

p =

reduces the Abel

equation in the previous problem to q

+ fJq +

+

= 0

which has the same form as equation (1.20). 11.

By calculating the quantity, ac1

2 ac1

at1

show directly, using the chain rule for partial derivatives, that the classical diffusion equation (1.21) remains invariant under the following transformations, (i)

x1 = x + et,

t1 = t,

c1c

exp[_ —

2

1

(ii)

x1 = (1—ct)

=

(1—ct)

—

4(1—ct)

11

2 One-parameter groups and Lie series

In

this chapter we introduce the concepts of one—parameter groups and Lie For one—parameter groups there are two important results.

series.

the method of obtaining the global

Firstly

of the group from the infiniteBimal

Secondly the existence of canonical

coordinateB

for the group.

For

Lie series the important and remarkable result is the so—called Coninutation

These concepts are discussed below.

theorem.

2. 1

ONE—PARAMETER TRANSFORMATION GROUPS

In the (x,y) plane we say that the transformation =

is

f(x,y,c)

group of

a one-parameter

(i)

(identity) the value x =

(ii)

f(x,y,O)

(2.1)

,

tran8forn?ation8

y

=

the following properties hold:

characterizes the identity transformation,

c = 0

,

if

g(x,y,O)

(inverse) the parameter —c characterizes the inverse transformation, x =

(iii)

g(x,y,c)

y1 =

,

f(x1,y1,—c) x2 =

(closure) if the

,

y = g(x1,y1,—c)

f(x1,y1,cS)

two transformations

,

y2 = g(x1,y1,cS)

then the product of

is also a member of the set of transformations

(2.1) and moreover is characterized by the parameter x2 =

f(x,y,c+5)

,

c+tS

,

that

is

y2 =

Again we remark that the usual associativity law for groups follows from the closure property.

12

(a)

x1 = x ,

(b)

x1 = eCx

Some simple examples of one—parameter groups are: y1 = y + c ,

y1 = eCy

(translation group), (stretching group),

y sin c

x1 = x cos c

(c)

,

y1 = x sin c + y cos c

x1 = x

In order to show (c) forms a group we have inmiediately

when

=0

c

and

y = y1 cos c — x1 sin c

characterizes

the inverse and (ii) is satisfied.

x2 = x1 cos

— y1 sin d

(x

x2 =

and

cos c — y sin

=

x cos(c+5) — y

=

(x cos c — y sin c)sin

=

x sin(c+5) + y cos(c+tS)

y1 = y

so that —c

For (iii) we see that if

y2 = x1 sin d + y1 cos d

c)cos

and

group).

On inverting we obtain

so (i) is satisfied.

x = x1 cos c + y1 sin c

(rotation

then we have

5 —

(x

d +

(x sin c + y cos c)cos d

sin c + y cos c) sin d

sin(c+tS)

and

and therefore (iii) is satisfied. The functions

=0

c

and

g(x,y,c)

are referred to as the global form

If for small values of the parameter

of the group. since

f(x,y,c)

we expand (2.1) then

c

gives the identity we have dx

x1 = x + c

+ 0(c2)

y1 = y + c

,

+ 0(c2)

where

If

indicates

0(c2)

(2.2)

,

c=O

c=O

terms involving only and

we introduce functions

and higher powers of

c2

c.

by

dy

dx c

=

= fl(x,y)

,

(2.3)

,

c=O

c=O

then we obtain x1 = x +

+ 0(c2) ,

y1 = y + cfl(x,y) + 0(c2) ,

and (2.4) is referred to as the infinitesimal form of the group. property

form of

of one-parameter tran8formatwn groups is the

group we can deduce

that

(2.4) The crucial

given the infinitesimal

the global form by integrating the following

13

of differential equations,

autonomous dx1

dy1 =

subject

=

,

(2.5)

,

to the initial conditions,

x1 = x

y1 = y

,

e = 0

when

(2.6)

.

A proof of this result can be found in Dickson [3].

Here we merely indicate

its validity by means of a simple example.

For the rotation group (c) we have dx1

dy1 ,

e = 0

and therefore on setting F(x,y) =

—y

=

,

we have from (2.3)

x

Thus in this case we need to integrate dy1

dx1 ,

subject

Introducing the complex variables

to the initial conditions (2.6).

z=x+iy and z1=x1+iy1 weobtain dz1

= iz1 and thus log z1 =

ic

+ log z

where we have used the initial conditions (2.6). z1 = e1Cz

Imaginary parts of rotation group (c). r =

(x

then we have

2

2½

+ y ) z =

On equating real and

we can readily deduce the global form of the

If we introduce polar coordinates

,

re 10

0 =

tan

and from

—1

(y/x)

z1 = e

ic

z

we see that the global form of

the rotation group (c) can be written alternatively as 01 = 0 + c. 14

That is, in terms of

defined by

(r, 0)

(r,0)

r1 =

r

and

coordinates the rotation group has

the appearance of the translation group.

This is a general property of one-'

parameter transformation groups. For

any given one-parameter transformation group (2.1) there extBtB

and v(x,y)

functwn8 u(x, y)

u(x1,y1) The function

u(x,y)

=

auch that the global form of the group becomes

v(x1,y1)

,

=

v(x,y)

(2.7)

+ c .

is said to be an invariant of the group while together

u(x,y)

are referred to as the canonical coordinateB of the group.

(u,v)

Methods for finding

—=

From (2.5) we obtain

dx1 dy1

(2.8)

n(x1,y1)

which we suppose integrates to yield

u(x1,y1) —

constant

so that from the

Initial conditions (2.6) we deduce the first equation of (2.7) and

Alternatively

known.

eliminating

c

u(x,y)

from

and (2.1)2.

We note that if

and suppose that from

u(x1,y1) =

u(x,y), namely a =

u(x,y)

we can deduce the explicit relation

Now for the purposes of integration in (2.5),

y1 =

is an

u(x,y)

In the integration of (2.8) let

v(x,y).

is

may be deduced directly from (2.1) simply by

invariant then so also Is any function of Method for finding

u(x,y)

a

is a

constant and from (2.5)i we have dx

(2.9)

.

=

If for some constant fX

x0

we define

by

dt

(2.10)

•

=

0

then from (2.6) and (2.9) we can deduce (2.7)2 where and hence

v(x,y)

v(x,y) =

Is known.

15

Example 1

Consider,

X1

(1+ex)

(1+cx)2y

=

'

(2.11)

.

The reader should verify that (i), (ii) and (iii) of the definition of a one— parameter are Indeed satisfied. x1 = x — cx2 + 0(c2) ,

so that from (2.4)

For small values of

and

fl(x,y) =

differentiatIng (2. 11) with respect to

-

(1+ex)

2

2xy.

Alternatively on

we have

c

dy1

2

= —x1

we have

y1 = y + 2cxy + 0(c2)

= —x2

dx1

c

-i-— = 2(1+cx)xy

,

and (2.5) confirms the given expressions for

=

2x1y1

E(x,y)

and

(2.12)

.

fl(x,y).

From

(2.12) we have X1

dx1 dy1

u(x,y) = x2y

which on integrating gives v(x,y) = x

we see that

-1

as an invariant while from (2.12)i

satisfies (2.7).

could be deduced directly from (2.11) by eliminating

Example

2

and

by the relations,

v

Show that

r

2

Alternatively the invariant

x y

C.

are related to the canonical coordinates

and

—

u

213

'

—

where the Jacobian is given by

On differentIating (2.7) with respect to dx1

r 16

dy1

+

dx1

=0

,

C

we obtain dy1

+

=1 -s----

,

(2.

14)

where

and

u1

denote

v1

and

u(x1,y1)

(2.5) and (2.14) we have on replacing

n=0

+

(x1,y1)

by

respectively.

From

(x,y),

Ti = 1

÷

,

v(x1,y1)

and (2.13) can be deduced immediately from these relations. A transformation in the

Example 3

(x,y)

plane is area preserving if

a(x1,y1) =

1

(2.15)

.

Show that (2.1) is area preserving if and only if —

u

only.

From (2.4) and (2.15) we can deduce on equating terms of order

c,

(2.16)

ax

From (2.13) and (2.16) we can deduce

1 a(x,y)

=o

and the required condition follows.

lit may be of Interest to note that since the set of area preserving transformations forms a group, the infinitesimal condition (2.16) is precisely the same as the global condition. respect to

c

That is, if we differentiate (2.15) with

we have dy1

dx1

y1

=0

and on multiplying this equation by

dx1

,y1,

we obtain

dy1

÷ ,

' —0

17

so that we have a

dx1

dy1

a

=0

+

(2.

.

17)

On using (2.5) we see that (2.17) is the same condition as (2.16)fl

2.2

LIE SERIES AND THE COMMUTATION THEOREM

Suppose we have the group (2.1) with infinitesimal version (2.4). the differential operator

We define

by

L

(2.18)

.

L =

Now for any function

$(x1,y1)

which does not depend explicitly on

c

we

have

dx

d41

de

dy1

de

ax1

denotes

where

L1

(x1,y1).

From (2.5) and (2.19) we obtain

(2.20)

,

denotes the differential operator

with

(x,y)

replaced by

d3

= L1 (L1($1))

,

—i-

= L1 [L1

(2. 21)

.

then by Maclaurin's expansion we have

•(c) =

=

4(O) +

c=O

de

and thus from (2.20) and (2.21) with

dc

c=O

c = 0

That is, we have

+

+

c=O

we obtain 3

2

=

18

L

Similarly we have

d2

If we let

(2.19)

de

4(x1,y1).

= L1(41) where

ay1

+

+ ...

(2.22)

,

= n=O

and we refer to such a series as a Lie series.

We notice that we can write

(2.22) as =

(2.23)

,

provided we interpret the differential operator

as the series operator,

)

in particular if we take

to be

4(x,y) e

x

cL

y

x

and

y

then from (2.23) we obtain,

(2.24)

.

On combining (2.23) and (2.24) we have the remarkable result, CL

x, e

CL

y) = e

CL

(2.25)

,

which is called the Commutation

theorem of Lie series (see

and Knapp

t121, page 17).

To illustrate (2.24) consider the rotation group (c). differential operator L = —y so that

a

L(x) =

In this case the

is given by

L

a

+ x

r

—y

and

L(y) = x

and the global form of the group can be

deduced from (2.24) using the expansions,

k2k

cos

£

= k=O

(—1) C (2k)!

k2k+l

(—1) C

sin c = k=O

(2k-i-1)i

It is worthwhile noting that using Lie series we can give a formal

Example 4

solution of any autonomous system of differential equations given initial values.

That is, consider =

F(X,Y)

,

= G(X,Y)

,

19

and

X

Y =

a,

at

t =

0.

The formal solution of this initial value

problem is x = etMa

y

,

F(a,8)

(2.26)

,

is defined by

M

where the operator N =

=

+

Consider two simple examples.

Firstly, the single differential equation,

dX

dt In this case we have M = -a2 —a-,

and

-a2,

M(a) =

M2(a) = 2a3

and in general

N(a)=(—1)n n!an+1 Hence

x =

etha =

and thus for

(a) = a

n0

lati
O,

a

c05(x—x0) ac ,

and

ox

x0

(x,t)

+ 0

as

x +

and

provided

the functions

satisfy = 0

Hence with

±00

denote arbitrary constants, show that the initial

condition remains invariant under (6.1) fl(x,t)

—°°<x