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Talk given at the ICNAAM Conference, Kos (Greece), Sept. 2012. Based on joint work with G. Gaeta and S. Walcher. To appear in Math. Meth. of Appl. Sci.
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ordinary differential equations; dynamical systems; $\sigma$-symmetries; orbital symmetries; reduction procedures
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\documentclass[10pt]{article}
\def\al{\alpha}
\def\be{\beta}
\def\ga{\gamma}
\def\de{\delta}
\def\phi{\varphi}
\def\la{\lambda}
\def\ka{\kappa}
\def\si{\sigma}
\def\ze{\zeta}
\def\om{\omega}
\def\th{\theta}
\def\E{{\mathcal E}}
\def\X{{\mathcal X}}
\def\ka{\kappa}
\def\pd{\partial}
\def\d{{\rm d}}
\def\na{\nabla}
\def\~#1{\widetilde #1}
\def\beq{\begin{equation}}
\def\eeq{\end{equation}}
\def\lb{\label}
\def \sy {symmetry}
\def \sys {symmetries}
\def \eq {equation}
\def\={\,=\,}
\def\q{\quad}
\def\d{{\rm d}}
\def\sk{\smallskip}
\def\bk{\bigskip}
\def\pd{\partial}
\def\vf{vector field}
\def\so{solution}
\def\ni{\noindent}
\def\EOP{\hfill {$\bullet$}}
\def\EOE{\hfill{$\triangle$}}
\def\.#1{\dot #1}
\def\^#1{\widehat #1}
\date{}
\begin{document}
%\parindent0pt
\title{Generalized notions of symmetry of ODE's and reduction procedures}
\author{
Giampaolo Cicogna\thanks{Email: cicogna@df.unipi.it}
\\~\\
Dipartimento di Fisica ``E.Fermi'' dell'Universit\`a di Pisa\\
and Istituto Nazionale di Fisica Nucleare, Sez. di Pisa \\~\\
Largo B. Pontecorvo 3, Ed. B-C, I-56127, Pisa, Italy }
\maketitle
\ni{\bf Abstract}
\ni
This paper describes the notion of $\sigma$-symmetry,
which extends the one of $\lambda$-symmetry, and its application to
reduction procedures of systems of ordinary differential equations and of dynamical systems as well.
We also consider orbital symmetries, which give rise
to a different form of reduction of dynamical systems. Finally,
we discuss how dynamical systems can be transformed into higher-order ordinary differential equations,
and how these symmetry properties of the dynamical systems can be
transferred into reduction properties of the corresponding ordinary differential equations.
Many examples illustrate the various situations.
\bigskip \ni
{\it PACS}: 02.20.Sv; 02.30.Hq, {\it MOS}: 34A05; 37C80
\bigskip \ni
{\it Keywords}: ordinary differential equations; dynamical systems; $\sigma$-symmetries; orbital symmetries; reduction procedures
\bigskip\ni
{\bf Talk given at the ICNAAM Conference, Kos (Greece), Sept. 2012. Based on joint work with G. Gaeta and S. Walcher}
\medskip
\section*{Introduction}
It is well known that if an ordinary differential \eq\ (ODE)
of order $q>1$ admits a Lie point-\sy ,
then the order of the \eq\ can be lowered by \emph{one} (\emph{two} in
some cases, e.g. when the \eq\ comes from a variational problem), see
e.g. \cite{Ovs,Olv,Ste,CRC,BA}.
It is also known that the same is true even if the \eq\ admits a
$\la$-\sy ,
a notion which has been introduced by C. Muriel and J. Romero in 2001
\cite{MR1,MR2} and which has received a number of applications and extensions
(see e.g. \cite{Gtw} with references therein, and \cite{Sig13}
for a more recent contribution).
We have recently further extended this result \cite{Sprol,CGW,SDS}.
Let us fix our
notations. We will always denote by $t$ the independent variable, in order
to unify the notations, as a large part of this paper
will be concerned with dynamical systems, where time $t$ is typically the
independent variable. The ODE will be denoted by
\[\E\=\E\big(t,u^{(k)}(t)\big)\=0\q\q \big(u^{(k)}(t)=\d^ku/\d t^k \ ,\
k=0,\ldots,q\big)\]
and the generators of Lie point-\sys\ will be written in the form
\[X\=\phi(t,u)\frac{\pd}{\pd u}+\tau(t,u)\frac{\pd}{\pd t}\ .\]
According to a by now standard abuse of language, we will denote by $X$
both the \sy\ and its Lie generator.
We will consider, instead of a {\it
single} \vf\ $X$, a set $\X$ of $s>1$ \vf s $X_\al$ in involution
\beq\lb{inv} [X_\al,X_\be]\=\nu_{\al \be \ga }X_\ga
\q\q (\al,\be,\ga=1,\ldots,s) \eeq
together with a system of ODE's $\E_a=0$, $a=1,\ldots,n$.
This leads to the introduction of the notion of ``combined''
\emph{joint-$\la$-\sys },
or \emph{$\si$-\sys\ } for short. The precise definition and its
application to the reduction of
systems of ODE's will be given in the next Section. Using the same idea,
we will show (Sect. 2) that also dynamical systems (DS), i.e. systems of
first-order ODE's, can be suitably reduced when they admit a $\si$-\sy .
In Sect. 3, we include the case of \emph{orbital} \sys , which give rise
to a different form of reduction of DS. Finally, in Sect.
4, we discuss how DS can be transformed into a higher-order ODE, and how
these \sy\ properties of the DS can be transferred into reduction
properties of the corresponding ODE. Several new examples will illustrate the
various situations. All the objects (functions, \vf s) considered in this
paper are assumed to be smooth enough.
The presence of $\si$-\sys\ admits interesting geometrical interpretations
and algebraic aspects: for a full discussion of these arguments and several other
details we refer to \cite{Sprol,CGW,SDS} and references therein.
This is a full paper presented within ICNAAM 2012; a very short and
preliminary sketch of part of these results can be found in the Enlarged
Abstracts of the Conference Proceedings \cite{EnAb}.
\section{Basic definitions and reduction of ODE's}
First of all, we need the two following definitions.
\sk\ni
{\bf Definitions}
\ni
i) \emph {Given $n>1$ variables $u\equiv \{u^a(t)\}$, $(a=1,\ldots,n)$, and
$s>1$ \vf s $\X\equiv \{X_\al\}$, $(\al=1,\ldots,s)$,
a $\si$-prolongation is a
deformed prolongation rule which involves
a given $s\times s$ matrix $\si=\si(t,u,\dot u)$:
the first $\si$-prolongation $Y_\al^{[1]}$ of
$X_\al=\phi_\al\cdot\nabla_u+\tau_\al\pd/\pd t$ is defined by
\[Y_\al^{[1]}:=X_\al^{[1],\si}\=X_\al^{[1]}+ \si_{\al\be}(\phi_\be^a-\dot
u^a\tau_\be)\frac{\pd}{\pd \dot u^a} \]
where $X_\al^{[1]}$ is the first standard prolongation.
Higher order prolongations $Y_\al^{[k]}$ can be easily obtained by
recursion.}
\bk\ni
ii) \emph{A system of $n$ ODE's $\E\equiv \{\E_a\big(t,u^{(k)}(t)\big)\}=0$
for the $n$ variables $u(t)$, of order $q>1$, is $\si$-symmetric under the
set $\X$ if
%\vskip-.2cm
\[Y_\al^{[q]}\E|_{\E=0}\=0\]
%\vskip-.2cm
i.e. if $\E$ is invariant under the $\si$-prolongations $Y_\al^{[q]}$ of
all the $X_\al$.}
\sk
It can be remarked that the case $s=1$ would correspond to $\la$-\sys .
Based on the above definitions, we can state the following result.
\sk\sk\ni
{\bf Theorem 1.} \emph {Let a system of $n$ ODE's $\E=0$ of order $q>1$
be $\si$-symmetric under a set $\X$ of \vf s $X_\al\, (\al=1,\ldots,s>1)$
in involution with constant rank $r$ ($r\le s;\, r\le n$); if the
involution relations are preserved in their $q$-th $\si$-prolongations
$Y^{[q]}_\al$, then -- under standard regularity and nondegeneracy
conditions -- the order of $r$ ODE's can be lowered by
one. This is obtained in terms of some $r$ new variables $\eta_\al$ which
are invariant under the $1^{st}$ $\si$-prolongations $Y^{[1]}_\al$. }
\sk\ni
{\it Sketch of the proof.} The main ingredient of the proof is the
following {\it completely algebraic} result, which holds for general
\vf s
$X_\al=\phi_\al\cdot\nabla_u+\tau_\al\pd/\pd t$
\beq\lb{DtY}
[D_t,Y_\al^{[k+1]}]\=-\si_{\al\be}Y^{[k]}_\be+(D_t\tau_\al+
\si_{\al\be}\tau_\be)D_t\eeq
where $D_t$ is the total derivative,
and its consequence
\beq\lb{IDP}Y_\al^{[k+1]}\,\frac{D_t\ze_1^{[k]}}{D_t\ze_2^{[k]}}\=0\eeq
where
$\ze_i^{[k]}$ is any $k$-order differential invariant under $Y_\al^{[k]}$.
Assume for simplicity (but the general result holds in general)
that the $X_\al$ are \emph{vertical} \vf s, i.e. that $\tau_\al=0$:
then, the time $t$ is a common invariant under all
the $X_\al$. Assume also, for the moment, that $n=r$. Then, no
other variable is admitted with this property.
Considering the first $\si$-prolonged \vf s $Y_\al^{[1]}$,
there exist, according to Frobenius theorem, exactly $n$ common
differential
invariants of order $1$ under $Y_\al^{[1]}$. Let us denote these by
$\eta_\al\, (\al=1,\ldots, r=n)$. Using (\ref{IDP}) with $k=1$,
choosing as $\ze_1$ any of these $\eta_\al$
and $\ze_2=t$, we deduce that $D_t\eta_\al=\dot\eta_\al$ are second-order
differential quantities which are common invariants under the second
$\si$-prolongation $Y_\al^{[2]}$, and so on. This is called
{\it invariance by differentiation property}.
The $\si$-invariance of the system $\E=0$, then
implies that all the \eq s of this system must contain, apart from $t$,
only the common invariant variables with their derivatives. Choosing
$\eta_\al$ as new variables, the \eq s of our system thus become \eq s of
order
$q-1$. If instead $n>r$, then, still thanks to Frobenius theorem, there
are, in addition to $t$, other $(n-r)$ variables $w_j\
(j=1,\ldots,n-r)$ of order zero which are common invariants under
$X_\al$. Therefore, thanks to (\ref{IDP}), also $\dot w_j$ are $(n-r)$ common
invariants under the first
$\si$-prolongation $Y_\al^{[1]}$, in addition to other $r$ invariants
$\eta_\al$, and so on. In other words, starting from the
invariants $w_j$ and $\eta_\al$, one obtains all higher-order
differential invariants. As before, our system must be written in terms
of these invariant quantities; then the
system of ODE's can be split into a subsystem of $r$ \eq s
of order $q-1$ in the variables $t$ and $\eta_\al$,
and another system of $n-r$ \eq s of order $q$.
\hfill$\bullet$
\sk\ni
{\it Example 1.}
Consider the system of ODE's (in the examples we will usually write
as $u_1,u_2,\ldots$ instead of $u^a$ to avoid confusion, and $\dot u_1=\d u_1/\d t$, etc.)
\begin{equation}
\left\{\displaystyle \begin{array}{ll}
\stackrel{\ldots}{u}_1\=t\ddot u_2+t\dot u_2+2\dot u_2+u_2+h_1(t,u)\\
\ddot u_2\=\dot u_1-\dot u_2+h_2(t,u)\\
\ddot u_3\=u_2+t\dot u_2+h_3(t,u)
\end{array}\right.
\end{equation}
where $h_a$ are arbitrary functions of $t$ and of the quantities
$u_1-u_2-u_3,u_1-u_2-\dot u_1+tu_2,u_1-u_2-\dot u_2$.
For generic $h_1,h_2,h_3$ there is no standard Lie \sy\ for this system,
but
it is $\si$-symmetric under the \vf s (then $n=3, r=2$)
\[X_1\=\frac{\pd}{\pd u_1} + \frac{\pd}{\pd u_2}\q,\q X_2\=\frac{\pd}{\pd
u_1} +\frac{\pd}{\pd u_3}\]
with
\[\si\=\pmatrix { 0 & t\cr 1 & 0 }\ .\]
The first $\si$-prolongations are
%\vskip-.2cm
\[Y_1^{[1]}\=X_1+t\frac{\pd} {\pd \dot u_1} +t\frac{\pd} {\pd \dot u_3}
\q,\q\
Y^{[1]}_2\=X_2+\frac{\pd} {\pd \dot u_1} + \frac{\pd} {\pd \dot u_2} \ .\]
%\vskip-.2cm
In the new $\si$-\sy\ adapted variables
%\vskip-.2cm
$w=u_1-u_2-u_3,\, \eta_1=u_1-u_2-\dot u_1+tu_2 ,\, \eta_2= u_1-u_2-\dot
u_2$
%\vskip-.2cm
the above \eq s become, in agreement with Theorem 1,
\[\ddot \eta_1=-\dot\eta_1+\dot\eta_2+h_1(\eta_1,\eta_2,w)\ ,\
\dot \eta_2=-h_2(\eta_1,\eta_2,w)\ ,\ \ddot
w=\dot\eta_2+\dot\eta_1-h_3(\eta_1,\eta_2,w)
\]\EOE
\sk
It can be observed that if one of the \eq s of the system of ODE's is of
order $1$ and this is
lowered according to Theorem 1, then one is left with an
\emph{algebraic} \eq\ for the variables $t$ and $\eta_\al$. This happens
for
instance if in Example above one of the \eq s is replaced by
\[ \dot u_1\=u_1-u_2+tu_2+h_0(t,u)\]
which is reduced to
\[\eta_1+h_0(\eta_1,\eta_2,w)\=0\ .\]
Notice that this algebraic \eq\ is actually a first-order differential
\eq\ for the initial variables $u^a$ (the presence of an ``auxiliary''
first-order differential \eq\ is indeed standard in $\la$-type \sys ).
This remark introduces the special and specially interesting case of
dynamical
systems, which will be considered in detail in the next sections.
\section{Reduction of Dynamical Systems}
Dynamical systems are systems of first-order time-evolution
differential \eq s of the form
\[\dot u^a\=f^a(t,u)\q\q a=1,\ldots,n\]
It is not too restrictive to consider \emph{autonomous} DS,
and vertical \vf s with $\phi_\al$
\emph{independent of time}, i.e.
\beq\lb{ass}\dot u=f(u) \q\q\ X_\al\=\phi_\al^a\frac{\pd}{\pd u^a}\equiv
\phi_\al\cdot\nabla_u\eeq
Given a DS, the $\si$-determining \eq s, i.e. the \eq s giving the
conditions for the DS to be invariant under the first $\si$-prolongations
$Y_\al^{[1]}$ of $X_\al$, when
restricted to the \so \ manifold of the DS, take the particularly simple
form
%\beq \lb{sideq1} [\phi_\al,f]\=\si_{\al\be}\phi_\be \eeq
\beq\lb{sideq} [X_\al,F]\=\si_{\al\be}X_\be \q\q
(\al,\be=1,\ldots,s)\eeq
having introduced the ``dynamical'' \vf
\[F\=f\cdot\nabla_u \ .\]
In particular, the restriction to the \so \ manifold of the DS $\dot
u=f(u)$, implies
that $\si$ may be chosen as a function of $t,u$ only, indeed
$\si\big(t,u,f(t,u)\big)=\overline{\si}(t,u)$. From (\ref{sideq}), one
may directly recover for this case the invariance by differentiation property: indeed,
if $w_j$ satisfies $X_\al w_j=0$, then
\[X_\al(D_tw_j)=X_\al(f\cdot\nabla_u)w_j=X_\al F\,w_j=
(FX_\al+\si_{\al\be}X_\be)w_j=0 \]
i.e. $D_tw_j$ is also invariant under all the $X_\al$.
\sk
As well known, given a set $\X$ of \vf s in involution, it is not granted
in general that their prolongations are still in involution (see \cite{Sprol,SDS}
for a discussion and some examples on this point). However, in the case of
DS, we have the following useful result (in the following, we will simply
write $Y_\al$ instead of $Y_\al^{[1]}$):
\sk\sk\ni
{\bf Lemma}. \emph{Let a DS satisfy (\ref{sideq}) with a set of \vf s
$X_\al$ in involution. Then, restricting to the \so \ manifold of the
DS, the first
$\si$-prolonged \vf s $Y_\al$ satisfy the same involution property.}
\sk\ni
{\tt Proof}. We first have
\[Y_\al\=X_\al+\big(D_t\phi^a_\al+\si_{\al\be}\phi^a_\be\big)
\frac{\pd}{\pd \dot u^a}=
X_\al+X_\al \,f\nabla_{\dot u}\]
thanks to (\ref{sideq}). Then
\[[Y_\al,Y_\be]\=[X_\al,X_\be]+\nu_{\al\be\ga}X_\ga f\nabla_{\dot u}
=\nu_{\al\be\ga}Y_\ga\]
using the involution properties of $X_\al$.
\hfill$\bullet$
\sk
Then we have:
\sk\ni
{\bf Theorem 2.} \emph{In the above simplifying assumptions (\ref{ass}),
let a DS be
$\si$-symmetric under a set $\X$ of \vf s $X_\al\, (\al=1,\ldots,s>~1)$ in
involution, with rank $r1$ can be transformed into a DS, as well
known.
Writing $u^{(n)}=p(t,u,\dot u,\ldots)$, if the ODE does not contain
explicitly the independent variable $t$, then one can put as usual
\[u=u_1\ ,\ \dot u_1=u_2\ ,\ldots, \ \dot u_n=p(u,\dot u,\ldots)\]
if instead $p$ depends on $t$, one simply includes the new variable
$u_0=t$ and the \eq\ $\dot u_0=1$.
The converse is ``in principle" (locally, and apart from degenerate cases) also
true (see \cite{NuLe,CGW}), but the transformation of a DS into an ODE
requires the inversion of some implicit expressions.
Let us show the procedure in the case of a DS with 3 dependent variables
$u^a$. If the DS is autonomous, $\dot u^a=f^a(u)$, then one puts
\[u_1:=y_1:=y\ ,\ \dot u_1=f_1(u):=y_2=\dot y, \]
\[\dot y_2=D_t f_1(u)=f\cdot\nabla_u\,f_1(u):=\Phi(u):= y_3=\ddot y\]
then one has to express $u_2$ and $u_3$ in terms of $y_1,y_2,y_3$ using
the two above definitions, and finally one gets
\[\dot y_3=\ \stackrel{\ldots}{y}=D_t\Phi\big(u(y)\big):=p(y)\]
which produces the ODE
\[ \stackrel{\ldots}{y}\=p(y,\dot y,\ddot y)\ . \]
If the DS is non-autonomous, then it can be ``autonomized'' introducing as
usual $u_0=t$, and the above procedure can be adapted accordingly.
The procedure of transforming a DS into a higher-order ODE opens
interesting possibilities of reducing the ODE. If indeed the DS admits
some \sy\ (including $\si$-\sys\ and orbital \sys ), then we have shown
that the DS can be reduced in terms of suitable \sy -adapted variables.
This reduction is immediately transferred, up to the change of variables
described above, to the higher-order ODE. Observing that not all \sys\
of the DS become
automatically Lie point-\sys\ of the ODE, we get a sort of ``reduction of
the ODE's without \sys ".
There are several possibilities in this direction, as we will show in the
following examples.
\sk
To illustrate the procedure, we give first an example of DS admitting a
standard \sy ; we construct the corresponding higher order ODE and
show how the \sy\ property of the DS can be used to obtain a reduced \eq\
for the ODE. In this example, the reduced \eq\ can be easily solved and this
procedure provides thus an alternative way to get the full \so\ of the ODE.
Examples 6 and 7 deal with ODE's deduced from DS admitting resp. a
$\la$-\sy\ and a $\si$-\sy .
\bk\ni
{\it Example 5.} The DS
\[\dot u_1=u_1+u_1^2u_2\q,\q \dot u_2=u_2+u_1u_2^2\]
admits the standard \sy\
%\[X\=u_1\frac{\pd}{\pd u_1}-u_2\frac{\pd}{\pd u_2}\]
$X=u_1\pd/\pd u_1-u_2\pd/\pd u_2$.
An invariant variable under $X$ is $w=u_1u_2$, which satisfies the reduced \eq\
$\dot w=2w+2w^2$. The ODE obtained through the positions $u_1=y,\,\dot
u_1=y_2=\dot y$ etc. is
\[\ddot y\=-2\dot y+3\frac{\dot y^2}{y}\ .\]
Integrating the reduced \eq\ for $w$ and passing to the new variable $y$
we obtain the reduced \eq\ for the ODE
\[\frac{\dot y}{y}\=\frac{c\, \exp(2t)}{1-c\,\exp(2t)}-1\]
and from this the full \so\ of the ODE
\[y\=\frac{c'\exp(t)}{\sqrt{c\,\exp(2t)-1}} \]
where $c,c'$ are constants. \EOE
\bk\ni
{\it Example 6.} This is a simple example with a $\la$-\sy . We start
from the DS
\[ \dot u_1\=u_2 \q,\q \dot u_2\=2u_2^2/u_1 \]
having a (standard) dilation \sy\ $X=u_1\pd/\pd u_1+u_2\pd/\pd u_2$.
Using Prop. 1 with $\mu=u_1$, the new DS
\beq\lb{exb} \dot u_1\=u_2+u_1^2 \q,\q \dot u_2\=2u_2^2/u_1+u_1u_2 \eeq
admits the above $X$ as a $\la$-\sy . With $u_1=y,\,u_2=\dot y-y^2$,
according to the above described procedure, we get the ODE
\[\ddot y\=2\frac{\dot y^2}{y}-y\dot y+y^3 \ .\]
The DS (\ref{exb}) can be reduced by the $X$-invariant variable
$w=u_2/u_1$, indeed $\dot w=w^2$; the same reduction holds for the new
variable $\~w=(\dot y/y)-y$, as easily checked. The reduced
\eq\ for $w$ can be immediately solved producing the
(time-dependent) first integral for the ODE $\kappa=t+y/(\dot y-y^2)$ = const.
This \eq\ for $y(t)$ can be further integrated
giving the general \so\ of the ODE
\[y(t)\=\ \Big((c-t)(c'+\log(c-t)\Big)^{-1} \]
where $c,c'$ are constants.
As above, the \so\ of the ODE could be obtained
(although not too easily) also by standard methods, but this example can
be useful to further illustrate this
\sy-based procedure. \EOE
\sk\sk\ni
{\it Example 7.} This is an example where an ODE is constructed starting
from a DS admitting a $\si$-\sy . The very simple DS
\[\dot u_1=1\q,\q \dot u_2=u_3\q ,\q \dot u_3=u_2\]
admits the two standard \sys
\[X_1\=\frac{\pd}{\pd u_1} \q ,\q X_2\=u_2\frac{\pd}{\pd u_2}
+u_3\frac{\pd}{\pd u_3}\ .\]
Using Prop.1 with $\mu_1=u_3,\,\mu_2=1/u_1$, we
obtain the new DS
\beq\lb{exc}\dot u_1=1+u_3\q,\q \dot u_2=u_3+u_2/u_1\q,\q
\dot u_3=u_2+u_3/u_1\eeq
which then admits the two \vf s $X_1,X_2$ as $\si$-\sy . A common invariant
under these \vf s is $w=u_2/u_3$ which satisfies the \eq\ $\dot w=1-w^2$.
The ODE which is deduced from the above DS (\ref{exc}) is
\[ \stackrel{\ldots}{y}\=\dot y-1+2\frac{\ddot y}{y}+\frac{(\dot y-1)^2}{y^2}\ .\]
After integration of the \eq\ for $w$, passing to the new
variable $y$, one obtains the reduced \eq\ for $y(t)$
\[\frac{y\ddot y-\dot y+1}{y(\dot y-1)}=\frac{\exp(2t)-c}{\exp(2t)+c} \]
where $c$ is a constant. \EOE
\sk\sk
The two next and final examples deal with the case of $\si$-orbital \sys .
According to Prop.2, we can construct orbitally symmetric DS starting
from a $\si$-symmetric (or standardly symmetric as well) by
multiplication by an arbitrary function $\rho(u)$. A good choice for this
function may be, e.g., $\rho=1/f_1(u)$ with usual notations, in such a
way that the new DS becomes, renaming for convenience the variables
$u_1,\ldots,u_n$ as $v_0,\ldots,v_{n-1}$
\beq\lb{uv}\dot v_0=1\q,\q \dot v_1=f_2(v)/f_1(v)\q,\ldots,\q\dot
v_{n-1}=f_n(v)/f_1(v)\eeq
i.e. in the form of an ``autonomized'' DS where $v_0=t$ and then $v_1=y$,
$\dot v_1=f_2(v)/f_1(v)=\dot y$, $\dot v_2=D_t\big(f_2(v)/f_1(v)\big)=\ddot y$, etc.
The ODE deduced
from this DS will then be of order $(n-1)$. We shall adopt this choice
for the function $\rho(u)$ in both the following examples.
As said above, in the case of orbital \sys , we need {\it at least two}
invariants $w_j$ under the \vf s $X_\al$ in order to have a reduced \eq .
This may be reached either with two invariants under a single \vf\ (hence in the
case of a single standard \sy , or also a $\la$-\sy\ as considered in Example 8 below),
or with two common invariants under two \vf s (standard, or also $\si$-\sy\ as
in Example 9).
\sk\sk\ni
{\it Example 8.} The DS
\[\dot u_1\=u_1u_2\q,\q\dot u_2\=u_1/u_3\q,\q\dot u_3=u_3\]
admits the standard \sy\
%\[X\=u_1\frac{\pd}{\pd u_1}+u_3\frac{\pd}{\pd u_3}\]
$X\=u_1 \pd/\pd u_1+u_3 \pd/\pd u_3$.
Using Prop.1 with $\mu=u_2$ and then Prop.2 with $\rho=1/(u_1u_2)$
we get the DS, using the notations introduced in (\ref{uv}),
\[
\dot v_0\=1\q,\q
\dot v_1\=\frac{1}{2v_1 v_2}\q,\q
\dot v_2\=v_2\frac{1+v_1}{2v_0 v_1}
\]
which admits the above \vf\ $X$ as an orbital $\si$- (actually, a $\la$)
-\sy . The ODE which can be deduced from this DS is
\[\ddot y\=-\frac{\dot y(2t\dot y+y+1)}{2ty}\ .\]
There are two
independent invariants under the above \vf\ $X$, namely
$w_1=u_2=v_1,\,w_2=u_1/u_3=v_0/v_2$; they satisfy the \eq s
\[\dot w_1=\frac{1}{v_0}\frac{w_2}{2w_1} \q,\q
\dot w_2=\frac{1}{v_0}w_2\frac{w_1-1}{2w_1}\]
and from these we obtain the reduced \eq\ $\d w_2/\d w_1=w_1-1$ which
can be easily integrated giving in the new variable $y$ the reduced \eq\
for $y(t)$ (and a first integral for the ODE)
\[2ty\dot y-\frac{1}{2}y^2+y\= {\rm const}\ .\]
\sk\ni
{\it Example 9.} The simple DS
\[\dot u_1=0\q,\q \dot u_2=u_3\q,\q \dot u_3=u_4 \q,\q \dot u_4=u_2\]
admits the two standard \sys
\[X_1\=\frac{\pd}{\pd u_1}\q,\q X_2\=u_2\frac{\pd}{\pd
u_2}+u_3\frac{\pd}{\pd u_3}+u_4 \frac{\pd}{\pd u_4}\ .\]
Thanks to Prop.1 and 2, with $\mu_1=u_2,\,\mu_2=u_1$ and $\rho=1/u_2$
we obtain the DS, with the notations as in (\ref{uv})
\[\dot v_0=1\q,\q \dot v_1=v_0+v_2/v_1\q,\q \dot v_2=
(v_3+v_0v_2)/v_1\q,\q\dot v_3=1+v_0v_3/v_1\ .\]
Two independent common invariants under $X_1,X_2$ are
$w_1=u_2/u_3=v_1/v_2,\,w_2=u_4/u_3=v_3/v_2$.
The corresponding ODE is
\[ \stackrel{\ldots}{y}=\frac{1}{y^2}\Big(1-(\dot y-t)^3-ty+3ty\ddot y-4y\dot y\ddot
y\Big)\]
which admits the reduced \eq
\[\frac{\d w_1}{\d w_2}\=\frac{1-w_1w_2}{w_1-w_2^2}\ .\]
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