Content-Type: multipart/mixed; boundary="-------------0111081229669"
This is a multi-part message in MIME format.
---------------0111081229669
Content-Type: text/plain; name="01-416.comments"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="01-416.comments"
AMS-Code: 34A30
PACS-Code: 02.30.Hq
---------------0111081229669
Content-Type: text/plain; name="01-416.keywords"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="01-416.keywords"
Linear ordinary differential equations, Lie symmetry algebra, Floquet theory
---------------0111081229669
Content-Type: application/x-tex; name="classic5.tex"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="classic5.tex"
\documentstyle[11pt]{article}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newtheorem{theorem}{Theorem}
\newtheorem{acknowledgement}[theorem]{Acknowledgement}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}
\begin{document}
\setlength{\parindent}{5 mm}
\catcode`\@=11
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\@addtoreset{equation}{section}
\catcode`\@=10
% ** Definitions **
\def\dsp{\displaystyle}
\def\d{\mbox{\rm d}}
\def\e{\mbox{\rm e}}
% ** math **
\def\r{\prime}
\def\p{\partial}
\def\ut#1{{\mathunderaccent\tilde #1}}
\def\cosec{{\rm cosec}}
\def\kd{{\delta _{ij}}}
\def\ke{{\epsilon _{ijk}}}
\def\half{\mbox{$\frac{1}{2}$}}
\def\tha{\mbox{$\frac{3}{2}$}}
\def\fha{\mbox{$\frac{5}{2}$}}
\def\nha{\mbox{$\frac{n}{2}$}}
\def\oqr{\mbox{$\frac{1}{4}$}}
\def\ofi{\mbox{$\frac{1}{5}$}}
\def\tqr{\mbox{$\frac{3}{4}$}}
\def\tth{\mbox{$\frac{2}{3}$}}
\def\mod{\mbox{\rm mod}}
\def\f{\frac}
\def\p{\dsp\partial}
\def\arcsinh{{\rm arcsinh}}
\def\sinh{{\rm sinh}}
\def\cosh{{\rm cosh}}
\def\arccosh{{\rm arccosh}}
\def\arctanh{{\rm arctanh}}
\def\tanh{{\rm tanh}}
\def\coth{{\rm coth}}
\def\sech{{\rm sech}}
\def\cosech{{\rm cosech}}
% ** greek **
\def\ga{\alpha}
\def\gb{\beta}
\def\gc{\chi}
\def\gd{\delta}
\def\ge{\epsilon}
\def\gf{\phi}
\def\gg{\gamma}
\def\gi{\iota}
\def\gk{\kappa}
\def\gl{\lambda}
\def\gp{\psi}
\def\gs{\sigma}
\def\gt{\theta}
\def\gu{\upsilon}
\def\gve{\varepsilon}
\def\gw{\omega}
\def\gz{\zeta}
\def\gG{\Gamma}
\def\gD{\Delta}
\def\gF{\Phi}
\def\gL{\Lambda}
\def\gP{\Psi}
\def\gS{\Sigma}
\def\gT{\Theta}
\def\gW{\Omega}
\def\bhw{\hat{\mbox{\boldmath $\omega$}}}
\def\bht{\hat{\mbox{\boldmath $\theta$}}}
\def\bhf{\hat{\mbox{\boldmath $\phi$}}}
\def\bhr{\hat{\mbox{\boldmath $r$}}}
\def\bhL{\hat{\mbox{\boldmath $L$}}}
% ** bold **
\def\bfA{{\bf A}}
\def\bfB{{\bf B}}
\def\bfC{{\bf C}}
\def\bfD{{\bf D}}
\def\bfE{{\bf E}}
\def\bfF{{\bf F}}
\def\bfG{{\bf G}}
\def\bfH{{\bf H}}
\def\bfI{{\bf I}}
\def\bfJ{{\bf J}}
\def\bfK{{\bf K}}
\def\bfL{{\bf L}}
\def\bfM{{\bf M}}
\def\bfN{{\bf N}}
\def\bfO{{\bf O}}
\def\bfP{{\bf P}}
\def\bfQ{{\bf Q}}
\def\bfR{{\bf R}}
\def\bfS{{\bf S}}
\def\bfT{{\bf T}}
\def\bfU{{\bf U}}
\def\bfV{{\bf V}}
\def\bfW{{\bf W}}
\def\bfX{{\bf X}}
\def\bfY{{\bf Y}}
\def\bfZ{{\bf Z}}
\def\bfa{{\bf a}}
\def\bfb{{\bf b}}
\def\bfc{{\bf c}}
\def\bfd{{\bf d}}
\def\bfe{{\bf e}}
\def\bff{{\bf f}}
\def\bfg{{\bf g}}
\def\bfh{{\bf h}}
\def\bfi{{\bf i}}
\def\bfj{{\bf j}}
\def\bfk{{\bf k}}
\def\bfl{{\bf l}}
\def\bfm{{\bf m}}
\def\bfn{{\bf n}}
\def\bfo{{\bf o}}
\def\bfp{{\bf p}}
\def\bfq{{\bf q}}
\def\bfr{{\bf r}}
\def\bfs{{\bf s}}
\def\bft{{\bf t}}
\def\bfu{{\bf u}}
\def\bfv{{\bf v}}
\def\bfw{{\bf w}}
\def\bfx{{\bf x}}
\def\bfy{{\bf y}}
\def\bfz{{\bf z}}
% ** differentials **
\def\p{\dsp \partial}
\def\pb{{\p\over\p b}}
\def\pc{{\p\over\p c}}
\def\pd{{\p\over\p d}}
\def\pe{{\p\over\p e}}
\def\pf{{\p\over\p f}}
\def\pg{{\p\over\p g}}
\def\ph{{\p\over\p h}}
\def\pj{{\p\over\p j}}
\def\pk{{\p\over\p k}}
\def\pl{{\p\over\p l}}
\def\pn{{\p\over\p n}}
\def\po{{\p\over\p o}}
\def\pq{{\p\over\p q}}
\def\pr{{\p\over\p r}}
\def\ps{{\p\over\p s}}
\def\pt{{\p\over\p t}}
\def\pu{{\p\over\p u}}
\def\pv{{\p\over\p v}}
\def\pw{{\p\over\p w}}
\def\px{{\p\over\p x}}
\def\py{{\p\over\p y}}
\def\pz{{\p\over\p z}}
\def\pA{{\p\over\p A}}
\def\pB{{\p\over\p B}}
\def\pC{{\p\over\p C}}
\def\pD{{\p\over\p D}}
\def\pE{{\p\over\p E}}
\def\pF{{\p\over\p F}}
\def\pG{{\p\over\p G}}
\def\pH{{\p\over\p H}}
\def\pI{{\p\over\p I}}
\def\pJ{{\p\over\p J}}
\def\pK{{\p\over\p K}}
\def\pL{{\p\over\p L}}
\def\pM{{\p\over\p M}}
\def\pN{{\p\over\p N}}
\def\pO{{\p\over\p O}}
\def\pQ{{\p\over\p Q}}
\def\pR{{\p\over\p R}}
\def\pS{{\p\over\p S}}
\def\pT{{\p\over\p T}}
\def\pU{{\p\over\p U}}
\def\pV{{\p\over\p V}}
\def\pW{{\p\over\p W}}
\def\pX{{\p\over\p X}}
\def\pY{{\p\over\p Y}}
\def\pZ{{\p\over\p Z}}
\def\da{{\d\over\d a}}
\def\db{{\d\over\d b}}
\def\dc{{\d\over\d c}}
% \def\dd{{\d\over\d d}}
\def\de{{\d\over\d e}}
\def\df{{\d\over\d f}}
\def\dg{{\d\over\d g}}
\def\dh{{\d\over\d h}}
\def\di{{\d\over\d i}}
\def\dj{{\d\over\d j}}
\def\dk{{\d\over\d k}}
\def\dl{{\d\over\d l}}
% \def\dm{{\d\over\d m}}
\def\dn{{\d\over\d n}}
\def\do{{\d\over\d o}}
\def\ddp{{\d\over\d p}}
\def\dq{{\d\over\d q}}
\def\dr{{\d\over\d r}}
\def\ds{{\d\over\d s}}
\def\\d t{{\d\over\d t}}
\def\du{{\d\over\d u}}
\def\dv{{\d\over\d v}}
\def\dw{{\d\over\d w}}
\def\dx{{\d\over\d x}}
\def\dy{{\d\over\d y}}
\def\dz{{\d\over\d z}}
\def\dA{{\d\over\d A}}
\def\dB{{\d\over\d B}}
\def\dC{{\d\over\d C}}
\def\dD{{\d\over\d D}}
\def\dE{{\d\over\d E}}
\def\dF{{\d\over\d F}}
\def\dG{{\d\over\d G}}
\def\dH{{\d\over\d H}}
\def\dI{{\d\over\d I}}
\def\dJ{{\d\over\d J}}
\def\dK{{\d\over\d K}}
\def\dL{{\d\over\d L}}
\def\dM{{\d\over\d M}}
\def\dN{{\d\over\d N}}
\def\dO{{\d\over\d O}}
\def\dP{{\d\over\d P}}
\def\dQ{{\d\over\d Q}}
\def\dR{{\d\over\d R}}
\def\dS{{\d\over\d S}}
\def\\d t{{\d\over\d T}}
\def\dU{{\d\over\d U}}
\def\dV{{\d\over\d V}}
\def\dW{{\d\over\d W}}
\def\dX{{\d\over\d X}}
\def\dY{{\d\over\d Y}}
\def\dZ{{\d\over\d Z}}
% ** derivatives **
\def\upt{\p_t}
\def\upr{\p_r}
\def\upv{\p_v}
\def\upy{\p_y}
\def\upf{\p_f}
\def\upx{\p_x}
\def\upz{\p_z}
\def\up#1{\p_{#1}}
\def\pa#1#2{\f{\dsp \p#1}{\dsp \p#2}}
\def\pp#1#2{\f{\dsp \p^2 #1}{\dsp \p #2^2}}
\def\pam#1#2#3{\f{\dsp \p^2 #1}{\dsp \p #2 \p #3}}
\def\dm#1#2{\f{\dsp \d #1}{\dsp \d #2}}
\def\dd#1#2{\f{\dsp \d^2 #1}{\dsp \d #2^2}}
\def\dddot#1{\mathinner{\buildrel\vbox{\kern5pt\hbox{...}}\over{#1}}}
% ** math misc **
\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\bq{\begin{eqnarray}}
\def\eq{\end{eqnarray}}
\def\beq{\begin{eqnarray*}}
\def\eeq{\end{eqnarray*}}
%\newtheorem{theorem}{Theorem}[section]
\def\ext#1#2{G^{[#1]} #2_{|_{\!\!|_{#2=3D0}}} =3D 0}
\def\exta#1{G^{[#1]}}
\def\extb#1#2{G^{[#1]} #2_{|_{\!\!|_{#2=3D0}}} }
\def\enter{$\longleftarrow\!\!\!^|$}
\def\heading#1{\begin{center}{\LARGE #1} \end{center} \vspace{10mm}}
\def\de{\d ifferential equation}
\def\des{\d ifferential equations}
\def\fode{first order ordinary differential equation}
\def\fodes{first order ordinary differential equations}
\def\pde{partial differential equation}
\def\pdes{partial differential equations}
\def\sodes{second order ordinary differential equations}
\def\sode{second order ordinary differential equation}
\def\odes{ordinary differential equations}
\def\ode{ordinary differential equation}
\def\todes{third order ordinary differential equations}
\def\tode{third order ordinary differential equation}
\def\node{$n$th order ordinary differential equation}
\def\nodes{$n$th order ordinary differential equations}
% ** really misc **
\def\ie{{\it ie }}
\def\cf{{\it cf }}
\def\viz{{\it viz }}
\def\etal{{\it et al }}
\def\n{\nonumber}
\def\({\left (}
\def\){\right )}
\def\bi{\begin{itemize}}
\def\ei{\end{itemize}}
\def\z{&=&}
\def\lb{\left[ }
\def\rb{\right] }
\def\pic#1#2#3{\begin{center}\input #1.tex #2 =
\end{center}\begin{center}{\it #3}\end{center}}
\def\lrl{Laplace-Runge-Lenz}
\def\lrlv{Laplace-Runge-Lenz vector}
\def\kp{Kepler Problem}
\def\kdp{Kepler-Dirac Problem}
\def\noe{Noether's Theorem}
\def\re#1{(\ref{#1})}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% End Preamble %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title{The essential harmony in the classical equations of mathematical physics}
\author{M C
Nucci$^{\dagger}$ and P G L Leach$^{\ddagger}$\footnote{permanent
address: School of Mathematical and Statistical Sciences,
University of Natal, Durban 4041, Republic of South Africa}}
\date{$^{\dagger}$Dipartimento di Matematica e Informatica,
Universit\`a di Perugia\newline 06123 Perugia,
Italia\\$^{\ddagger}$GEODYSYC, Department of Mathematics,
University of the Aegean\newline Karlovassi 83 200, Greece}
\maketitle
\begin{abstract}
The possibility to transform any system of linear ordinary
differential equations into a system of constant coefficient
equations is demonstrated using Lie theory. Some examples relate
the classical equations of mathematical physics to the simple
harmonic oscillator. The r\^oles of the third order form of the
Ermakov-Pinney equation and of Fleischen-von Weltunter systems are
explained.
\end{abstract}
\section{Introduction \label{sect1}}
Transformation methods to facilitate the solution of differential
equations have a history just slightly shorter than that of the subject
of differential equations.
The development of transformation methods was, as for all methods
of solving differential equations, something of an {\it ad hoc} affair
for some two centuries.
Then Sophus Lie, applying the geometric ideas of transformation
groups which he developed \cite{Lie 1,Lie 2, Lie 3},
established a systematic approach by determining the point \cite{Lie:dgl}
and contact \cite{Lie:bht} symmetries of differential equations and the
simplifying transformations which follow from the knowledge of
the symmetries. Noether's treatment of the symmetries of the
Action Integral \cite{Noether 18} paved the way for generalised symmetries
as a common tool. The final decades of the last millennium saw the
introduction of nonlocal symmetries to provide some sense of completeness
to the class of symmetries of differential equations. In the subfield of
partial differential equations a number of specialised symmetries, such as
potential \cite{Bluman 88, Bluman 89} and nonclassical
\cite{Bluman 69, Winternitz 80} were introduced, although the latter
were shown to be equivalent to standard symmetries in terms of
the method of determination \cite{Nucci 00}.
In the Floquet theory of systems of periodic differential
equations \cite{Arscott 64,MacLachlan 45} a corollary to the theorem on the Floquet
representation of the solution of such a system establishes that
for a system of periodic differential equations there exists a
nonsingular periodic transformation of variables transforming the
system into one of constant coefficients. Central to this demonstration
is the existence of the monodromy matrix which arises from the periodic
nature of the differential equation.
A question which arises naturally when one sees the results for
periodic systems is whether some transformation having a similar
effect exists for nonperiodic systems. Since systems are often
derived from higher order equations, the question may be extended
to the existence of a transformation which converts a
nonautonomous scalar equation to an autonomous scalar equation.
For second order linear equations this is known to be the case for the time-dependent
oscillator
\begin{equation}
{q}'' +\omega^2 (t)q = 0\Leftrightarrow H =
\half\(p^2+\omega^2q^2\),\quad p = {q}'\label{1.1}
\end{equation}
(the prime denotes differentiation with respect to the independent variable time, $t $)
which has been the subject of numerous investigations.
Lewis \cite{Lewis 67, Lewis 68a, Lewis 68b, Lewis 69} provided the
modern impetus by showing, using Kruskal's asymptotic method
\cite{Kruskal 62}, that there exists an invariant
\begin{equation}
I = \frac{1}{ 2\rho^2}\lb\(\rho p-\rho' q\)^2+\frac{q^2}{\rho^2}\rb\label{1.2}
\end{equation}
for the Hamiltonian \re{1.1}, where $\rho (t) $ is a solution of the auxiliary equation
\begin{equation}
{\rho}'' +\omega^2\rho = \frac{1}{\rho^3},\label{1.3}
\end{equation}
commonly known as the Ermakov-Pinney equation after Ermakov who, as far as it
is known, in 1880 was the first to treat the problem of the integration
of \re{1.1} \cite{Ermakov 80} and Pinney who provided the statement of the
solution of \re{1.3} in 1950 \cite{Pinney 50}. Subsequently \re{1.2} was
shown to be the representation, \re{1.1}, in the original coordinates of the Hamiltonian
\begin{equation}
\bar{H} = \half\(P^2+Q^2\)\label{1.4}
\end{equation}
by means of a time-dependent linear canonical transformation coupled with a change of time variable
\cite{Gunther 77, Leach 77},
\begin{equation}
Q = \frac{q}{\rho},\quad P = \rho p- \rho'q,\quad T = \int\rho^{- 2} (t)\d t,\label{1.5}
\end{equation}
an instance of the generalised canonical transformation of Burgan \etal \cite{Burgan 78}. Instead of the generalised canonical transformation, \re{1.5}, one could transform the original nonautonomous equation to
\begin{equation}
\ddot{Q}+ Q = 0\label{1.6}
\end{equation}
by means of the Kummer-Liouville transformation \cite{Kummer 34, Liouville 37}
\begin{equation}
Q = \frac{q}{\rho},\quad T = \int\rho^{-2} (t)\d t\label{1.7}
\end{equation}
(overdot means differentiation with respect to the `new time', $T $) so that the solution of \re{1.1} follows from that of \re{1.7} as
\begin{equation}
q (t) = \rho (t)\left[A\sin T (t) + B\cos T (t)\right],\label{1.8}
\end{equation}
where $A $ and $B $ are the required two arbitrary constants of
integration. Some examples of classes of nonautonomous higher
order scalar equations have been shown to be rendered autonomous
by transformations of the type \re{1.7} by Hill \cite{Hill 92} who
used a method based on group rather than algebraic
considerations. As has been demonstrated by Aguirre and Krause
\cite{agu88a,agu88b}, generally the use of the finite transformations of
the group is more cumbersome than the infinitesimal
transformations associated with the corresponding algebra.
In this paper we address two major questions. The first is the possibility of
the existence of a transformation based on the presence of symmetries which
transforms a nonautonomous system of first order ordinary differential
equations into an autonomous system, thereby complementing the results following
from the Floquet theory of systems of periodic differential equations. The
second question is to determine the algebraic conditions which allow the
possibility of the transformation of a nonautonomous higher order scalar
equation to an autonomous one of the same order. Given that the higher order
equation can be written in the form of a system of first order linear
equations,
the connection between the two results becomes also a matter of interest.
An $n $th order linear scalar equation of maximal symmetry ($n+ 4 $ Lie point
symmetries) is shown always to be transformable to an autonomous equation of
the same order which is what one might expect from studies of other aspects of $n $th
order linear ordinary differential equations \cite{Mahomed 90, Flessas 95}. In
the deductions we find that there is an intermediate class of nonautonomous
equations which can be transformed to autonomous form. Members of the class
have $n+ 2 $ Lie point symmetries, not the $n+ 4 $ of the equations of maximal
order, and we find that they have a specific structure which demarcates them
from the generic $n $th order equations possessing $n+ 1 $ symmetries. This
explanation of the reason for the differing number of symmetries is one of the
new results featured in this paper.
This second class
of scalar $n $th order ordinary differential equations shows that the third
order Ermakov-Pinney equation
\begin{equation}
{\xi}''' + 2\omega^2{\xi}' + 2\omega{\omega}'\xi = 0\label{1.9}
\end{equation}
-- formally obtained by
differentiation of \re{1.3} with respect to time and setting $\xi =\rho^2 $;
but in fact of more fundamental importance in the study of symmetries of
differential equations \cite{Mahomed 90} -- plays a determining role. We see
that this equation arises not only with scalar equations of higher order but
also in systems of first order equations. Its integral, {\it videlicet}
\begin{equation}
\xi{\xi}'' -\half{\xi}'^2+ \omega^2\xi^2 = K,\label{1.10}
\end{equation}
has an interesting role in the transformation of nonautonomous
second order ordinary differential equations via two-dimensional
linear systems. We show, firstly through demonstration by example
and then in general, that the classical equations of Mathematical
Physics, {\it videlicet} those scalar linear second order
equations which arise from separation of variables in partial
differential equations arising in field theory and quantum
mechanics, reduce to the first order system for the simple
harmonic oscillator with the frequency determined by the constant
of integration, $K $, of the relevant Ermakov-Pinney equation.
This process of transformation and solution provides the potential
of unexpected relations for many special functions.
\section{Transformations of nonautonomous systems to
autonomous systems \label{sect2}}
We consider the $n $-dimensional homogeneous system of nonautonomous first order equations
\begin{equation}
{\bfu}' = A (t)\bfu\Leftrightarrow {u}'_i = a_{ij} (t)u_j,\label{2.1}
\end{equation}
where $A (t) $ is an $n\times n $ matrix the elements of which, $a_{ij} $,
can be nontrivial functions of time and summation over repeated
indices is implied, as always in the sequel unless specifically
excluded. (Hereinafter we suppress the explicit dependence $(t) $
unless the context demands it.) Since \re{2.1} is a first order
system, it possesses an infinite number of Lie point symmetries,
which are equally generalised symmetries,
a feature to be born in mind when making the transition from scalar $n $th order
equation to $n $-dimensional first order system. Apart from any
other Lie point symmetries \re{2.1} possesses the obvious
symmetries of
\begin{eqnarray}
G_{H} & = & u_i\up{u_i}\nonumber\\
G_{Si} & = & b_{ij} (t)\up{u_j}\quad j = 1,n\nonumber\\
G_{sl} & = & s_0\upt + s_{ij}u_j\up{u_i},\quad i,j = 1,n,\label{2.2}
\end{eqnarray}
where $G_{H} $ and $G_{Si} $ represent the homogeneity and $n $ solution
symmetries respectively and $G_{sl} $ the type of symmetry of the form
associated with the rescaling symmetry in the algebra $sl (2,R) $
characteristic of linear systems of maximal order \cite{Gorringe 87,
Mahomed 99}. When the matrix $A(t)$ has some specific substructure, there are
additional homogeneity symmetries. For example, if $A(t)$ is diagonal, there
exist $n$ homogeneity symmetries of the form $x_i\up{x_i}$ with no summation
on $i$.
The most suitable candidate for the determination of a linear transformation
to render \re{2.1} autonomous is one of the type $G_{sl} $. The first
extension is
\begin{equation}
G_{sl}^{\lb 1\rb} = s_0\upt + s_{ij}u_j\up{u_i}
+ \({s}'_{ij}u_j+ s_{ij}{u}'_j- {s}'_0 {u}'_j\)\up{{u}'_i}\label{2.3}
\end{equation}
and the application of \re{2.3} to \re{2.1} with \re{2.1}
taken into account gives
\begin{equation}
{s}'_{kj}u_j+ s_{kl}a_{lj}u_{j} - {s}'_0a_{kj}u_j =
s_0{a}'_{kj}u_j+ a_{ki}s_{ij}u_j,\quad k = 1,n
\label{2.4}
\end{equation}
so that each independent variable yields the equation
\begin{equation}
{s}'_{kj} = s_0{a}'_{kj} +{s}'_0a_{kj} -s_{kl}a_{lj} +a_{ki}s_{ij},\quad k,j = 1,n\label{to buy}
\end{equation}
and the system of $n^2 $ equations to be solved to determine the symmetry is
\begin{equation}
{S}' =\(s_0 A\)' + AS -S A.\label{2.6}
\end{equation}
The system \re{2.6} of $n^2$ independent equations contains
$n^2+1$ dependent variables. Evidently one of these,
logically/\ae sthetically $s_0$, plays the role of a parametric
function and may well be set at zero (or 1 if so desired). In this
we already note a contrast with the transformation \re{1.7} in
which the introduction of new time is mandatory. We see below
that the type of relation required in \re{1.7} is a consequence of
the {\it ab initio} restriction of $S$ to be zero in its upper
triangle.
The canonical variables are found from the characteristics of $G_{sl}$ once the
solution of \re{2.6} is obtained -- not necessarily a trivial task. In general
the associated Lagrange's system is
\begin{equation}
\frac{\d t}{s_0} =\frac{\d u_1}{s_{1j}u_j} =\frac{\d
u_2}{s_{2j}u_j} =\ldots =\frac{\d u_n}{s_{nj}u_j}, \label{2.7}
\end{equation}
\ie
\begin{equation}
0 = \frac{s_0\d u_i}{\d t} - s_{ij}u_j,\quad i = 1,n.\label{2.8}
\end{equation}
We take the solution to be
\begin{equation}
\bfu (t) = B^{- 1}\bfu_0,\quad T = \int \frac{\d
t}{s_0},\label{2.9}
\end{equation}
where $\bfu_0 $ is a constant vector and $B $ is the solution of
\begin{equation}
s_0B' + BS = 0\quad\Leftrightarrow\quad \dot{B}+BS=0,\label{2.10}
\end{equation}
so that the new variables are given by
\begin{equation}
\bfw = B\bfu,\quad T = \int \frac{\d t}{s_0}.\label{2.11}
\end{equation}
The system of equations in the new variables is
\begin{eqnarray}
\dot{\bfw}& = & s_0^{- 1}\(B{\bfu}' +{B}'\bfu\)\nonumber\\ & = &
s_0^{- 1}B (A -S)B^{- 1}\bfw\nonumber\\ & = & C\bfw,\label{2.12}
\end{eqnarray}
where overdot means differentiation with respect to the new time,
$T$, and the constancy of $C $ may be verified by differentiation
with respect to $T $ when\re{2.6}, \re{2.9} and the \re{2.10} are
taken into account.
The existence of these solutions of \re{2.6} and \re{2.8} is
guaranteed provided the elements of $A $ are continuous \cite{Ince
27} [p 72]. Thus we have\newline
{\bf Theorem 1}:
{\it The system of linear equations ${\bfu}' = A (t)\bfu $,
where the elements, $a_{ij} (t) $, of $A $ are continuous
functions of the independent variable $t $ may be transformed to
the autonomous system $\dot{\bfw} = C\bfw $ by means of the
Kummer-Liouville transformation
\begin{equation}
\bfw = B\bfu,\quad T = \int \frac{\d t}{s_0},\nonumber
\end{equation}
where $B $ is the matrix determined by the coefficient functions of the Lie point symmetry
\begin{equation}
\Gamma = s_0\upt + s_{ij}u_j\up{u_i}\nonumber
\end{equation}
of the original system given by the solution of the system
\begin{equation}
{S}' = (s_0A)' + AS -S A.\nonumber
\end{equation}}
Without loss of generality $s_0 $ can be set at 0 or 1 in which
case there is no need for a transformation of the independent
variable. The result for periodic systems employing the monodromy
matrix of Floquet theory is now extended to all systems using a
particular class of the Lie point symmetries of the system. These
are the generalised rescaling symmetries. (Conventionally a
rescaling symmetry has the form $\alpha_0t\upt +
\alpha_{ij}u_i\up{u_j} $ with the $\alpha_{ij} $ constants. Here
the coefficients have been extended to include functions of time.
Even nonlocal generalised rescaling symmetries have found
application \cite{Krause 94, Pillay 98}, although they may be
local symmetries of an equivalent system of first order equations
\cite{Nucci 96b}.)
We can now envisage the process of the solution of a nonautonomous equation as
a series of processes which are summarised in the diagram contained in
Figure 1.
The $n $th order nonautonomous equation is converted to what is commonly called
the equivalent $n $-dimensional system of linear equations although in the
language of symmetries one should always be sensitive to the nonlocal change of
variables which means that the type of symmetries, say point, is not
necessarily preserved. The generalised symmetries of the $n $th order equation
are now all point symmetries of the first-order system. The generalised
self-similar symmetries of the system are calculated and their canonical
variables are used to construct the autonomous system corresponding to the
nonautonomous system. The solution of the autonomous system is trivial and
with these solutions the solution to the original nonautonomous $n $th order
equation is obtained through the inverse transformation which, since the
original transformation is linear, is easily accomplished.
%\newpage
\begin{center}
\unitlength=1.00mm \special{em:linewidth 0.4pt} %
\linethickness{0.4pt} \thicklines
\begin{picture}(159.00,155.00)
\put(10.00,145.00){\framebox(30.00,10.00)[cc]{$\bfu' = A(t)\bfu$}}
\put(40.00,150.00){\vector(1,0){45.00}}
\put(85.00,150.00){\vector(-1,0){45.00}}
\put(85.00,145.00){\framebox(70.00,10.00)[cc] {$u_1^{(n)}
+p(t)u_1^{(n- 2)} +\sum_{i = 0}^{n- 3}q_i(t)u_1^{(i)}=0$}}
\put(25.00,145.00){\line(0,-1){10}}
\put(25.00,130.00){\makebox(0,0)[cc]{$s_{ij}(t)u_j\p_{u_i}$}}
\put(25.00,125.00){\vector(0,-1){10.00}}
\put(25.00,110.00){\makebox(0,0)[cc]{$\bfs_{j+1} = \bfs'_j +
A^T\bfs_j$}} \put(25.00,100.00){\makebox(0,0)[cc]{$j=1,n-1$}}
\put(25.00,90.00){\makebox(0,0)[cc]{$\bfs_1$ satisfies nth-order
K-vW eqs}} \put(25.00,85.00){\vector(0,-1){10.00}}
\put(10.00,65.00){\framebox(30.00,10.00)[cc]{$\dot{\bfw}= C\bfw
$}} \put(25.00,65.00){\vector(0,-1){20.00}}
\put(10.00,35.00){\framebox(30.00,10.00)[cc]{$\bfu=\bfF(t,\bfc)$}}
\put(120.00,145.00){\line(0,-1){10}}
\put(120.00,130.00){\makebox(0,0)[cc]{$s_0(t)\p_t+s_0'(t)^{(n-1)/2}u_1\p_{u_1}$}}
\put(120.00,125.00){\vector(0,-1){10.00}}
\put(120.00,110.00){\makebox(0,0)[cc]{${(n+1)!\over 4!(n-2)!}
s_0'''+ p s_0'+ {1\over 2} p's_0 = 0$}}
\put(120.00,100.00){\makebox(0,0)[cc]{$q_i(t)$ special }}
\put(120.00,95.00){\vector(0,-1){10.00}}
\put(85.00,75.00){\framebox(70.00,10.00)[cc]{${{\rm d}^n v_1
\over{\rm d}T^n }+K{{\rm d}^{n-2}v_1 \over{\rm d}T^{n-2}}
+\sum_{i=0}^{n-3}D_i{{\rm d}^i v_1 \over{\rm d}T^{i}} = 0$}}
\put(120.00,75.00){\vector(0,-1){20}}
\put(105.00,45.00){\framebox(30.00,10.00)[cc]{$u_1=f(t,\bfc)$}}
\put(120.00,45.00){\line(0,-1){5}}
\put(44.00,40.00){\line(1,0){76}}
\put(43.00,40.00){\makebox(0,0)[cc]{\LARGE$\ni$}}
\put(65.00,0.00){\makebox(0,0)[cc]{Figure 1:}}
\put(65.00,-10.00){\makebox(0,0)[cc]{Nth order equation.}}
\end{picture}
\end{center}
\newpage
\section{Second order equations \label{sect3}}
We commence our considerations of second order equations with two examples.
The first of these is Bessel's equation
\begin{equation}
x^2y''+xy'+\(x^2-\mu^2\)y = 0\label{3.1}
\end{equation}
the solution set of which, Bessel's functions, is probably the
most tabulated of all the higher transcendental functions.
Bessel's functions arise from diverse sources apart from the
differential equation \re{3.1}, such as from the group
representation of the Euclidean group, $E_2 $, of translations and
dilations in the plane \cite{Chuk 91}. Apart from preceding
Bessel's systematic study of their properties by about a century,
the use of particular Bessel's functions to solve specific
problems is probably the first application of special functions to
the solution of physical problems. The earliest known instance in
Mechanics was in the investigation of transverse oscillations of a
hanging heavy chain fixed only at its point of suspension by
Daniel Bernoulli in 1732, but an earlier instance of application
in the class of Riccati equations is reported in the
correspondence of Leibniz (03:10:1702) \cite{Dutka 95}. (Colwell
gives a date of 1703 without any details \cite{Colwell 92}.) The
work of Euler on what became known as elliptic functions was a few
years later in the period 1756-1757. Curiously both areas were
seriously established at about the same time \cite{Legendre,
Jacobi, Rosati, Bessel 24, Abel 26, Gauss 76}. By the way,
Hermite, who began his career with Jacobi's help, proposed a
generalization of Floquet theory for double periodic functions in
a series of papers on applications of elliptic functions
\cite{Hermite}.
We write \re{3.1} as the two-dimensional system
\begin{equation}
\(\begin{array}{c}y\\z\end{array}\)'= \(\begin{array}{ccc} 0 &\, & 1\\
\dsp{\frac{\mu^2}{x^2}} - 1 &\, &\dsp{-\frac{1}{x}}
\end{array}\)\(\begin{array}{c}y\\z\end{array}\),\label{3.2}
\end{equation}
where now the prime denotes differentiation with respect to the independent
variable, $x $, by the definition of the second variable $z =y'$. If we take $s_0 = 1 $ in \re{2.6},
we obtain the system
\begin{eqnarray}
\(\begin{array}{c}s_{11}\\s_{12}\\s_{21}\\s_{22}\end{array}\)'&=&
\(\begin{array}{cccc} 0 & \(1-\dsp{\frac{\mu^2}{x^2}}\) & 1 & 0\\
- 1 &\dsp{\frac{1}{x}} & 0 & 1\\ -\(1-\dsp{\frac{\mu^2}{x^2}}\) &
0 & \dsp{\frac{1}{x}} & \(1-\dsp{\frac{\mu^2}{x^2}}\)\\ 0 &
-\(1-\dsp{\frac{\mu^2}{x^2}}\) & - 1 & 0
\end{array}\)\(\begin{array}{c}s_{11}\\s_{12}\\s_{21}\\s_{22}\\
\end{array}\) \nonumber \\
&&+\(\begin{array}{c} 0\\0\\-2\dsp{\frac{\mu^2}{x^3}}\\
\dsp{\frac{1}{x^2}}\end{array}\)\label{3.3}
\end{eqnarray}
from which we may write $s_{11} $, $s_{21} $ and $s_{22} $ in terms of $s_{12}
$ which satisfies the nonhomogeneous third order ordinary differential equation
\begin{equation}
s'''_{12}+\(4-\frac{4\mu^2-1}{x^2}\)s'_{12}+\frac{4\mu^2-1}{x^3}s_{12} =
\frac{4\mu^2-x^2}{x^3}.\label{3.4}
\end{equation}
Equation \re{3.4} has three linearly independent solutions and so
there are three independent transformations available to render
\re{3.2} into autonomous form. However, the determination of the
characteristics is a by no means trivial task and we employ a
different stratagem. We set $s_{12} = 0 $ and allow $s_0 $ to remain
unspecified. Now \re{2.6} with \re{3.2} gives
\begin{eqnarray}
\(\begin{array}{c}s_{11}\\0\\s_{21}\\s_{22}\end{array}\)'&=&
\(\begin{array}{cccc} 0 & 0 & 1 & 0\\ - 1 &0 & 0 & 1\\
-\(1-\dsp{\frac{\mu^2}{x^2}}\) & 0 & -\dsp{\frac{1}{x}} &
\(1-\dsp{\frac{\mu^2}{x^2}}\)\\ 0 & 0 & -1& 0
\end{array}\)\(\begin{array}{c}s_{11}\\0\\s_{21}\\s_{22}\\ \end{array}\)
\nonumber \\&& +\(\begin{array}{c}
0\\0\\-2\dsp{\frac{\mu^2}{x^3}}\\
\dsp{\frac{1}{x^2}}\end{array}\)s_0 +\(\begin{array}{c}
0\\1\\-\(1-\dsp{\frac{\mu^2}{x^2}}\)\\
-\dsp{\frac{1}{x}}\end{array}\)s'_0 \label{3.5}
\end{eqnarray}
and we find that the governing equation is that for $s_0 $, {\it videlicet}
\begin{equation}
s'''_0+\(4+\frac{1-4\mu^2}{x^2}\)s'_0-\frac{1- 4\mu^2}{x^3}s_0 = 0,\label{3.6}
\end{equation}
which we recognise as the third-order Ermakov-Pinney equation, and
\begin{equation}
s_{11} =\half\(s'_0-\frac{s_0}{x}\),\quad s_{21}
=\half\(s'_0-\frac{s_0}{x}\)',\quad s_{22} = -\half\(s'_0+\frac{s_0}{x}\).
\label{3.7}
\end{equation}
The canonical variables are found from the associated Lagrange's system
\begin{equation}
\frac{\d x}{s_0} = \frac{\d y}{\half\(s'_0-\frac{s_0}{x}\)y} = \frac{\d z}{\half\(s'_0-\frac{s_0}{x}\)'y
-\half\(s'_0+\frac{s_0}{x}\)z}\label{3.77}
\end{equation}
the solution of which is particularly simple due to the point nature of the
symmetry. We find that
\begin{eqnarray}
v_1 & = & y\(\frac{x}{s_0}\)^{\half}\nonumber\\
v_2 & = &\(s_0x\)^{\half}z-\lb\(\frac{x}{s_0}\)^{\half}\rb'y\nonumber\\
T & = & \int\frac{\d x}{s_0 (x)}.\label{3.8}
\end{eqnarray}
It is easy enough to show that
\begin{equation}
\dm{v_1}{T} = v_2\label{3.9}
\end{equation}
and somewhat less easy to show that
\begin{equation}
\dm{v_2}{T} = -\half v_1\lb s_0s''_0-\half s'_0{}^2+\half s_0^2\(4-\frac{4\mu^2-1}{x^2}\)\rb.\label{3.10}
\end{equation}
The Ermakov-Pinney equation admits the integrating factor $s_0 $ and we obtain
\begin{equation}
s_0s''_0-\half s'_0{}^2+\half s_0^2\(4-\frac{4\mu^2-1}{x^2}\) =
2\Omega^2\label{3.11}
\end{equation}
so that the autonomous system is
\begin{equation}
\frac{\d\phantom{T}}{\d T}\(\begin{array}{c}v_1\\v_2\end{array}\) =
\(\begin{array}{cc} 0 & 1\\-\Omega^2 & 0\end{array}\)
\(\begin{array}{c}v_1\\v_2\end{array}\)\label{3.12}
\end{equation}
which is the easily solved equation for the simple harmonic oscillator
provided $\Omega^2 > 0 $. Under the standard
change of variable $s_0 =\rho^2 $, \re{3.11} takes the usual form for the second order Ermakov-Pinney equation
\begin{equation}
\rho''+\(1-\frac{\mu^2-\oqr}{x^2}\)\rho = \frac{\Omega^2}{\rho^3}\label{3.13}
\end{equation}
which has been given fruitful interpretation \cite{Eliezer 76} as the radial component of the equation of motion
\begin{equation}
\bfr'' +\(1-\frac{\mu^2-\oqr}{x^2}\)\bfr = 0,\label{3.14}
\end{equation}
where $\rho =|\bfr| $. The term on the right side of \re{3.13} is related to
the conservation of the square of angular momentum and so one
expects $\Omega^2\geq 0 $. (One notes that quantum mechanically
this requirement is reduced to $\Omega^2 > -\oqr $ to prevent
`collapse into the centre' \cite{Camiz 71}, but that should not be
the case here for which \re{3.14} is quite classical.)
Before we move to an interpretation of these results we consider a second
example which is that of the quantal oscillator with Schr\"odinger equation
\begin{equation}
\(-\half\pp{}{x} +\half x^2\)\psi =
i\pa{\psi}{t}\label{3.15}
\end{equation}
for which the spatial part of the wavefunction, $\psi $, satisfies
\begin{equation}
u''_1+ \(\lambda -x^2\)u_1 = 0\label{3.60}
\end{equation}
in the usual process of separation of variables. For the two-dimensional
system,
\begin{equation}
\(\begin{array}{c}u_1\\u_2\end{array}\)'=\(\begin{array}{ccc} 0 &\, & 1\\
x^2-\lambda &\, & 0\end{array}\)
\(\begin{array}{c}u_1\\u_2\end{array}\),\label{3.17}
\end{equation}
we use a generalised self-similarity symmetry of the type \re{2.2} with $s_{12} = 0 $ and $s_0\neq 0 $. We obtain
\begin{equation}
\Gamma = s_0\upx +\half s'_0u_1\up{u_1} +\half\(s''_0u_1-s'_0u_2\)\up{u_2},\label{3.18}
\end{equation}
where
\begin{equation}
s'''_0+ 4\(\lambda -x^2\)s'_0-4xs_0 = 0\label{3.19}
\end{equation}
which is of Ermakov-Pinney type with the integrated form
\begin{equation}
s_0s''_0-\half s'_0{}^2+ 2\(\lambda -x^2\)s_0^2 = 2\Omega^2.\label{3.20}
\end{equation}
The canonical variables are
\begin{equation}
v_1 = \frac{u_1}{s_0^{1/2}},\quad v_2 =
s_2^{1/2}u_2-\(s_0^{1/2}\)'u_1, \quad T =\int\frac{\d x}{s_0}
\label{3.21}
\end{equation}
and the autonomous system is the simple harmonic oscillator
\re{3.12}.
If we again write $s_0 = \rho^2 $, the solution of \re{3.12} gives
for the solution of \re{3.60}
\begin{equation}
u_1 = \rho\lb A_1\sin\Omega T+ A_2\cos\Omega T\rb\label{3.23}
\end{equation}
which is a very classical way of writing the solution of \re{3.60}! We note
that the solution of \re{3.19} is (courtesy of Maple V)
\begin{equation}
s_0 = \frac{1}{x}\left\{C_1W_W^2+C_2W_WW_M+C_3W_M^2\right\},\label{3.24}
\end{equation}
where $W_W\(\oqr\lambda,\oqr,x^2\) $ and $W_M\(\oqr\lambda,\oqr,x^2\) $ are
Whittaker's $W $ and $M $ functions \cite{Abramowitz 65} [Ch 13, p. 504].
We observe that the transformation takes no note of the
eigenvalues of the original equation \re{3.60} which, of course,
are a consequence of imposed boundary conditions. We are now in a
position to state\newline
{\bf Theorem 2}: {\it A second order ordinary differential equation is equivalent
under a point transformation to the simple harmonic oscillator
except in special circumstances.}
\noindent{\bf Proof:} Without loss of generality the second order ordinary differential
equation may be taken in the normal form
\begin{equation}
u''_1+q (t)u_1 = 0\label{3.25}
\end{equation}
(the prime denotes differentiation with respect to the independent
variable time). The corresponding system of first-order equations
\begin{equation}
\(\begin{array}{c}u_1\\u_2\end{array}\)'=\(\begin{array}{ccc} 0 &,
& 1\\-q &, & 0\end{array}\)
\(\begin{array}{c}u_1\\u_2\end{array}\)\label{3.26}
\end{equation}
always has a Lie symmetry of the form
\begin{equation}
\Gamma = s_0\upx +\half s'_0u_1\up{u_1} +\half\(s''_0u_1-s'_0u_2\)\up{u_2},\label{3.27}
\end{equation}
where $s_0 $ is a solution of the third order Ermakov-Pinney equation
\begin{equation}
s'''_0+ 4qs'_0+ 2q's_0 = 0.\label{3.28}
\end{equation}
Under the transformation
\begin{equation}
v_1 = \frac{u_1}{s_0^{1/2}},\quad v_2 =
s_0^{1/2}u_2-\(s_0^{1/2}\)'u_1, \quad T = \int\frac{\d x}{s_0}
\label{3.29}
\end{equation}
\re{3.26} becomes the autonomous system
\begin{equation}
\frac{\d\phantom{T}}{\d T}\(\begin{array}{c}v_1\\v_2\end{array}\)
= \(\begin{array}{cc} 0 & 1\\-\Omega^2 & 0\end{array}\)
\(\begin{array}{c}v_1\\v_2\end{array}\),\label{3.30}
\end{equation}
where the constant $\Omega^2$ is obtained from the integration of
\re{3.28} once using the integrating factor $s_0 $ and is given by
\begin{equation}
s_0s''_0-\half s'_0{}^2+ 2qs_0^2 = 2\Omega^2.\label{3.31}
\end{equation}
The solution of the original problem is given by
\begin{equation}
u_1 = \rho\lb A_1\sin\Omega T+ A_2\cos\Omega T\rb,\label{3.32}
\end{equation}
where $s_0 = \rho^2 $.\newline
{\bf Corollary}: {\it The transformation to autonomous form can be achieved without
a change in the independent variable by demanding the symmetry
\begin{equation}
\Gamma = s_{ij}u_j\up{u_i},\label{3.33}
\end{equation}
\ie by making $s_0 = 0 $.}
The price to pay is a
more complicated transformation of the dependent variables. The
essential feature of either method of transformation is a
nonconstant rescaling of the dependent variable. In the new
coordinates, $v_1 $ varies sinusoidally. The effect of the
transformation can make $u_1 $ vary quite erratically \cite{Leach
93}. The schema for second order equations and two-dimensional
systems is illustrated in Figure 2.
We note that we have achieved, albeit in a formal sense, the
unity of all scalar second order order differential equations of
maximal symmetry which is presaged by their possession of the
common algebra of Lie point symmetries, $sl (2,R) $. There may be
some possibility to derive usable relationships between various
special functions. A curious point is that naturally the
transformations considered here lead to a simple harmonic
oscillator rather than the free particle equation which is the
simplest representative of all scalar second order ordinary
differential equations. This is a consequence of the general
integral of the third order Ermakov-Pinney equation. The presence
of the oscillator constant guarantees that there is no singularity
in the rescaling due to an accidental zero in $s_0$. In the case
of Bessel's equation it is amusing to note that one of the first
applications \cite{Dutka 95} of Bessel's functions was in the
solution of Kepler's equation
\begin{equation}
M = E-e\sin E,\label{3.34}
\end{equation}
where $E $ is the eccentric anomaly, $M $ the mean anomaly and $e $ the
eccentricity of an inverse square orbit.
The Kepler Problem is also reducible to a simple harmonic oscillator by means
of Lie symmetries \cite{Nucci 01a}.
%\newpage
\begin{center}
\unitlength=1.00mm \special{em:linewidth 0.4pt} %
\linethickness{0.4pt} \thicklines
\begin{picture}(159.00,155.00)
\put(10.00,145.00){\framebox(30.00,10.00)[cc]{$\bfu' = A(t)\bfu$}}
\put(40.00,150.00){\vector(1,0){60.00}}
\put(85.00,150.00){\vector(-1,0){45.00}}
\put(100.00,145.00){\framebox(40.00,10.00)[cc]
{$u_1''+q(t)u_1=0$}} \put(25.00,145.00){\line(0,-1){10}}
\put(25.00,130.00){\makebox(0,0)[cc]{$s_{ij}(t)u_j\p_{u_i}$}}
\put(25.00,125.00){\vector(0,-1){10.00}}
\put(25.00,110.00){\makebox(0,0)[cc]{$\bfs_2 = \bfs'_1 +
A^T\bfs_1$}}
\put(25.00,100.00){\makebox(0,0)[cc]{$\bfs_1=(-s_2'/2,s_2)$}}
\put(25.00,90.00){\makebox(0,0)[cc]{$s_2'''+4q s_2'+2q's_2=0 $}}
\put(25.00,85.00){\vector(0,-1){10.00}}
\put(10.00,65.00){\framebox(30.00,10.00)[cc]{$\dot{\bfw}= C\bfw
$}} \put(25.00,65.00){\vector(0,-1){20.00}}
\put(10.00,35.00){\framebox(30.00,10.00)[cc]{$\bfu=\bfF(t,\bfc)$}}
\put(120.00,145.00){\line(0,-1){10}}
\put(120.00,130.00){\makebox(0,0)[cc]{$s_0(t)\p_t+{1\over 2}
s_0'(t)u_1\p_{u_1}$}} \put(120.00,125.00){\vector(0,-1){15.00}}
\put(120.00,105.00){\makebox(0,0)[cc]{$ s_0'''+ 4q s_0'+ 2 q's_0
= 0$}} \put(120.00,100.00){\vector(0,-1){15.00}}
\put(100.00,75.00){\framebox(40.00,10.00)[cc]{${{\rm d}^2 v_1
\over{\rm d}T^2 }+Dv_1 = 0$}}
\put(120.00,75.00){\vector(0,-1){30}}
\put(105.00,35.00){\framebox(30.00,10.00)[cc]{$u_1=F_1(t,\bfc)$}}
\put(105.00,40.00){\vector(-1,0){65}}
\put(40.00,40.00){\vector(1,0){65}}
\put(65.00,0.00){\makebox(0,0)[cc]{Figure 2:}}
\put(65.00,-10.00){\makebox(0,0)[cc]{Second order equation.}}
\end{picture}
\end{center}
\newpage
\section{Equations of higher order \label{sect4}}
We have dwelt upon second order equations at some length for two reasons apart
from the frequency of their occurrences in applications. In the first place
linear second order ordinary differential equations are exceptional in that
all have the same Lie algebra, $sl (3,R) $, of point symmetries. This is not the case with higher order linear equations for which the number of Lie point symmetries can be $n+ 1 $, $n+ 2 $ and $n+ 4 $, where $n $ is the order of the equation. Consequently one would expect some differences in the ability to find a
transformation to convert a nonautonomous $n $th order ordinary differential
equation to an autonomous $n $th order ordinary differential equation. The
second point is that the happy facility
one had for the second order linear equations in the choice of symmetry,
\ie putting $s_0 = 0 $ and $s_2\neq 0 $ or vice versa, does not have the same impact.
In both instances the symmetry selected for the linear system is a generalised symmetry of
the $n $th order ordinary differential equation.
We consider firstly the third order equation written in normal form, {\it videlicet}
\begin{equation}
{u}'''_1+p (t){u}'_1+q (t)u_1 = 0.\label{4.1}
\end{equation}
The corresponding system of first order equations is
\begin{equation}
{\bfu}' = A(t)\bfu,\label{4.2}
\end{equation}
where
\begin{equation}
\bfu = \lb\begin{array}{l}u_1\\u_2\\u_3\end{array}\rb,\quad
A = \lb\begin{array}{rrr} 0 & 1 & 0\\0 & 0 & 1\\-q & -p & 0\end{array}\rb.\label{4.3}
\end{equation}
In the case that $s_0 = 0 $, \re{2.6} leads to the system of
determining equations for the symmetry, $s_{ij}u_j
\up{u_i} $,
\begin{eqnarray}
{\bfs}'_1 & = & -A^T\bfs_1+\bfs_2\nonumber\\ {\bfs}'_2 & = &
-A^T\bfs_2+\bfs_3\nonumber\\ \bfs'_3 & = & -q\bfs_1- p\bfs_2-
A^T\bfs_3,\label{4.4}
\end{eqnarray}
where the vectors $\bfs_1 $, $\bfs_2 $ and $\bfs_3 $ consist of the elements
$s_{1j} $, $s_{2j} $ and $s_{3j} $ in
$s_{ij}u_j\up{u_i} $ of the symmetry. It is immediately apparent that the task of determining the $s_{ij} $ is reducible to that of determining $\bfs_1 $ from the system of three third order equations obtained by the elimination of $\bfs_2 $ and $\bfs_3 $ and then obtaining $\bfs_2 $ and $\bfs_3 $ recursively using (\ref{4.4}a,b).
We note that this procedure works {\it mutatis mutandis} for higher order
equations. For an $n $th order equation the three vectors $\bfs_i,i = 1,3 $
become $n $-vectors $\bfs_i,i = 1,n $, the $3\times 3 $ matrix $A $ becomes
$n\times n $ and the number of equations in \re{4.4} $n $ with the addition of more of the type of (\ref{4.4}a,b). For the $n $th order differential equation
\begin{equation}
u_1^{(n)} +pu_1^{(n- 2)} +\sum_{i = 0}^{n- 3}q_iu_1^{(i)}=0\label{4.5}
\end{equation}
the bottom row of $A $ is
\begin{equation}
-q_0, -q_1,\ldots, -q_{n- 3}, -p, 0.\label{4.6}
\end{equation}
The inclusion of $s_0 $ requires the addition of $(010)^T{s}'_0 $,
$(001)^T{s}'_0 $ and $-\lb (qp 0)^T s_0\rb' $ to (\ref{4.4}a,b,c)
respectively with suitable extensions for higher orders.
In determining the Lie point symmetries of \re{4.1} which have the form
\begin{equation}
\Gamma = s_0\upt + s_1u_1\up{u_1}\label{4.7}
\end{equation}
we arrive at two equations for $s_0 $, {\it videlicet}
\begin{eqnarray}
2{s}'''_0+ 2p{s}'_0+{p}'_0 & = & 0\nonumber\\ s^{\mbox{{\small\rm
IV}}}_0+p{s}''_0+ 3q{s}'_0+{q}'s_0 & = & 0. \label{4.8}
\end{eqnarray}
The form of (\ref{4.8}a) is the familiar third order Ermakov-Pinney equation which determines $s_0 $ given $p $. Differentiation of the
former of \re{4.8} and substitution into the latter of \re{4.8} gives the compatibility condition
\begin{equation}
6q{s}'_0+ 2{q}'s_0 = 3{p}'{s}'_0+{p}''s_0\label{4.9}
\end{equation}
which, with the integrating factor $s_0^2 $, is easily integrated to give
\begin{equation}
q = \half {p}'-\frac{ D}{s_0^3}. \label{4.10}
\end{equation}
Consequently for \re{4.1} to be transformable to an autonomous equation it must have the form
\begin{equation}
{u}'''_1+pu'_1+\(\half {p}'-\frac{D}{s_0^3}\)u_1 = 0,\label{4.11}
\end{equation}
where $s_0 $ is given by the solution of (\ref{4.8}a). Alternatively one could write
\begin{eqnarray}
&&{u}'''_1 -\frac{1}{s_0^2}\lb 2s_0{s}''_0-{s}'_0{}^2-K\rb
{u}'_1\nonumber\\ & &-\frac{1}{s_0^3}\lb
s_0^2{s}'''_0-2s_0{s}'_0{s}''_0+ {s}'_0{}^3+K{s}'_0+D\rb u_1 =
0\label{4.12}
\end{eqnarray}
in which $s_0 $ is now be to taken as an arbitrary differentiable
function and $K $, which was the value of the integral $2
s_0{s}''_0-{s}'_0{}^2+ps_0^2 $, is now some parameter. The symmetry is
\begin{equation}
\Gamma = s_0\upt +\dot{s}_0u_1\up{u_1}\label{4.13}
\end{equation}
for which the canonical variables are
\begin{equation}
T = \int\frac{\d t}{s_0}\quad\mbox{\rm and}\quad v_1 =
\frac{u_1}{s_0}.\label{4.14}
\end{equation}
and \re{4.11} ({\it \ae q} \re{4.4}) becomes the constant coefficient equation
\begin{equation}
\dddot{v_1}+K\dot{v_1}+Dv_1 = 0.\label{4.15}
\end{equation}
(As usual overdot means differentiation with respect to new time.)
Here we note an interesting result. When $D = 0 $, \re{4.1}, and so
\re{4.15}, is self-adjoint and possesses seven Lie point
symmetries, the maximum for a third order ordinary differential
equation. When $D\neq 0 $, the equation is not self-adjoint and
has only five Lie point symmetries as it loses two of the elements
of the $sl (2,R) $ subalgebra. Thus we have the curious situation
that the number of Lie point symmetries, five or seven, does not
affect the possibility to transform an equation of the form of
\re{4.1} to autonomous form. The two additional symmetries in the
maximal symmetry case are not necessary for this purpose. We
conclude that \re{4.12} with $D\neq 0 $ represents the most
general form of a third order equation in normal form which can be
rendered autonomous by a point transformation (as in \re{4.14}).
We do observe, however, that there exist three possible
transformations of the third order equation of maximal symmetry to
render it autonomous since any one of the three solutions of
(\ref{4.8}a) can be used. In the case that $D\neq 0$ there is only
one transformation and that is the one with the $s_0$ of the
equation \re{4.12}.
When we determine the generalised self-similar symmetries
\begin{equation}
\Gamma = s_{ij}u_j\up{u_i},\label{4.16}
\end{equation}
in which $s_0 $ is put at zero without loss of generality, of the
linear system \re{4.4}, we obtain three third order equations for
$s_{11} $, $s_{12} $ and $s_{13} $, {\it videlicet} the so-called
Fleischen-van Weltunter (F-vW) system mentioned in the Abstract
\begin{eqnarray}
s_{11}^{\prime\prime\prime}&=& q^{\prime\prime}s_{13} + 3
q^{\prime}s_{13}^{\prime}+ q^{\prime}s_{12} - s_{11}^{\prime}p + 3
s_{12}^{\prime}q + 3 s_{13}^{\prime\prime}q \label{4.17} \\
s_{12}^{\prime\prime\prime}&=& p^{\prime\prime}s_{13} + 3
p^{\prime}s_{13}^{\prime}+ p^{\prime}s_{12} + 2 q^{\prime}s_{13} - 3
s_{11}^{\prime\prime}+ 2 s_{12}^{\prime}p \nonumber \\
&&+ 3 s_{13}^{\prime\prime}p + 3 s_{13}^{\prime}q \label{4.18} \\
s_{13}^{\prime\prime\prime}&=& 2 p^{\prime}s_{13} - 3 s_{11}^{\prime}- 3
s_{12}^{\prime\prime}+ 2 s_{13}^{\prime}p \label{4.19}
\end{eqnarray}
(using the interactive code of Nucci \cite{Nucci 90, Nucci 96}).
These equations are compatible and so all third order ordinary
differential equations can be rendered autonomous in this way.
There is no contradiction in terms of the numbers of Lie point
symmetries in the case of a \re{4.1} with only four Lie point
symmetries as the transformation is generalised and so
conservation of the number of point symmetries is not assured.
The same procedure applies to higher order equations. Only in the case of
second order equations is there a difference due to the peculiar nature of
the Lie point symmetries of scalar second order linear differential equations.
One has a choice of applying a symmetry as in \re{4.7} to obtain a point
transformation to be able to render autonomous equations having $n+ 4 $
or $n+ 2 $ Lie point symmetries or the generalised symmetry \re{4.16} to
render autonomous all linear differential equations through the underlying
linear system. As the pattern is established with the third order equations,
we summarise the procedure in Figure 3.
We noted that our procedure enabled us to identify the general form of a
linear third order ordinary differential equation with five Lie point
symmetries (\re{4.11}, or equivalently \re{4.12}). This same process can be
applied to higher order linear equations almost as easily using the interactive
REDUCE code of Nucci \cite{Nucci 90, Nucci 96}. As Hill \cite{Hill 92}
has already given the result, obtained by a different approach, for the class
of fourth
order equations, we give the result for the fifth order equation
\begin{equation}
u_1^{\mbox{{\rm\small V}}} +p (t) {u}'''_1+q_2 (t){u}''_1+q_1 (t)
{u}'_1+q_0 (t)u_1 = 0.\label{4.20}
\end{equation}
\newpage
\begin{center}
\unitlength=1.00mm \special{em:linewidth 0.4pt} %
\linethickness{0.4pt} \thicklines
%\newpage
\begin{picture}(159.00,155.00)
\put(10.00,145.00){\framebox(30.00,10.00)[cc]{$\bfu' = A(t)\bfu$}}
\put(40.00,150.00){\vector(1,0){50.00}}
\put(90.00,150.00){\vector(-1,0){50.00}}
\put(90.00,145.00){\framebox(60.00,10.00)[cc]{$u'''_1+p (t)u'_1+q
(t)u_1 = 0$}} \put(25.00,145.00){\line(0,-1){10}}
\put(25.00,130.00){\makebox(0,0)[cc]{$s_{ij}(t)u_j\p_{u_i}$}}
\put(25.00,125.00){\vector(0,-1){10.00}}
\put(25.00,110.00){\makebox(0,0)[cc]{$\bfs_{2} = \bfs'_1 +
A^T\bfs_1$}} \put(25.00,100.00){\makebox(0,0)[cc]{$\bfs_{3} =
\bfs'_2 + A^T\bfs_2$}}
\put(25.00,90.00){\makebox(0,0)[cc]{$\bfs_1$ satisfies third order
K-vW eqs (4.17)-(4.19)}} \put(25.00,85.00){\vector(0,-1){10.00}}
\put(10.00,65.00){\framebox(30.00,10.00)[cc]{$\dot{\bfw}= C\bfw
$}} \put(25.00,65.00){\vector(0,-1){20.00}}
\put(10.00,35.00){\framebox(30.00,10.00)[cc]{$\bfu=\bfF(t,\bfc)$}}
\put(120.00,145.00){\line(0,-1){10}}
\put(120.00,130.00){\makebox(0,0)[cc]{$s_0(t)\p_t+s_0'(t)u_1\p_{u_1}$}}
\put(120.00,125.00){\vector(0,-1){10.00}}
\put(120.00,110.00){\makebox(0,0)[cc]{$2s'''_0+ 2ps'_0+p's_0 =
0$}} \put(120.00,100.00){\makebox(0,0)[cc]{$q(t) = {1\over 2}
p'-\frac{ D}{s_0^3} $}} \put(120.00,95.00){\vector(0,-1){10.00}}
\put(90.00,75.00){\framebox(60.00,10.00)[cc]{${{\rm d}^3 v_1\over
{\rm d}T^3} +K{{\rm d}v_1\over {\rm d}T}+Dv_1 = 0$}}
\put(120.00,75.00){\vector(0,-1){20}}
\put(105.00,45.00){\framebox(30.00,10.00)[cc]{$u_1=f(t,\bfc)$}}
\put(120.00,45.00){\line(0,-1){5}}
\put(44.00,40.00){\line(1,0){76}}
\put(43.00,40.00){\makebox(0,0)[cc]{\LARGE$\ni$}}
\put(65.00,0.00){\makebox(0,0)[cc]{Figure 3:}}
\put(65.00,-10.00){\makebox(0,0)[cc]{Third order equation.}}
\end{picture}
\end{center}
\vspace{30mm}
We find that the conditions on the functions $q_i (t),i = 0,2 $ for \re{4.20}
to be rendered autonomous by this type of transformation are
\begin{eqnarray}
q_2 & = & \frac{3}{2} {p}'-\frac{1}{2}\frac{D_2}{{s}_{0}^3}\nonumber\\ q_1 &
=
&\frac{9}{10}{p}''+\frac{4}{25}p^2+\frac{3}{2}\frac{{s}'_0D_2}{s_0^4}
+\frac{D_1}{50s_0^4}\nonumber\\ q_0 & = & \frac{1}{5}
{p}'''+\frac{4}{25}p p'-\(\frac{3}{2}\frac{{s}'_0{}^2}{{s}_{0}^5}
-\frac{p}{10s_0^3}\)D_2 -\frac{{s}'_0D_1}{25s_0^5}
+\frac{D_0}{50s_0^5}\label{4.21}
\end{eqnarray}
and the canonical variables for the transformation are found from the Lie point symmetries
\begin{equation}
\Gamma = s_0\upt + 2s'_0u_1\up{u_1},\label{4.22}
\end{equation}
where $s_0 $ is the solution of the Ermakov-Pinney equation
\begin{equation}
10{s}'''_0+ 2p{s}'_0+p's_0 = 0. \label{4.23}
\end{equation}
The fifth order equations of maximal symmetry are obtained when all the
arbitrary constants $D_i,i = 0,2 $, are zero, \cf the results given in Mahomed and Leach \cite{Mahomed 90}. If any of the constants, $D_i $, are nonzero, the number of point symmetries is reduced to seven. If any of the functions $q_i,i = 0,2 $, departs from the prescriptions of \re{4.21}-\re{4.23}, the number of symmetries is reduced to six. When the conditions are satisfied, the autonomous form of \re{4.20} is
\begin{equation}
{{\rm d}^5 v_1\over{\rm d}T^5} +K{{\rm d}^3 v_1\over{\rm d}T^3}
+D_2{{\rm d}^2v_1\over{\rm d}T^2}+\(\frac{4}{25}K+D_1\){{\rm d}
v_1 \over{\rm d}T}+D_0v_1 = 0.\label{4.24}
\end{equation}
The F-vW system for \re{4.20} is
\begin{eqnarray}
s_{11}^{V} &=& - 6 p^{\prime\prime}q_0 s_{15} - 4
p^{\prime}q_0^{\prime}s_{15} - 15 p^{\prime}s_{15}^{\prime}q_0 - 3
p^{\prime}q_0 s_{14} + q_0^{IV} s_{15} \nonumber \\
&& + 5 q_0^{\prime\prime\prime}s_{15}^{\prime}+
q_0^{\prime\prime\prime}s_{14}+ 5 q_0^{\prime\prime}s_{14}^{\prime}+ 10
q_0^{\prime\prime}s_{15}^{\prime\prime}+ q_0^{\prime\prime}s_{13} + 5
q_0^{\prime}s_{13}^{\prime} \nonumber \\
&& + 10 q_0^{\prime}s_{14}^{\prime\prime}+ 10
q_0^{\prime}s_{15}^{\prime\prime\prime}- 2 q_0^{\prime}s_{15}^{\prime}p +
q_0^{\prime}s_{12} - 4 q_2^{\prime}q_0 s_{15} \nonumber \\
&& - s_{11}^{\prime\prime\prime}p - s_{11}^{\prime\prime}q_2 -
s_{11}^{\prime}q_1 + 5 s_{12}^{\prime}q_0 + 10 s_{13}^{\prime\prime}q_0 + 10
s_{14}^{\prime\prime\prime}q_0 \nonumber \\
&& - 2 s_{14}^{\prime}p q_0 + 5 s_{15}^{IV} q_0 - 7 s_{15}^{\prime\prime}p
q_0 - 3 s_{15}^{\prime}q_0 q_2 \label{5.1}
\end{eqnarray}
\begin{eqnarray}
s_{12}^{V} &=& - 6 p^{\prime\prime}q_1 s_{15} - 4
p^{\prime}q_1^{\prime}s_{15} - 15 p^{\prime}s_{15}^{\prime}q_1 - 4
p^{\prime}q_0 s_{15} - 3 p^{\prime}q_1 s_{14} \nonumber \\
&& + 4 q_0^{\prime\prime\prime}s_{15} + 15
q_0^{\prime\prime}s_{15}^{\prime}+ 3 q_0^{\prime\prime}s_{14} + 10
q_0^{\prime}s_{14}^{\prime}+ 20 q_0^{\prime}s_{15}^{\prime\prime}+ 2
q_0^{\prime}s_{13} \nonumber \\
&&+ q_1^{IV} s_{15} + 5 q_1^{\prime\prime\prime}s_{15}^{\prime}+
q_1^{\prime\prime\prime}s_{14} + 5 q_1^{\prime\prime}s_{14}^{\prime}+ 10
q_1^{\prime\prime}s_{15}^{\prime\prime}+ q_1^{\prime\prime}s_{13} \nonumber
\\
&& + 5 q_1^{\prime}s_{13}^{\prime}+ 10 q_1^{\prime}s_{14}^{\prime\prime}+ 10
q_1^{\prime}s_{15}^{\prime\prime\prime}- 2 q_1^{\prime}s_{15}^{\prime}p +
q_1^{\prime}s_{12} - 4 q_2^{\prime}q_1 s_{15} \nonumber \\
&& - 5 s_{11}^{IV} - 3 s_{11}^{\prime\prime}p - 2 s_{11}^{\prime}q_2 -
s_{12}^{\prime\prime\prime}p - s_{12}^{\prime\prime}q_2 + 4
s_{12}^{\prime}q_1 + 10 s_{13}^{\prime\prime}q_1 \nonumber \\
&& + 5 s_{13}^{\prime}q_0 + 10 s_{14}^{\prime\prime\prime}q_1 + 10
s_{14}^{\prime\prime}q_0 - 2 s_{14}^{\prime}p q_1 + 5 s_{15}^{IV} q_1
\nonumber \\
&&+ 10 s_{15}^{\prime\prime\prime}q_0 - 7 s_{15}^{\prime\prime}p q_1 - 2
s_{15}^{\prime}p q_0 - 3 s_{15}^{\prime}q_1 q_2 \label{5.2}
\end{eqnarray}
\begin{eqnarray}
s_{13}^{V}&=& - 6 p^{\prime\prime}q_2 s_{15} - 4
p^{\prime}q_2^{\prime}s_{15} - 15 p^{\prime}s_{15}^{\prime}q_2 - 4
p^{\prime}q_1 s_{15} - 3 p^{\prime}q_2 s_{14} \nonumber \\
&& + 6 q_0^{\prime\prime}s_{15} + 15 q_0^{\prime}s_{15}^{\prime}+ 3
q_0^{\prime}s_{14} + 4 q_1^{\prime\prime\prime}s_{15} + 15
q_1^{\prime\prime}s_{15}^{\prime}+ 3 q_1^{\prime\prime}s_{14} \nonumber \\
&& + 10 q_1^{\prime}s_{14}^{\prime}+ 20 q_1^{\prime}s_{15}^{\prime\prime}+ 2
q_1^{\prime}s_{13} + q_2^{IV} s_{15} + 5
q_2^{\prime\prime\prime}s_{15}^{\prime} \nonumber \\
&& + q_2^{\prime\prime\prime}s_{14} + 5 q_2^{\prime\prime}s_{14}^{\prime}+
10 q_2^{\prime\prime}s_{15}^{\prime\prime}+ q_2^{\prime\prime}s_{13} + 5
q_2^{\prime}s_{13}^{\prime} \nonumber \\
&& + 10 q_2^{\prime}s_{14}^{\prime\prime}+ 10
q_2^{\prime}s_{15}^{\prime\prime\prime}- 2 q_2^{\prime}s_{15}^{\prime}p - 4
q_2^{\prime}q_2 s_{15} + q_2^{\prime}s_{12} \nonumber \\
&& - 10 s_{11}^{\prime\prime\prime}- 3 s_{11}^{\prime}p - 5 s_{12}^{IV} - 3
s_{12}^{\prime\prime}p + 3 s_{12}^{\prime}q_2 \nonumber \\
&& - s_{13}^{\prime\prime\prime}p + 9 s_{13}^{\prime\prime}q_2 + 4
s_{13}^{\prime}q_1 + 10 s_{14}^{\prime\prime\prime}q_2 + 10
s_{14}^{\prime\prime}q_1 \nonumber \\
&& - 2 s_{14}^{\prime}p q_2 + 5 s_{14}^{\prime}q_0 + 5 s_{15}^{IV} q_2 + 10
s_{15}^{\prime\prime\prime}q_1 - 7 s_{15}^{\prime\prime}p q_2 \nonumber \\
&& + 10 s_{15}^{\prime\prime}q_0 - 2 s_{15}^{\prime}p q_1 - 3
s_{15}^{\prime}q_2^2 \label{5.3}
\end{eqnarray}
\begin{eqnarray}
s_{14}^{V} &=& p^{IV} s_{15} + 5 p^{\prime\prime\prime}s_{15}^{\prime}+
p^{\prime\prime\prime}s_{14} + 5 p^{\prime\prime}s_{14}^{\prime}+ 10
p^{\prime\prime}s_{15}^{\prime\prime}- 6 p^{\prime\prime}p s_{15} \nonumber
\\
&&+ p^{\prime\prime}s_{13} - 4 p^{\prime 2} s_{15} + 5
p^{\prime}s_{13}^{\prime}+ 10 p^{\prime}s_{14}^{\prime\prime}+ 10
p^{\prime}s_{15}^{\prime\prime\prime}- 17
p^{\prime}s_{15}^{\prime}p \nonumber \\ &&- 3 p^{\prime}p s_{14} -
4 p^{\prime}q_2 s_{15} + p^{\prime}s_{12} + 4 q_0^{\prime}s_{15} +
6 q_1^{\prime\prime}s_{15} + 15 q_1^{\prime}s_{15}^{\prime}
\nonumber \\ && + 3 q_1^{\prime}s_{14} + 4
q_2^{\prime\prime\prime}s_{15} + 15
q_2^{\prime\prime}s_{15}^{\prime}+ 3 q_2^{\prime\prime}s_{14} + 10
q_2^{\prime}s_{14}^{\prime}+ 20 q_2^{\prime}s_{15}^{\prime\prime}
\nonumber
\\
&& - 4 q_2^{\prime}p s_{15} + 2 q_2^{\prime}s_{13} - 10
s_{11}^{\prime\prime}- 10 s_{12}^{\prime\prime\prime}+ 2 s_{12}^{\prime}p -
5 s_{13}^{IV} \nonumber \\
&&+ 7 s_{13}^{\prime\prime}p + 3 s_{13}^{\prime}q_2 + 9
s_{14}^{\prime\prime\prime}p + 9 s_{14}^{\prime\prime}q_2 - 2
s_{14}^{\prime}p^2 + 4 s_{14}^{\prime}q_1 \nonumber \\
&& + 5 s_{15}^{IV} p + 10 s_{15}^{\prime\prime\prime}q_2 - 7
s_{15}^{\prime\prime}p^2 + 10 s_{15}^{\prime\prime}q_1 - 5 s_{15}^{\prime}p
q_2 + 5 s_{15}^{\prime}q_0 \label{5.4}
\end{eqnarray}
\begin{eqnarray}
s_{15}^{V}&=& 4 p^{\prime\prime\prime}s_{15} + 15
p^{\prime\prime}s_{15}^{\prime}+ 3 p^{\prime\prime}s_{14} + 10
p^{\prime}s_{14}^{\prime}+ 20 p^{\prime}s_{15}^{\prime\prime} \nonumber \\
&&- 4 p^{\prime}p s_{15} + 2 p^{\prime}s_{13} + 4 q_1^{\prime}s_{15} + 6
q_2^{\prime\prime}s_{15} + 15 q_2^{\prime}s_{15}^{\prime}+ 3
q_2^{\prime}s_{14} \nonumber \\
&& - 5 s_{11}^{\prime}- 10 s_{12}^{\prime\prime}- 10
s_{13}^{\prime\prime\prime}+ 2 s_{13}^{\prime}p - 5 s_{14}^{IV} + 7
s_{14}^{\prime\prime}p + 3 s_{14}^{\prime}q_2 \nonumber \\
&& + 9 s_{15}^{\prime\prime\prime}p + 9 s_{15}^{\prime\prime}q_2 - 2
s_{15}^{\prime}p^2 + 4 s_{15}^{\prime}q_1. \label{5.5}
\end{eqnarray}
The outstanding question is whether there is any feature distinguishing
equations of $n+ 4 $ and $n+ 2 $ Lie point symmetries. In the case of the
third order equation \re{4.1} the answer is to be found in its F-vW system
\re{4.17}-\re{4.19}. As a set of higher order equations, the F-vW system can
be analysed for its Lie point symmetries. The maximum number of Lie point symmetries for a
system of $n $ $m $th order ordinary differential equations ($m\geq 3 $) is $n^2+mn + 3 $ \cite{Nucci 01b}, comprising the three symmetries of $sl (2,R) $, $mn $ solution symmetries and $n^2 $ homogeneity symmetries. The structure of the general symmetry is
\begin{equation}
\Gamma = \xi\upt +\sum_{i = 1}^m\(\eta_iu_i+\zeta_i\)\up{u_i}.\label{4.25}
\end{equation}
For \re{4.1} the number of Lie point symmetries of its F-vW system is either the maximal 21 or 18. The difference is determined by the relation \re{4.9} which arises in the simplification of the determining equations for the symmetries of the system.
The F-vW system \re{4.17}-\re{4.19} is the simplest example of that type as the size and order of the system increases with the order of the original equation. Thus for the fourth order equation in normal form
\begin{equation}
u_1^{\mbox{{\small\rm IV}}} +p{u}''_1+q_1{u}'_1+q_0u_1 =
0\label{4.25a}
\end{equation}
its F-vW system comprises the four fourth order equations
\begin{eqnarray}
s_{11}^{IV} &=& - 3 p^{\prime}q_0 s_{14} + q_0^{\prime\prime\prime}s_{14} +
4 q_0^{\prime\prime}s_{14}^{\prime}+ q_0^{\prime\prime}s_{13} + 4
q_0^{\prime}s_{13}^{\prime}+ 6 q_0^{\prime}s_{14}^{\prime\prime}+
q_0^{\prime}s_{12} \nonumber \\
&& - s_{11}^{\prime\prime}p - s_{11}^{\prime}q_1 + 4 s_{12}^{\prime}q_0 + 6
s_{13}^{\prime\prime}q_0 + 4 s_{14}^{\prime\prime\prime}q_0 - 2
s_{14}^{\prime}p q_0 \label{4.26} \\
s_{12}^{IV}&=& - 3 p^{\prime}q_1 s_{14} + q_1^{\prime\prime\prime}s_{14} + 4
q_1^{\prime\prime}s_{14}^{\prime}+ q_1^{\prime\prime}s_{13} + 4
q_1^{\prime}s_{13}^{\prime}+ 6 q_1^{\prime}s_{14}^{\prime\prime} \nonumber
\\
&&+ q_1^{\prime}s_{12}+ 3 q_0^{\prime\prime}s_{14} + 8
q_0^{\prime}s_{14}^{\prime}+ 2 q_0^{\prime}s_{13} - 4
s_{11}^{\prime\prime\prime}- 2 s_{11}^{\prime}p - s_{12}^{\prime\prime}p
\nonumber \\
&&+ 3 s_{12}^{\prime}q_1+ 6 s_{13}^{\prime\prime}q_1 + 4 s_{13}^{\prime}q_0
+ 4 s_{14}^{\prime\prime\prime}q_1 + 6 s_{14}^{\prime\prime}q_0 - 2
s_{14}^{\prime}p q_1 \label{4.27} \\
s_{13}^{IV} &=& p^{\prime\prime\prime}s_{14} + 4
p^{\prime\prime}s_{14}^{\prime}+ p^{\prime\prime}s_{13} + 4
p^{\prime}s_{13}^{\prime}+ 6 p^{\prime}s_{14}^{\prime\prime}- 3 p^{\prime}p
s_{14}+ p^{\prime}s_{12} \nonumber \\
&& + 3 q_1^{\prime\prime}s_{14}+ 8 q_1^{\prime}s_{14}^{\prime}+ 2
q_1^{\prime}s_{13} + 3 q_0^{\prime}s_{14} - 6 s_{11}^{\prime\prime}- 4
s_{12}^{\prime\prime\prime}+ 2 s_{12}^{\prime}p \nonumber \\
&& + 5 s_{13}^{\prime\prime}p + 3 s_{13}^{\prime}q_1 + + 4
s_{14}^{\prime\prime\prime}p + 6 s_{14}^{\prime\prime}q_1 - 2
s_{14}^{\prime}p^2 + 4 s_{14}^{\prime}q_0 \label{4.28} \\
s_{14}^{IV} &=& 3 p^{\prime\prime}s_{14} + 8 p^{\prime}s_{14}^{\prime}+ 2
p^{\prime}s_{13} + 3 q_1^{\prime}s_{14} - 4 s_{11}^{\prime}- 6
s_{12}^{\prime\prime}- 4 s_{13}^{\prime\prime\prime} \nonumber \\
&&+ 2 s_{13}^{\prime}p + 5 s_{14}^{\prime\prime}p + 3 s_{14}^{\prime}q_1.
\label{4.29}
\end{eqnarray}
The maximum number of Lie point symmetries of this system is 31 and is
achieved only if the condition that (\ref{4.25a}) have eight Lie point symmetries, {\it videlicet}
\begin{equation}
q_1 = p\quad\mbox{\rm and}\quad q_0
=\frac{3}{10}{p}''+\frac{9}{100}p^2\label{4.30}
\end{equation}
is achieved. (The symmetry condition is stronger than the requirement that
the equation be self-adjoint. The latter places no constraint on $q_0 $.) If
(\ref{4.25a}) can be rendered autonomous by a point transformation,
\ie $q_1 $ and $q_0 $ are such as to permit six Lie point symmetries in
(\ref{4.25a}), the Lie point symmetries of the F-vW system \re{4.26}-\re{4.29}
number 28.
We infer that this property holds for all F-vW systems arising from the analysis
of linear ordinary differential equations of order $m (\geq 3) $ for the generalised
self-similar symmetry.% that produces the
%linear Kummer-Liouville transformation which renders the equation autonomous.
\section{Concluding remarks}
The results reported in this paper were motivated by a corollary
in Floquet theory which states that there exists a linear
transformation from a first-order linear system with periodic
coefficient matrix to one with a constant coefficient matrix. We
have shown that this result is general and can be extended to
scalar linear equations of higher order -- equally linear systems
of higher order equations -- by means of the common technique of
writing a higher order linear equation as a first-order system.
In the case of linear scalar higher order equations we saw that
equations with both $n+ 4 $ and $n+ 2 $ Lie point symmetries could
be rendered autonomous by a Kummer-Liouville transformation linear
in the dependent variable. This is not possible for equations
with only $n+ 1 $ Lie point symmetries. The F-vW system which
arises from the Lie point symmetries of these equations was shown
to maintain the distinction in symmetry between the class of
equations with $n+4$ Lie point symmetries and that with $n+2$ Lie
point symmetries.
In this analysis the system of determining equations to be solved
quickly becomes manually unhandable and the success of the
analysis is dependent upon the use of a suitable code.
Interpretation of the results also requires a knowledge of the Lie
point symmetries and the algebras for systems of higher order
linear differential equations, a subject which is yet to receive
extensive attention.
\section*{Acknowledgements}
PGLL expresses his deep appreciation of the hospitality of MCN and the
provision of facilities by the Dipartimento di
Matematica e Informatica, Universit\`a di Perugia, during the
period in which this work was initiated, and of GEODYSYC,
Department of Mathematics, University of the Aegean,
during the period in which this work was
finalised and acknowledges the support of the National Research
Foundation of South Africa and the University of Natal.
\begin{thebibliography}{99}
\bibitem{Abel 26}
Abel NH (1826-27) Recherches sur les fonctions elliptiques {\it
J f\"ur Math} {\bf II} 101-196
\bibitem{Abramowitz 65}
Abramowitz M \& Stegun IA (1970) {\it Handbook of
Mathematical Functions} (Dover, New York)
\bibitem{agu88a}
Aguirre M \& Krause J (1988) Finite point transformations and linearisation of
$\ddot{x} = f(x)$ {\it J Phys A: Math Gen} {\bf 21} 2841-2845
\bibitem{agu88b}
Aguirre M \& Krause J (1988) $SL(3,R)$ as the group of symmetry transformations
for all one-dimensional linear systems. II Realisations of the Lie algebra
{\it J Math Phys} {\bf 29} 1746-1752
\bibitem{Arscott 64}
Arscott FM (1964) {\it Periodic Differential Equations} (Pergamon, London)
\bibitem{Bessel 24}
Bessel FW (1824) Untersuchung des Teils der planetarischen
St\"orungen, welcher aus der Bewegung der Sonne entsteht {\it
Abhandlungen der Berliner Akademie} (pub 1826) 1-52
\bibitem{Bluman 69}
Bluman GW \& Cole JD (1969) The general similarity solution of the heat
equation {\it J Math Mech} {\bf 18} 1025-1042
\bibitem{Bluman 88}
Bluman GW \& Reid GJ (1988) New symmetries for ordinary differential equations
{\it IMA J Appl Math} {\bf 40} 87-94
\bibitem{Bluman 89}
Bluman GW \& Kumei S (1989) {\it Symmetries and Differential Equations} (Springer-Verlag, New York)
\bibitem{Burgan 78}
Burgan J-R, Feix MR, Fijalkow E, Gutierrez J \& Munier A (1978) Utilization
des groupes de transformation pour la resolution des \'equations aux
deriv\'ees partielles in {\it Applied Inverse Problems} ed Sabatier PC (Springer-Verlag, Berlin)
\bibitem{Camiz 71}
Camiz P, Geradi A, Marchioro C, Presutti E \& Scacciatelli E (1971) Exact
solution of a time-dependent quantal harmonic oscillator with a singular
perturbation {\it J Math Phys} {\bf 12} 2040-2043
\bibitem{Chuk 91}
Chukwumah GC (1991) Lie group theory of the Bessel function of the first kind
of integral order {\it Int J Theoret Phys} {\bf 30} 1011-1031
\bibitem{Colwell 92}
Colwell P (1992) Bessel functions and Kepler's equation {\it Am
Math Monthly} {\bf 99} 45-48
\bibitem{Dutka 95}
Dutka J (1995) On the early history of Bessel functions
{\it Arch Hist Exact Sci} {\bf 49} 105-134
\bibitem{Eliezer 76}
Eliezer CJ \& Grey A (1976) A note on the time-dependent harmonic oscillator
{\it SIAM J Appl Math} {\bf 30} 463-468
\bibitem{Ermakov 80}
Ermakov V (1880) Second order differential equations. Conditions of complete
integrability {\it Univ Izvestia Kiev Ser III} {\bf 9} 1-25 (trans Harin AO)
\bibitem{Flessas 95}
Flessas GP, Govinder KS \& Leach PGL (1997) Characterisation of the algebraic
properties of first integrals of scalar ordinary differential equations
{\it J Math Anal Appl} {\bf 212} 349-374
\bibitem{Gauss 76}
Gauss CF (1876) {\it Werke} {\it III} (K\"{o}niglichen Gesellschaft der
Wissenschaften, G\"{o}ttingen) 404-460
\bibitem{Gorringe 87}
Gorringe VM \& Leach PGL (1988) Lie point symmetries for systems of second
order ordinary differential equations {\it Qu\ae st Math}
{\bf 14} 277-289
\bibitem{Gunther 77}
G\"unther NJ \& Leach PGL (1977) Generalised invariants for the time-dependent
harmonic oscillator {\it J Math Phys} {\bf 18} 572-576
\bibitem{Hermite} Hermite C (1877) Sur quelques applications des fonctions ellliptiques
in {\it Oeuvres de Charles Hermite. Tome III} (Gauthier-Villars,
Paris, 1908) 267-418
\bibitem{Hill 92}
Hill JM (1992) {\it Differential Equations and Group Methods for
Scientists and Engineers} (CRC Press, Boca Raton)
\bibitem{Ince 27}
Ince EL (1927) {\it Ordinary Differential Equations} (Longmans, Green \& Co, London)
\bibitem{Jacobi}
Jacobi KGJ (1829) {\it Fundamenta Nova Theori\ae\ Functionum
Ellipticarum} (Fratrum Borntraeger, Regiomonti)
\bibitem{Mahomed 99}
Kara AH \& Mahomed FM (2000) Relationship between symmetries and
conservation laws {\it Int J Theoret Phys} {\bf 39} 23-40
\bibitem{Krause 94}
Krause J \& (1984) On the complete symmetry group of the classical Kepler system
{\it J Math Phys} {\bf 35} 5734-5748
\bibitem{Kruskal 62}
Kruskal MJ (1962) Asymptotic theory of Hamiltonian and other systems with all
solutions nearly periodic {\it J Math Phys} {\bf 3} 806-828
\bibitem{Kummer 34}
Kummer EE (1887) De generali quadam \ae quatione differentiali
tertii ordinis {\it J f\"ur Math} {\bf 100} 1-9 (reprinted from
the Programm des evangelischen K\"onigl und Stadgymnaziums in
Liegnitz for the year 1834)
\bibitem{Leach 77}
Leach PGL (1977) On the theory of time-dependent linear canonical
transformations as applied to Hamiltonians of harmonic oscillator type
{\it J Math Phys} {\bf 18} 1608-1611
\bibitem{Leach 93}
Leach PGL, Govinder KS, Brazier TI \& Schei CS (1993) The ubiquitious
time-dependent simple harmonic oscillator {\it S Afr J Sci} {\bf 89} 126-130
\bibitem{Legendre}
Legendre A-M (1825) {\it Trait\'e des Fonctions Elliptiques}
(Gauthier, Paris)
\bibitem{Winternitz 80}
Levi D \& Winternitz P (1989) Nonclassical symmetry reduction:
example of the Boussinesq equation {\it J Phys A: Math Gen} {\bf
22} 2915-2924
\bibitem{Lewis 67}
Lewis HR Jr (1967) Classical and quantum systems with
time-dependent harmonic oscillator-type Hamiltonians {\it Phys Rev
Lett} {\bf 18} 510-512
\bibitem{Lewis 68a}
Lewis HR Jr (1968) Motion of a time-dependent harmonic oscillator
and of a charged particle in a time-dependent, axially symmetric,
electromagnetic field {\it Phys Rev} {\bf 172} 1313-1315
\bibitem{Lewis 68b}
Lewis HR Jr (1968) Class of exact invariants for classical and
quantum time-dependent harmonic oscillators {\it J Math Phys} {\bf
9} 1976-1986
\bibitem{Lewis 69}
Lewis HR Jr \& Riesenfeld WB (1969) An exact quantum theory of the
time-dependent harmonic oscillator and of a charged particle in a
time-dependent electromagnetic field {\it J Math Phys} {\bf 10}
1458-1473
\bibitem{Lie 1}
Lie S (1888) {\it Theorie der Transformationsgruppen. Erster
Abschnitt} (Teubner,Leipzig)
\bibitem{Lie 2}
Lie S (1890) {\it Theorie der Transformationsgruppen. Zweiter
Abschnitt} (Teubner, Leipzig)
\bibitem{Lie 3}
Lie S (1893) {\it Theorie der Transformationsgruppen. Dritter und
Letzter Abschnitt} (Teubner, Leipzig)
\bibitem{Lie:bht}
Lie S (1896) {\it Geometrie der Ber\"uhrungstransformationen}
(Teubner, Leipzig)
\bibitem{Lie:dgl}
Lie S (1912) {\it Vorlesungen \"uber Differentialgleichungen}
(Teubner, Leipzig)
\bibitem{Liouville 37}
Liouville J (1837) Sur le d\'eveloppement des fonctions ou parties de fonctions
en s\'eries dont les divers termes sont assujettis \`a satisfaire \`a une
m\^eme \'equatione diff\'erentielle du second ordre contenent un param\`etre
variable {\it J de Math} {\bf II} 16-35
\bibitem{MacLachlan 45}
MacLachlan NW (1947) {\it The Theory and Application of Mathieu Functions} (Clarendon, Oxford)
\bibitem{Mahomed 90}
Mahomed FM \& Leach PGL (1990) Symmetry Lie algebras of $n$th order ordinary
differential equations {\it J Math Anal Appl} {\bf 151} 80-107
\bibitem{Noether 18}
Noether E (1918) Invariante Variationsprobleme {\it Nach K\"onig
Gesell Wissen G\"ottingen Math-phys Kl} 235-257
\bibitem{Nucci 90}
Nucci MC (1990) Interactive REDUCE programs for calculating
classical, non-classical and Lie-B\"{a}cklund symmetries of
differential equations {\it Preprint GT Math: 062090-051}
\bibitem{Nucci 96}
Nucci MC (1996) Interactive REDUCE programs for calculating Lie
point, non-classical, Lie-B\"{a}cklund, and approximate symmetries
of differential equations: manual and floppy disk {\it CRC
Handbook of Lie Group Analysis of Differential Equations. Vol. 3:
New Trends} ed N H Ibragimov (Boca Raton: CRC Press) 415-481.
\bibitem{Nucci 96b}
Nucci MC (1996) The complete Kepler group can be derived by Lie
group analysis {\it J Math Phys} {\bf 37} 1772-1775
\bibitem{Nucci 00}
Nucci MC (2000) Nonclassical symmetries as special solutions of
heir-equations (preprint: nlin.SI/0011008)
\bibitem{Nucci 01a}
Nucci MC \& Leach PGL (2001) The harmony in the Kepler and related
problems {\it J Math Phys} {\bf 43} 746-764
\bibitem{Nucci 01b}
Nucci MC \& Leach PGL (2001) Maximal algebras of Lie point symmetries for
systems of ordinary differential equations (preprint: Dipartimento di
Matematica e Informatica, Universit\`a di Perugia, 06123 Perugia, Italia)
\bibitem{Pillay 98}
Pillay T \& Leach PGL (1999) Generalized Laplace-Runge-Lenz
vectors and nonlocal symmetries {\it S Afr J Sci} {\bf 95} 403-407
\bibitem{Pinney 50}
Pinney E (1950) The nonlinear differential equation
$y''(x)+p(x)y+cy^{-3}=0$ {\it Proc Amer Math Soc} {\bf 1} 681
\bibitem{Rosati} Rosati L, Nucci MC \& Mezzanotte F (2001)
Italian edition and rereading of FUNDAMENTA NOVA by K.G.J. Jacobi 150
years since his death (February 18, 1851 - February 18, 2001) [in Italian]
{\it Preprint RT n. 2001-4}
\end{thebibliography}
\end{document}
---------------0111081229669--