text:
Culture Obituaries»
Film Obituaries
Claude Chabrol, who died on September 12 aged 80, was one of that group of critics-turned-film-makers who revolutionised French cinema in the late Fifties and came to be known as the "new wave".
Claude Chabrol in 1979 Photo: AFP/GETTY
Claude Chabrol (seated) with actors (left to right) Jean Carmet, Stephane Audran and Isabelle Huppert in Cannes, 1978 Photo: AFP/GETTY
Stéphane Audran and Michel Bouquet in Chabrol's Just Before Nightfall (1971)
1:20PM BST 12 Sep 2010
Their careers developed in different directions, but they were united at the outset in rejecting the hidebound traditions of contemporary French films, with their formulaic, "well-made" scripts, and in calling for a new kind of cinema – realistic, and free of studio constraints and the shackles of the star system.
When Chabrol made his first film, Le Beau Serge, in 1958, using a small inheritance of his first wife's, he put these principles into practice, shooting it in black and white, on location in his home town of Sardent, with the then unknown players Gérard Blain and Jean-Claude Brialy.
The first feature film of the new wave, it was both a critical and commercial success, enabling Chabrol immediately to shoot a second, Les Cousins (1959), and to act as "godfather" to his friends, who were also eager to make their first films.
Chabrol used the profits from Le Beau Serge and Les Cousins to fund Eric Rohmer's Le Signe du Lion (1959) and Philippe de Broca's Les Jeux de l'Amour (1960) and Le Farceur (1961), and to help pay for Jacques Rivette's Paris Nous Appartient (1960). He also acted as "technical adviser" on Jean-Luc Godard's first feature, A Bout de Souffle (1959), but this was a token credit to enable Godard to piggyback into the cinema on Chabrol's shoulders; in practice, Godard was an innovator who needed no technical advice.
Squat, bespectacled and rotund, Chabrol played the joker in the pack, resembling nothing so much as a startled owl. It was an image that did not always redound to his advantage, casting doubt on his seriousness of purpose. He was a bon vivant and something of a gourmet, losing no opportunity in his films for a feast or a banquet. Wags said that they would not recognise a Chabrol film without, at the very least, a good domestic "blow-out".
Brilliant movies, huge lunches
Though in his work he pilloried the bourgeoisie – its foibles and petty cruelties – he was himself of this class and shared many of its values: property, wealth and domesticity. While two of his marriages failed, he retained a touching faith in the institution itself.
Born in Paris on June 24 1930, the son of a pharmacist, Claude Chabrol grew up at Sardent, 150 miles south of the capital, and always regarded himself more of a country boy than a Parisian. Indeed, his first two films were variants on the theme of the town mouse and the country mouse.
Movie-mad as a boy, he ran a film club from a barn at the age of 12, developing an early taste for thrillers and detective stories. To please his father, he studied Pharmacy at the Sorbonne and was expected to enter the family business. His military service was duly undertaken in the medical corps, where he reached the rank of sergeant.
When he was demobbed, however, he abandoned medicine and sought an independent career related to his first love, cinema. In 1955, aged 25, he landed what should have been a plum job as head of publicity for Twentieth Century-Fox in Paris. It did not work out. By their lights, Chabrol proved "the worst press officer they'd ever seen" and he was fired after a year, taking consolation from the fact that he was succeeded by a worse one, Jean-Luc Godard.
At this time he began to associate with film buffs at the Cinémathèque Française in the Avenue de Messina and at the Ciné-Club des Quartiers Latins, where they chewed the fat and argued over the finer points of film technique. The "gang", all of whom became film-makers in their own right, included Eric Rohmer, Jacques Rivette, Jean-Luc Godard and François Truffaut.
They were all of a similar mind and began to write regularly for the film magazine Cahiers du Cinéma, founded in 1950 by the critic and theorist André Bazin. Chabrol contributed pieces under his own name and as Charles Eitel and Jean-Yves Goute (the latter incorporating his wife's maiden name). Between them they developed a philosophy of cinema eventually labelled the politique des auteurs, arguing that no film should be judged in isolation but in the context of its director's previous work and of the conventions of its genre.
Impersonal French films of the time were dismissed in favour of what they regarded as the more individual ones of Roberto Rossellini, Carl Dreyer and Fritz Lang and of a whole slate of Hollywood film-makers hitherto barely taken seriously, such as Howard Hawks, Nicholas Ray and Otto Preminger.
Head of the pantheon in their eyes was Alfred Hitchcock, and in 1957 – with Eric Rohmer – Chabrol published a pioneering study that for the first time analysed Hitch's work in the context of his Catholic upbringing. Thrillers that had formerly been admired only as superior entertainments were reinterpreted in terms of such theological concepts as the transference of guilt. It was so persuasively argued that for many years this was regarded as the definitive work on Hitchcock, though never translated into English or updated to reflect the many films he made after 1957. Hitchcock remained a prime influence on Chabrol, many of whose films had themes and plots that could as easily have been filmed by him.
After the initial enthusiasm, Chabrol's critical standing went sharply into reverse, while that of his Cahiers colleagues continued to climb. Through the early and mid-Sixties, he seemed incapable of putting a foot right with critics or the public. Self-indulgent art-house fare such as A Double Tour (1959), Les Bonnes Femmes (1960) and Les Godelureaux (1961) was followed by palpably commercial potboilers with lip-smackingly lurid titles: The Tiger Likes Fresh Blood (1964), Marie-Chantal v Dr Kah and An Orchid for the Tiger (both 1965). These in turn were succeeded by awkward bilingual co-productions with Hollywood stars whose best years were behind them. In 1967, for example, Anthony Perkins appeared in Chabrol's The Champagne Murders and Jean Seberg in La Route de Corinthe, which was belatedly released outside France as Who's Got the Black Box?
Chabrol did himself no favours by seeming to relish this work. "In drivel like the Tiger series," he said, "I really wanted to get the full extent of the drivel. They were drivel so, OK, let's get into it up to our necks." Not surprisingly, questions began to be asked about him – whether he was simply a one-string-fiddle man who had happened to be in the right place at the right time with just enough money to shake up the addled French film industry of 1958 but without the staying power and originality of Truffaut and Godard. Was he, indeed, so very different from the older school of film makers he professed to despise?
It was a premature judgment. In 1968 Chabrol met the producer André Génovès, who gave him a free hand to make the film of his choice. He picked Les Biches, an intense psychological study of a three-way struggle for sexual domination between a bisexual woman, her protégée and the handsome architect who becomes the lover of both. More stylish and disciplined than anything Chabrol had made before, with a fat part for the actress Stéphane Audran, whom he had married in 1962, Les Biches restored his critical reputation and ushered in the most sustained creative phase of his career.
Between 1967 and 1973 he produced a stream of outstanding dramas, many starring his wife, in which he put the bourgeoisie under the microscope, exposing its prejudices and hypocrisies yet defending the strength of family ties. They included La Femme Infidèle and Que la Bête Meure (1969), La Rupture (1970), Just Before Nightfall (1971) and Les Noces Rouges (1973).
Best of all was Le Boucher (1970), about the love between a schoolteacher and the village butcher with a compulsion to murder young women. It had a depth and emotional commitment new in Chabrol's work, prompting Le Figaro, in a moment of unaccustomed hyperbole, to call it the best French film since the Liberation. Many of these films had a thriller structure reminiscent of Hitchcock and macabre touches that would not have been out of place in his work, such as the scene in Le Boucher when a murder is signalled by the victim's blood dripping from an overhanging rock into a schoolgirl's sandwich.
In general, Chabrol avoided what he called "big subjects", preferring little themes to which he could give the big treatment. When he departed from this principle, the results were usually unfortunate, as in Ten Days' Wonder (1972), an Ellery Queen story with Orson Welles and Anthony Perkins that he inflated into a study of comparative religion; and Nada (1974), an ill-considered foray into the field of politics and international terrorism. Of Ten Days' Wonder it was said that he let his belly rule his brain, accepting the assignment only because it could be shot in Alsace, where he was eager to sample the cuisine.
Chabrol never recaptured the consistency of his best years but remained a prolific director, making around 60 films in all. Occasional high spots stood out from the many pedestrian works he undertook merely to keep in practice, among which was a shoddy remake of Fritz Lang's Dr Mabuse called Dr M (1987). It was doubly disappointing in that Chabrol had named Lang, along with two other German directors, Ernst Lubitsch and FW Murnau, as one of the formative influences on his work. He was under no illusions about these films and cheerfully nominated one of them, The Twist (1976), with Ann-Margret and Bruce Dern, as the second-worst film ever made.
Some of his best work in later years was with the actress Isabelle Huppert, whom he first directed in Violette Nozière (1978), the story of a notorious murderess who poisoned her family. For Chabrol she also played the last woman to be guillotined in France (for carrying out abortions) in Une Affaire de Femmes (1988); the title role in a version of Madame Bovary (1991); and the village postmistress with a shady past in La Cérémonie (1995), an adaptation of a Ruth Rendell thriller.
Partially successful, too, was L'Enfer (1994), with Emmanuelle Béart, based on a 30-year-old script that the veteran film-maker Henri-Georges Clouzot had had to abandon when he suffered a heart attack in 1964.
Chabrol was also active in television, for which he made two Henry James short stories in 1973; four episodes of a series called Histoires Insolites in 1974; and, in 1993, an ambitious documentary about occupied France, The Eye of Vichy.
With the beginning of the new century, Chabrol seemed to regain his earlier form, making a number of films which were quite well-received. They included Merci pour le Chocolat (2000); The Bridesmaid (2004); and The Comedy of Power (2006). A Girl Cut in Two (2007) turned on a rich family's attempt to freeze out their son's widow. His last film was Bellamy (2009), with Gerard Depardieu.
He played cameo parts in numerous films, both his own and his friends', beginning with Jacques Rivette's short Le Coup du Berger (1956). His most fully-rounded role was as a parody of himself, opposite his wife, Stéphane Audran, in a 20-minute short, La Muette, that he contributed to the portmanteau picture Six in Paris (1966). They played gluttonous parents whose constant bickering leads to a sticky end when their small daughter plugs her ears against the din and subsequently fails to hear their cries for help.
Chabrol was married three times: to Agnès Goute, Colette Dacheville (the actress Stéphane Audran) and Aurore Pajot. He had four children.
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text:
\section{Introduction}

The main objective of this paper is to find solutions of the
relativistic {\it d}-brane system in $(d+1,1)$ dimensions in connection
with solutions of a Galileo invariant system in $(d,1)$ dimensions, as
recently considered in \cite{bja98,jpo98}. Although
we are mainly interested in the cases of $d=1$, $2$ and $3$ spatial
dimensions, some of the results will be naturally extended to $d>1$, arbitrary.
We remark that the Galileo invariant system is defined in $(d,1)$ dimensions,
and is related to the relativistic {\it d}-brane system in one higher spatial
dimension, in $(d+1,1)$ dimensions. The relativistic {\it d}-brane is an
extended object described by $(\phi^0,\phi^1,...,\phi^d)$, where $\phi^0$
is the evolution parameter. The motion of the {\it d}-brane is governed
by the Nambu-Goto action in $(d+1,1)$ space-time dimensions, with
$x^{\mu}=(x^0,{\bf x},x^{d+1})$ and ${\bf x}=(x^1,...,x^d)$. The Galileo
invariant system was recently investigated in the work \cite{bja98,jpo98},
and shows very interesting features, among them direct relation to fluid
mechanics \cite{lli}, to the hydrodynamical formulation of quantum
mechanics \cite{mad26,mer}, and to relativistic membrane \cite{hop82,bho93}
and its generalization to {\it d}-brane in $(d+1,1)$ dimensions.
Other interesting issues are investigated in \cite{hop94,hop95},
in \cite{jev98} and also in \cite{sch96,oga98,sch98}.

The Galileo invariant system is a Lagrangian system governed by the pair of
fields $\theta=\theta(t,{\bf x})$ and $\rho=\rho(t,{\bf x})$. Its dynamics
is first-order in time, and the corresponding symplectic structure shows
that $\theta(t,{\bf x})$ and $\rho(t,{\bf x})$ are canonically
conjugate \cite{fja88}. There are two equations of motion, one is a
continuity equation for the density $\rho$ and the current
${\bf J}=\rho\,\nabla\theta$ and the other describes first-order time
evolution of $\theta$, and depends on the presence of density-dependent
interactions. The connection between such dynamical system and the membrane
and its generalization to the {\it d}-brane system only appears under the very
specific density-dependent interaction $V(\rho)=\lambda/\rho$.

In the work \cite{bja98} some solutions of the dynamical system were
presented in $(1,1)$ dimensions and in $(d,1)$ for $d>1$. Also, in
\cite{jpo98} the $(1,1)$ dimensional case was shown to be exactly solvable
and solved explicitly. A lesson to be learned here is that this dynamical
system seems to present very specific behavior in $(1,1)$ dimensions,
differently of its general behavior in other dimensions $(d,1)$ for $d>1$.
This fact appears in the former investigations \cite{bja98,jpo98} and also
in the present investigations, which follows an alternate route to find
solutions. We believe that the distinct behavior in $d=1$ is somehow
related to the fact that the current density looses its vectorial character
in one spatial dimension. In this work instead of dealing
directly with the equations of motion that follows from the dynamical system
we solve the {\it d}-brane system in $(d+1,1)$ dimensions. After solving the
{\it d}-brane system we show how the solutions solve the corresponding
Galileo invariant system. Although we shall be concerned mainly with the
{\it d}-brane formulation of the problem, which is of direct interest to
theoretical particle physics, we emphasize that the subject is of broader
interest since it also offers connections to fluid dynamics \cite{lli},
quantum mechanics \cite{mer}, and other
subjects \cite{jev98,sch96,oga98,sch98} of interest to physics.

The present work is organized as follows. In the next
Sec.~{\ref{sec:general}} we introduce the
Galileo invariant system in $(d,1)$ dimensions and the relativistic
general {\it d}-brane system in $(d+1,1)$ dimensions, and we show the interesting
connection between these two apparently distinct systems.
We deal with specific issues in Sec.~{\ref{sec:solutions}}, where we split
the subject into finding solutions of {\it d}-brane systems in $(4,1)$,
$(3,1)$ and in $(2,1)$ dimensions, and showing how they solve the corresponding
Galileo invariant system in $(3,1)$, $(2,1)$, and $(1,1)$ dimensions. In
Sec.~{\ref{sec:generalization}} we generalize results obtained in $d=1,2,3$
to $d>1$, arbitrary. Comments and conclusions are briefly presented in
Sec.~{\ref{sec:comments}}.

\section{General considerations}
\label{sec:general}

Here we introduce the Galileo invariant
system and the relativistic {\it d}-brane system in $(d,1)$ and in $(d+1,1)$
dimensions, respectively. The informations collected below follow in
accordance with Refs.~{\cite{bja98,jpo98}}, and are important for the
issues that compose the rest of the paper. For simplicity we split the
subject into the two subsections that follow.

\subsection{The Galileo invariant system in ({\it d},1) dimensions}

The Galileo invariant system is governed by the action \cite{bja98}
\begin{equation}
\label{actiong}
I_{\lambda}=\int dt d{\bf x}\left(\theta\,\frac{\partial\rho}{\partial t}-
\frac{1}{2}\,\rho\,\nabla\theta\cdot\nabla\theta-\frac{\lambda}{\rho}\right)
\end{equation}
Here we have ${\bf x}=(x^1,...,x^d)$, and in this case we are working in
$(d,1)$ dimensions. This system is first-order in time \cite{fja88} and the
fields $\theta(t,{\bf x})$ and $\rho(t,{\bf x})$ are canonical variables,
with the Poisson bracket
\begin{eqnarray}
\{\rho(t,{\bf x}),\theta(t,{\bf y})\}=\delta({\bf x}-{\bf y})
\een
The Lagrangian and Hamiltonian are given by
\begin{eqnarray}
L&=&\int d{\bf x}\left(\theta\,\frac{\partial\rho}{\partial t}-
\frac{1}{2}\rho\nabla\theta\cdot\nabla\theta-\frac{\lambda}{\rho}\right)
\\
H&=&\int d{\bf x} \left(\frac{1}{2}\rho\nabla\theta\cdot\nabla\theta+
\frac{\lambda}{\rho}\right)
\een
There are two equations of motion. They can be obtained for instance via
\begin{eqnarray}
\frac{\partial\theta}{\partial t}&=&\{\theta, H\}
\\
\frac{\partial\rho}{\partial t}&=&\{\rho,H\}
\een
and they have the explicit form
\begin{eqnarray}
\label{mg1}
\frac{\partial\theta}{\partial t}+
\frac{1}{2}\,\nabla\theta\cdot\nabla\theta&=&\frac{\lambda}{\rho^2}
\\
\label{mg2}
\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\,\nabla\theta)&=&0
\een
The first equation of motion couples $\theta$ to $\rho$ via the specific
density-dependent interaction $V(\rho)=\lambda/\rho$. This equation decouples
$\theta$ from $\rho$ in the absence of interactions, in the limit
$\lambda\to0$. The second equation, Eq.~{(\ref{mg2})}, is a continuity
equation linking the density $\rho$ with the current density
${\bf J}=\rho\nabla\theta$. Here the current density necessarily
couples the two fields $\rho$ and $\theta$. In one spatial
dimension this coupling is non-vectorial, and we believe that this
may be behind the very specific behavior this system presents
in the $(1,1)$ dimensional case.

The hydrodynamical description of quantum mechanics follows with
the Schr\"odinger action
\begin{equation}
I_S=\int\,dt\,d{\bf x}\left( i\Psi^{*}{\dot\Psi}-
\frac{1}{2}\nabla\Psi^{*}\cdot\nabla\Psi-V_S\right)
\end{equation}
where $V_S=V(t,{\bf x})+{\bar V}(\Psi^{*}\Psi)$ is the potential.
We use
\begin{equation}
\Psi(t,{\bf x})=\sqrt{\rho}\,e^{i\theta}
\end{equation}
and the specific potential
\begin{equation}
\label{sp}
V_S=\frac{\lambda}{\rho}-\frac{1}{8}\,\frac{(\nabla\rho)^2}{\rho}
\end{equation}
to make $I_S$ to collapse to $I_{\lambda}$, reproducing the two equations
of motion $(\ref{mg1})$ and $(\ref{mg2})$. The quantum mechanical description
of the system just introduced is then governed by the dynamical system with
the action (\ref{actiong}). This hydrodynamical description is restricted
to pure states \cite{fan57} and one may envisage another route, relying on
the phase-space description of quantum mechanics \cite{wig32}, which seems
appropriate to generalize the above description -- recall that the
(probability) density and the (probability) current density are the
first two velocity moments of the Wigner distribution \cite{cza83}.
On the other hand, we remark that potentials like the one in Eq.~(\ref{sp})
have been considered for instance in Ref.~\cite{dgo94}, and appear under the
assumption of local probability conservation for the usual density and a
current density that is extended to account for diffusion.

Although the dynamical system $(\ref{actiong})$ is
manifestaly invariant under Galileo transformations, it also presents
\cite{bja98} symmetry algebra that can be identified with that of the
Poincar\'e group in $(d+1,1)$ dimensions. To see this, in $(d+1,1)$
dimensions we use light-cone coordinates to identify $(x^0,x^1,...,x^{d+1})$
with $(t,\theta,{\bf x})$, where ${\bf x}$ stands for the transverse
coordinates and
\begin{eqnarray}
\label{x+}
t&\equiv&\frac{1}{\sqrt{2}}(x^0+x^{d+1})=x^{+}
\\
\label{x-}
\theta&\equiv&\frac{1}{\sqrt{2}}(x^0-
x^{d+1})=x^{-}
\een
We now introduce the light-cone components of the Poincar\'e
generators $P^{\mu}$ and $M^{\mu\nu}$ in the usual form
\begin{eqnarray}
P^{\mu}&=&(P^-,P^+,P^i)
\\
M^{\mu\nu}&=&(M^{+-},M^{+i},M^{-i},M^{ij})
\een
To identify the generators of the Poincar\'e symmetry we recall that the
Poincar\'e group contains the extended Galileo group as a
subgroup \cite{sus68}. We write
\begin{eqnarray}
\label{p+}
P^-&\equiv&H=\int d{\bf x}\,{\cal E}
\\
\label{p-}
P^+&\equiv&N=\int d{\bf x} \,\rho
\\
\label{pi}
P^i&=&\int d{\bf x} \, J^i
\een
and
\begin{eqnarray}
\label{+-}
M^{+-}&\equiv&D=\int d{\bf x}\,(t\,{\cal E} - \,\theta\rho)
\\
\label{b+}
M^{+i}&\equiv&B^i=\int d{\bf x}\,\left(t\,J^i- \,\rho\, x^i\right)
\\
\label{b-}
M^{-i}&\equiv&G^i=\int d{\bf x}\,({\cal E} x^i-\,\theta\,J^i)
\\
\label{ang}
M^{ij}&=&\int d{\bf x}\,(x^i\,J^j-x^j\,J^i)
\een
The first set identifies generators of translations, which introduces the
Hamiltonian $H$, charge $N$ and linear momentum ${\bf P}$. The second set
identifies generators of rotations, which introduces the dilation $D$,
Galileo boost ${\bf B}$, ${\bf G}$ and the angular momentum ${\bf M}$.
Here we are using
\begin{eqnarray}
J^i&=&\rho\,\frac{\partial\theta}{\partial x^i}
\\
{\cal E}&=&\frac{1}{2}\,
\rho\,\nabla\theta\cdot\nabla\theta+\frac{\lambda}{\rho}
\een
It is not hard to check that the generators $(\ref{p+})$-$(\ref{ang})$ close
the Poincar\'e algebra in $(d+1,1)$ dimensions.

It is interesting to realize that interchangeability of the light-cone
coordinates $(+)$ and $(-)$ allows interchanging $t$ and $\theta$ in
$(\ref{x+})$ and $(\ref{x-})$, and this further implies interchanging
$P^{+}$ and $P^{-}$ and $M^{+i}$ and $M^{-i}$. We interchange $P^{+}$
and $P^{-}$ with $\rho\leftrightarrow{\cal E}$, which implies
$H\leftrightarrow N$ and ${\bf B}\leftrightarrow-{\bf G}$.
One verifies that under the interchange of light-cone coordinate the dilation
changes as $D\leftrightarrow -D$, picking up the minus sign in the same
way the other rotation generators ${\bf B}$ and ${\bf G}$ do. These
properties make the identifications $(\ref{p+})$-$(\ref{p-})$
and $(\ref{+-})$-$(\ref{b-})$ very natural. Also they are important
for constructing the explicit transformations: We offer a simple
illustration by considering the generator of dilation symmetry; here we
know that time changes according to
\begin{equation}
t\to e^{w}\,t
\end{equation}
To make the interchangeability of light-cone coordinates to work correctly
we must change the $\theta$ field according to
\begin{equation}
\theta(t,{\bf x})\to e^w\theta(e^w t,{\bf x})
\end{equation}
which is indeed the correct transformation \cite{bja98}.
This reasoning is inspired on unpublished notes \cite{dho98},
and it helps building all the transformations explicitly, in particular
the apparently misterious field-dependent coordinate transformations
introduced in \cite{bja98}, which now follow naturally from the standard
coordinate and field transformations that appear in a Galileo boost.

There is an alternate way to see the naturalness of the identification of
the generators of the Lorentz symmetry. This was briefly introduced
in \cite{bja98}, and here we make the argument explicit. We introduce the
infinitesimal Lorentz transformations
\begin{equation}
\delta x^{\mu}=w^{\mu\nu}\,x_{\nu}
\end{equation}
In the $(d+1,1)$ dimensional space-time we change to the former light-cone
coordinates to rewrite the infinitesimal Lorentz transformations in the form
\begin{eqnarray}
\delta\,x^{+}&=&w^{+-}\,x^{+}+w^{i+}\,x^{i}
\\
\delta\,x^{-}&=&-\,w^{+-}\,x^{-}+w^{i-}\,x^{i}
\\
\delta\,x^{i}&=&w^{i-}\,x^{+}+w^{i+}\,x^{-}-w^{ij}\,x^{j}
\een
The parameters $w^{+-}$, $w^{i-}$, $w^{i+}$ and $w^{ij}$ respond for dilation,
Galileo boost, field-dependent coordinate transformations and rotations,
respectively. To see this explicitly we notice that the transformations
involving only $w^{+-}$ obey
\begin{eqnarray}
\delta\,x^{+}&=&w^{+-}\,x^{+}
\\
\delta\,x^{-}&=&-\,w^{+-}\,x^{-}
\een
We use $w^{+-}=w$ to see that $\delta^{(n)}\,x^{\pm}=(\pm w)^{n}\,x^{\pm}$.
The finite transformations are
\begin{eqnarray}
{\tilde x}^{+}&=&e^{w}\,x^{+}
\\
{\tilde x}^{-}&=&e^{-w}\,x^{-}
\een
We use the light-cone identifications $x^+=t$ and
$x^-=\theta_{w}(t,{\bf x})$ and write the new coordinates as ${\tilde x}^{+}=T$
and ${\tilde x}^{-}=\theta(T,{\bf X})$. We then get
\begin{eqnarray}
t&\to& T=e^{w}\,t
\\
\theta&\to&\theta_{w}(t,{\bf x})=e^{w}\theta(e^{w}t,{\bf x})
\een
in agreement with the dilation transformations introduced in
Ref.~{\cite{bja98}}.

Let us now focus attention on the transformations introduced by $w^{i-}$,
identifying $w^{i-}=w^{i}$. We obtain
\begin{eqnarray}
\delta\,x^{+}&=&0
\\
\delta\,x^{-}&=&w^{i}\,x^{i}
\\
\delta\,x^{i}&=&w^{i}\,x^{+}
\een
We see that
\begin{eqnarray}
\delta^{(2)}\,x^{+}&=&0
\\
\delta^{(2)}\,x^{-}&=&\,w^{2}\,x^{+};\hspace{1cm}\delta^{(3)}\,x^{-}=0
\\
\delta^{(2)}\,x^{i}&=&0
\een
The finite transformations are then given by
\begin{eqnarray}
T&=&t
\\
X^i&=&x^i+ w^i\,t
\een
and
\begin{equation}
\theta(T,{\bf X})=\theta_{w}(t,{\bf x})+ w^i x^i+\frac{1}{2}\,w^2\,t
\end{equation}
These transformations identify the Galileo boost -- see Ref.~{\cite{bja98}}.

We now examine the transformations introduced by $w^{i+}$, making the
identification $w^{i+}=w^i$. We get
\begin{eqnarray}
\delta\,x^{+}&=&w^{i}\,x^{i}
\\
\delta\,x^{-}&=&0
\\
\delta\,x^{i}&=&w^{i}\,x^{-}
\een
We see that
\begin{eqnarray}
\delta^{(2)}\,x^{+}&=&w^{2}\,x^{-};\hspace{1cm}\delta^{(3)}\,x^{+}=0
\\
\delta^{(2)}\,x^{-}&=&0
\\
\delta^{(2)}\,x^{i}&=&0
\een
The finite transformations are given by
\begin{eqnarray}
T&=&t+w^i x^i+\frac{1}{2}\,w^2\theta_{w}(t,{\bf x})
\\
X^i&=&x^i+w^i\, \theta_{w}(t,{\bf x})
\een
and also
\begin{equation}
\theta(T,{\bf X})=\theta_{w}(t,{\bf x})
\end{equation}
In this case we can write
\begin{eqnarray}
T&=&t+w^ix^i+\frac{1}{2}\,w^2\theta(T,{\bf X})
\\
X^i&=&x^i+w^i\, \theta(T,{\bf X})
\een
which identify the field-dependent coordinate transformations introduced
in Ref.~{\cite{bja98}}.

The Galileo invariant system also engenders interesting solutions. The
equations of motion $(\ref{mg1})$ and $(\ref{mg2})$ possess
dilation-invariant solutions that can be written in the form
\begin{eqnarray}
\label{sthetad}
\theta(t,r)&=&-\frac{r^2}{2(d-1)t}
\\
\label{srhod}
\rho(t,r)&=&\sqrt{\frac{2}{d}\lambda\,}\,(d-1)\,\frac{|t|}{r}
\een
This pair of solutions appears in $(d,1)$ dimensions, for $d>1$, and here
we have set $r=\sqrt{{\bf x}\cdot{\bf x}}$, which identifies the length
of the vector ${\bf x}=(x^1,x^2,...,x^d)$ in $d>1$ spatial dimensions.
In Ref.~{\cite{bja98}} the pair of solutions for $d=2$ and the corresponding
field-dependent coordinate transformations induced by the generators
of ${\bf G}$ were used to obtain other solutions to the dynamical system
in $(2,1)$ dimensions. Here, however, we explore other issues and get the
pair of solutions $(\ref{sthetad})$ and $(\ref{srhod})$ by following an
alternate route, which relies on solving the {\it d}-brane problem in
$(d+1,1)$ dimensions, for $d>1$.

This Galileo invariant system can also be solved in one spatial
dimension, and some solutions were already presented in \cite{bja98}.
Also, in \cite{jpo98} it was shown that the $(1,1)$ dimensional system
is integrable, and the explicit solutions were presented. This result
is confirmed by the investigation that appears in Sec.~{\ref{sec:solutions}}
for the 1-brane system in $(2,1)$ dimensions.

\subsection{The ({\it d}+1,1) dimensional {\it d}-brane system}

We follow Ref.~{\cite{hop82}} to introduce the relativistic
{\it d}-brane system in $(d+1,1)$ dimensions. The {\it d}-brane system is an
extended object described by the coordinates $(\phi^0,\phi^1,...,\phi^d)$,
where $\phi^0$ is the evolution parameter and $(\phi^1,...,\phi^d)$ constitute
the {\it d}-dimensional space that parametrizes the {\it d}-brane. This
object is governed by the Nambu-Goto action
\begin{equation}
\label{adb}
I_d=-\int d\phi^0 d{\phi^1}...d{\phi^d} \,\, \sqrt{G}
\end{equation}
where $G$ is $(-1)^d$ times the determinant of the induced metric
\begin{equation}
G_{\alpha\beta}\equiv\frac{\partial x^\mu}{\partial\phi^{\alpha}}\,
\frac{\partial x_\mu}{\partial\phi^{\beta}}
\end{equation}
where $\alpha,\beta=0,1,...,d$. Here the {\it d}-brane is submersed in the
$(d+1,1)$ spacetime and we use light-cone coordinates to represent
$x^{\mu}=(x^0,x^1,...,x^d,x^{(d+1)})$ as $(\tau,\theta,{\bf x})$, where
\begin{eqnarray}
\label{tau}
x^{+}&=&\frac{1}{\sqrt{2}}\,(x^0+x^{(d+1)})\equiv\tau
\\
\label{theta}
x^{-}&=&\frac{1}{\sqrt{2}}\,(x^0-x^{(d+1)})\equiv\theta
\een
We use light-cone coordinates with the same motivation of the former case,
where we have shown that the Galileo invariant system in $(d,1)$ dimensions
presents symmetry algebra that can be identified with that of the Poincar\'e
group in $(d+1,1)$ dimensions. This is convenient because light-cone
coordinates introduce the tranverse spatial components
${\bf x}=(x^1,...,x^d)$ very naturally, which we shall identify
directly with the spatial components of the Galileo invariant system
in $(d,1)$ dimensions.

We identify the evolution parameter $\phi^0$ with the light-cone time $\tau$
$(\tau\equiv\phi^0)$ to write the elements of $G_{\alpha\beta}$ in the form
\begin{eqnarray}
G_{00}&=&2\,\frac{\partial\theta}{\partial\tau}-
\frac{\partial{\bf x}}{\partial\tau}\cdot
\frac{\partial{\bf x}}{\partial\tau}
\\
G_{0i}&=&G_{i0}=\frac{\partial\theta}{\partial\phi^i}-
\frac{\partial{\bf x}}{\partial\tau}\cdot
\frac{\partial{\bf x}}{\partial\phi^i}
\\
\label{matrixg}
G_{ij}&=&-g_{ij}=\frac{\partial{\bf x}}{\partial\phi^i}\cdot
\frac{\partial{\bf x}}{\partial\phi^j}
\een
We write
\begin{equation}
g\equiv\det\left(g_{ij}\right)
\end{equation}
to get
\begin{equation}
G=g\,\left(2\frac{\partial\theta}{\partial\tau}-
\frac{\partial{\bf x}}{\partial\tau}\cdot
\frac{\partial{\bf x}}{\partial\tau}+ g^{ij}u_i u_j\right)
\end{equation}
where $g^{ik}g_{kj}=\delta^i_j$ and $u_i$ is defined as
\begin{equation}
\label{defu}
u_i\equiv -\frac{\partial\theta}{\partial\phi^i}+
\frac{\partial{\bf x}}{\partial\tau}\cdot
\frac{\partial{\bf x}}{\partial\phi^i}
\end{equation}
The equations of motion for the {\it d}-brane can be written in the
form \cite{hop82}
\begin{eqnarray}
\frac{\partial{\bf x}}{\partial\tau}&=&-\frac{\bf p}{\Pi}+
u^i\frac{\partial{\bf x}}{\partial\phi^i}
\\
\frac{\partial\theta}{\partial\tau}&=&\frac{1}{2\Pi^2}\,
(\,{\bf p}\cdot{\bf p}+g\,)+u^i\frac{\partial\theta}{\partial\phi^i}
\\
\frac{\partial{\bf p}}{\partial\tau}&=&-\frac{\partial}{\partial\phi^i}
\left(\frac{1}{\Pi}\,g\,g^{ij}\frac{\partial{\bf x}}{\partial\phi^j}\right)+
\frac{\partial}{\partial\phi^i}({\bf p}\,u^i)
\\
\frac{\partial\Pi}{\partial\tau}&=&\frac{\partial}{\partial\phi^i}(\Pi\,u^i)
\een
where ${\bf p}$ and $\Pi$ are canonical momenta conjugate to ${\bf x}$
and $\theta$, respectively. The motion is constrained to obey
\begin{equation}
\label{cg}
{\bf p}\cdot\frac{\partial{\bf x}}{\partial\phi^i}+
\Pi\frac{\partial\theta}{\partial\phi^i}=0
\end{equation}
This condition appears as a consequence of gauge symmetry, which comes
from the freedom to parametrize the {\it d}-brane.

In the light-cone coordinates with the time identification
$\tau\equiv\phi^0$ we can further investigate the physical contents of the
{\it d}-brane after seting $u_i=0$. This condition identifies the
(light-cone) time dependence of the parametrization and allows rewriting
the equations of motion in the simpler form
\begin{eqnarray}
{\bf {p}}&=&-\Pi\,\frac{\partial{\bf{x}}}{\partial\tau}
\\
\label{pitau}
\frac{\partial\Pi}{\partial\tau}&=&0
\een
and
\begin{eqnarray}
\label{gpi1}
\frac{\partial\theta}{\partial\tau}&=&\frac{1}{2}\,
\left(\frac{\partial{\bf x}}{\partial\tau}\cdot
\frac{\partial{\bf x}}{\partial\tau}+\frac{1}{\Pi^2}\,g\,\right)
\\
\label{gpi2}
\frac{\partial^2{\bf x}}{\partial\tau^2}&=&\frac{1}{\Pi}\,
\frac{\partial}{\partial\phi^i}
\left(\frac{1}{\Pi}\,g\,g^{ij}\frac{\partial{\bf x}}{\partial\phi^j}\right)
\een
Instead of Eq.~(\ref{cg}) the constrained motion now obeys
\begin{equation}
\label{c}
\frac{\partial\theta}{\partial\phi^i}=\frac{\partial{\bf x}}{\partial\tau}
\cdot\frac{\partial{\bf x}}{\partial\phi^i}
\end{equation}
which reproduces Eq.~(\ref{defu}) for $u_i=0$ in a self-consistent way.
The constraint Eq.~(\ref{c}) is independent of the explicit form of the
momentum $\Pi=\Pi({\phi^1,...,\phi^d})$, now $\tau$-independent in
accordance with Eq.~(\ref{pitau}). This fact shows that one is still free
to make $\tau$-independent reparametrization
$(\phi^1,...,\phi^d)\to({\bar\phi}^1,...,{\bar\phi}^d)$ of the {\it d}-brane.
Different choices of the momentum $\Pi$ are related to different choices of
$\tau$-independent parametrization and then we can choose $\Pi$ at
convenience. For instance, the choice $\Pi=-c\,w(\phi^1,...,\phi^d)$
envolving a constant $c$ times a specified function of $\phi^1,...,\phi^d$ is
known \cite{hop82} as the orthonormal gauge. We can illustrate this point by
going from $\Pi({\phi^1,...,\phi^d})$ to $\Pi=-c$, constant, for simplicity.
In this case the above Eqs.~(\ref{gpi1}) and (\ref{gpi2}) become
\begin{eqnarray}
\label{g1c}
\frac{\partial^2{\bf x}}{\partial\tau^2}&=&\frac{\partial}{\partial\phi^i}
\left(\frac{1}{c^2}\,{g}\,g^{ij}\frac{\partial{\bf x}}{\partial\phi^j}\right)
\\
\label{g2c}
\frac{\partial\theta}{\partial\tau}&=&\frac{1}{2}
\left(\frac{\partial{\bf x}}{\partial\tau}\cdot
\frac{\partial{\bf x}}{\partial\tau}+ \frac{1}{c^2}\,{g}\,\right)
\een
We now change parametrization by allowing
\begin{equation}
\frac{\partial{\bf x}}{\partial\phi^i}\to
\frac{\partial{\bf x}}{\partial{\bar\phi}^j}\,
\frac{\partial{\bar\phi}^j}{\partial\phi^i}
\end{equation}
with a similar change in $\theta(\tau,\phi^1,...,\phi^d)$, in accordance
with the constraint Eq.~(\ref{c}). We use
\begin{equation}
\frac{\partial{\bar\phi}^i}{\partial\phi^j}=c\,\delta^i_j
\end{equation}
as the choice to get rid of the factor $1/c^2$ in
Eqs.~(\ref{g1c}) and (\ref{g2c}). For simplicity we then consider
$\Pi=-1$, which further implies ${\bf p}=\partial{\bf x}/\partial\tau$.
In this case the equations of motion simplify to the equations
\begin{eqnarray}
\label{m1}
\frac{\partial^2{\bf x}}{\partial\tau^2}&=&\frac{\partial}{\partial\phi^i}
\left(g\,g^{ij}\frac{\partial{\bf x}}{\partial\phi^j}\right)
\\
\label{m2}
\frac{\partial\theta}{\partial\tau}&=&\frac{1}{2}
\left(\frac{\partial{\bf x}}{\partial\tau}\cdot
\frac{\partial{\bf x}}{\partial\tau}+ g\,\right)
\een

The constraint Eq.~$(\ref{c})$ was first solved \cite{bho93} in the $d=2$
case, the membrane case and it can also be solved in the general {\it d}-brane
case. We follow Ref.~{\cite{jpo98}} and here the main step concerns
inverting ${\bf x}={\bf x}(\tau,\phi^1,...,\phi^d)$ to get
${\phi^i}={\phi^i}(t,{\bf x})$, after renaming $\tau$ as time $t$.
Here we have
\begin{equation}
\label{taut}
\frac{\partial}{\partial\tau}=\frac{\partial}{\partial t}+
\nabla\theta\cdot\nabla
\end{equation}
and the constraint Eq.~(\ref{c}) is solved by
\begin{equation}
\label{csol}
\frac{\partial{\bf x}}{\partial\tau}=\nabla\theta
\end{equation}
In this case Eq.~(\ref{m2}) becomes
\begin{equation}
\frac{\partial\theta}{\partial t}+
\frac{1}{2}\nabla\theta\cdot\nabla\theta=\frac{1}{2}g
\end{equation}
and reproduces Eq.~(\ref{mg1}) if and only if we further define
\begin{equation}
\label{defg}
g\equiv\frac{2\lambda}{\rho^2}
\end{equation}

On the other hand the Eq.~(\ref{m1}) can be rewritten as
\begin{equation}
\label{m1'}
\frac{\partial^2{x^i}}{\partial\tau^2}=\frac{1}{2}
\frac{\partial g}{\partial x^i}
\end{equation}
The proof follows after recognizing that
\begin{equation}
gg^{ij}=\frac{1}{(d-1)!}\epsilon^{ii_2...i_d}\epsilon^{jj_2...j_d}
\frac{\partial x^{k_2}}{\partial\phi^{i_2}}\cdots
\frac{\partial x^{k_d}}{\partial\phi^{i_d}}
\frac{\partial x^{k_2}}{\partial\phi^{j_2}}\cdots
\frac{\partial x^{k_d}}{\partial\phi^{j_d}}
\end{equation}
In this case we can use
\begin{equation}
\epsilon^{jj_2...j_d}\frac{\partial x^{k_2}}{\partial\phi^{j_2}}\cdots
\frac{\partial x^{k_d}}{\partial\phi^{j_d}}
\frac{\partial x^{k}}{\partial\phi^{j}}
\equiv\epsilon^{kk_2...k_d}\sqrt{g}
\end{equation}
to get to
\begin{equation}
gg^{ij}\frac{\partial x^{k}}{\partial\phi^{j}}=\frac{1}{(d-1)!}
\epsilon^{ii_2...i_d}
\epsilon^{kk_2...k_d}\frac{\partial x^{k_2}}{\partial\phi^{i_2}}\cdots
\frac{\partial x^{k_d}}{\partial\phi^{i_d}}\sqrt{g}
\end{equation}
and now we can write, recalling the smoothness of
${\bf x}={\bf x}(\tau,\phi^1,...,\phi^d)$ and (anti) symmetry of the
Levi-Civita symbol,
\begin{eqnarray}
\frac{\partial}{\partial\phi^i}\left(gg^{ij}
\frac{\partial x^{k}}{\partial\phi^{j}}\right)&=&
\frac{1}{(d-1)!}\epsilon^{ii_2...i_d}\epsilon^{kk_2...k_d}
\frac{\partial x^{k_2}}{\partial\phi^{i_2}}\cdots
\frac{\partial x^{k_d}}{\partial\phi^{i_d}}
\Biggl[\frac{\partial}{\partial\phi^{i}}\left(\sqrt{g}\right)\Biggr]
\nonumber\\
&=&\frac{1}{(d-1)!}\epsilon^{ii_2...i_d}\epsilon^{kk_2...k_d}
\frac{\partial x^{k_2}}{\partial\phi^{i_2}}\cdots
\frac{\partial x^{k_d}}{\partial\phi^{i_d}}\Biggl[
\frac{\partial}{\partial x^{l}}\left(\sqrt{g}\right)\Biggr]
\frac{\partial x^{l}}{\partial\phi^{i}}\nonumber\\
&=&\frac{1}{(d-1)!}\epsilon^{lk_2...k_d}\epsilon^{kk_2...k_d}
\sqrt{g}\Biggl[\frac{\partial}{\partial x^{l}}
\left(\sqrt{g}\right)\Biggr]
\een
We use the identity
\begin{equation}
\frac{1}{(d-1)!}\epsilon^{ik_2...k_d}
\epsilon^{jk_2...k_d}\equiv{\delta^{ij}}
\end{equation}
to obtain
\begin{equation}
\frac{\partial}{\partial\phi^i}\left(gg^{ij}
\frac{\partial x^{k}}{\partial\phi^{j}}\right)
= \frac{1}{2}\frac{\partial g}{\partial x^{k}}
\end{equation}
which ends the proof. We then use Eqs.~(\ref{taut}) and (\ref{csol}) to write
\begin{eqnarray}
\frac{\partial^2{\bf x}}{\partial\tau^2}&=&
\frac{\partial}{\partial\tau}\nabla\theta\nonumber
\\
&=&\nabla\frac{\partial\theta}{\partial t}+
\nabla\theta\cdot\nabla(\nabla\theta)\nonumber
\\
&=&\nabla\left(\frac{\partial\theta}{\partial t}+
\frac{1}{2}\nabla\theta\cdot\nabla\theta\right)
\een
We use this together with Eq.~(\ref{m1'}) to obtain, discarding an
unimportant constant,
\begin{equation}
\frac{\partial\theta}{\partial t}+
\frac{1}{2}\nabla\theta\cdot\nabla\theta=\frac{1}{2}g
\end{equation}
This equation also reproduces the Eq.~(\ref{mg1}) for $g$ defined by
Eq.~(\ref{defg}).

The Galileo invariant system $(\ref{actiong})$ contains another equation of
motion. It is Eq.~(\ref{mg2}), the continuity equation. It is
obtained from the relativistic {\it d}-brane system as follows. We have,
by definition
\begin{equation}
\frac{\partial g}{\partial\tau}\equiv
g\,g^{ij}\frac{\partial}{\partial\tau}g_{ij}
\end{equation}
Also
\begin{equation}
\frac{\partial g}{\partial\tau}=\frac{\partial g}{\partial t}+
\nabla\theta\cdot\nabla g
\end{equation}
We use these results together with
\begin{eqnarray}
g^{ij}\frac{\partial}{\partial\tau}g_{ij}&=&
\frac{\partial\phi^i}{\partial x^k}\frac{\partial\phi^j}{\partial x^k}
\frac{\partial}{\partial\tau}\left(\frac{\partial x^l}{\partial\phi^i}
\frac{\partial x^l}{\partial\phi^j}\right)\nonumber\\
&=&2\frac{\partial\phi^j}{\partial x^k}\frac{\partial}{\partial\phi^j}
\frac{\partial x^k}{\partial\tau}\nonumber\\
&=&2\frac{\partial}{\partial x^k}\frac{\partial\theta}{\partial x^k}
\een
to obtain
\begin{equation}
\frac{\partial g}{\partial t}+\nabla\theta\cdot\nabla g=2g\nabla^2\theta
\end{equation}
which reproduces the continuity equation after choosing $g=2\lambda/\rho^2$,
as given by Eq.~{(\ref{defg})}. The motion of the {\it d}-brane in $(d+1,1)$
dimensions is then governed by the equations of motion that describe the
dynamical system $(\ref{actiong})$ in $(d,1)$ dimensions. We remark
that the above proof only works under the definition introduced in
Eq.~{(\ref{defg})}, with the density-dependent interaction described
by the very specific potential $V(\rho)=\lambda/\rho$.

There is an alternate way to make the {\it d}-brane system in $(d+1,1)$
dimensions to reproduce the Galileo invariant system in $(d,1)$ dimensions.
It follows like in Ref.~{\cite{hop94}}, for instance, but now in $d+1$
spatial dimensions. It relies on reformulating the description of the
{\it d}-brane according to the identifications \cite{hpri}
\begin{eqnarray}
x^{+}&=&\frac{1}{\sqrt{2}}\,(x^0+x^{d+1})\equiv t=\phi^0
\\
x^{-}&=&\frac{1}{\sqrt{2}}\,(x^0-x^{d+1})\equiv\theta
\een
and also
\begin{equation}
\phi^i\equiv x^i~,\hspace{1cm}i=1,2,...,d
\end{equation}
This is interesting since now $\theta=\theta(t,{\bf x})$ is directly identified
with the $\theta$ field of the Galileo invariant system in $(d,1)$ dimensions.
To see this explictly we recall from Eq.~(\ref{adb}) that the Lagrangian
density can be written as
\begin{equation}
{\cal L}_d=\sqrt{G}
\end{equation}
and now
\begin{equation}
G=2\,\frac{\partial\theta}{\partial t}+\nabla\theta\cdot\nabla\theta
\end{equation}
The momentum conjugate to $\theta$ is given by
\begin{equation}
\Pi=\frac{1}{\sqrt{2\,\frac{\partial\theta}{\partial t}+
\nabla\theta\cdot\nabla\theta}}
\end{equation}
and this gives the Hamiltonian density
\begin{equation}
{\cal H}=-\frac{1}{2\Pi}-\frac{1}{2}\,\Pi\,\nabla\theta\cdot\nabla\theta
\end{equation}
We use this Hamiltonian density to write the first-order Lagrangian density,
after defining $\Pi\equiv-\rho/\sqrt{2\lambda}$. We discard a total
time derivative and ignore an unimportant multiplicative constant to obtain
\begin{equation}
{\cal L}=\theta\,\frac{\partial\rho}{\partial t}-
\frac{1}{2}\,\rho\,\nabla\theta\cdot\nabla\theta-\frac{\lambda}{\rho}
\end{equation}
This is exactly the Lagrangian density that follows from the action
$(\ref{actiong})$ that defines the Galileo invariant system in $(d,1)$
dimensions.

\section{Some brane solutions}
\label{sec:solutions}

Let us now deal with solutions of {\it d}-brane systems in
$(2,1)$, $(3,1)$ and $(4,1)$ dimensions, and with the relations between such
solutions and solutions of the Galileo invariant system in one, two and
three spatial dimensions. We split the subject into the three
subsections that follow, which deal with $d=1,2,3$ separately.

\subsection{Solutions for {\it d}=1}

Here we consider the relativistic 1-brane system in $(2,1)$ dimensions.
We parametrize the system with $\phi$, and the unidimensional transverse
coordinate is given by $x=x(\tau,\phi)$. Also, the matrix $g_{ij}=
(\partial{\bf x}/\partial\phi^i)\cdot(\partial{\bf x}/\partial\phi^j)$
and its determinant $g$ now become the very same thing, explicitly
\begin{equation}
\label{g}
g=\left(\frac{\partial x}{\partial\phi}\right)^2
\end{equation}
Also, $g\,g^{ij}\to1$ and the equation of motion $(\ref{m1})$ becomes
\begin{equation}
\label{wave}
\frac{\partial^2 x}{\partial\tau^2}-\frac{\partial^2 x}{\partial \phi^2}=0
\end{equation}
The other Eqs.~(\ref{c}) and (\ref{m2}) give
\begin{eqnarray}
\label{thetat}
\frac{\partial\theta}{\partial\phi}&=&
\frac{\partial x}{\partial\tau}\,\frac{\partial x}{\partial\phi}
\\
\label{thetaphi}
\frac{\partial\theta}{\partial\tau}&=&
\frac{1}{2}\Biggl[\left(\frac{\partial x}{\partial\tau}\right)^2+
\left(\frac{\partial x}{\partial\phi}\right)^2\Biggr]
\end{eqnarray}

We can solve Eq.~$(\ref{wave})$ directly. It is a wave equation and presents
the general solution
\begin{equation}
\label{sol}
x(\tau,\phi)=f_{+}(\tau+\phi)+f_{-}(\tau-\phi)
\end{equation}
The solution is written in terms of two arbitrary functions $f_{\pm}$, and
here we have
\begin{equation}
\label{xtau}
\frac{\partial x}{\partial\tau}=f'_{+}(\tau+\phi) + f'_{-}(\tau-\phi)
\end{equation}
and
\begin{equation}
\label{xphi}
\frac{\partial x}{\partial\phi}=f'_{+}(\tau+\phi) - f'_{-}(\tau-\phi)
\end{equation}
where we are using the notation $f'(z)=(\partial f/\partial z)$. We use these
expressions to rewrite Eqs.~$(\ref{thetat})$ and $(\ref{thetaphi})$ as
\begin{eqnarray}
\frac{\partial\theta}{\partial\tau}&=&[f'_{+}(\tau+\phi)]^2+
[f'_{-}(\tau-\phi)]^2
\\
\label{thetaphi1}
\frac{\partial\theta}{\partial\phi}&=&[f'_{+}(\tau+\phi)]^2-
[f'_{-}(\tau-\phi)]^2
\een
in order to get
\begin{eqnarray}
\frac{1}{2}\left(\frac{\partial}{\partial\tau}+
\frac{\partial}{\partial\phi}\right)\theta(\tau,\phi) &=&
[f'_{+}(\tau+\phi)]^2
\\
\frac{1}{2}\left(\frac{\partial}{\partial\tau}-
\frac{\partial}{\partial\phi}\right)\theta(\tau,\phi) &=&
[f'_{-}(\tau-\phi)]^2
\end{eqnarray}
These results allow writing $\theta(\tau,\phi)=\theta_+(\tau+\phi)+
\theta_-(\tau-\phi)$, and the general solution for $\theta$ can be written
in the form
\begin{equation}
\label{thetatauphi1}
\theta(\tau,\phi)=\int^{(\tau+\phi)}\,[f'_{+}(z)]^2\,dz+
\int^{(\tau-\phi)}\, [f'_{-}(z)]^2\,dz
\end{equation}
The results given by Eqs.~(\ref{sol}) and (\ref{thetatauphi1}) constitute the
pair of general solutions ${\bf x}(\tau,\phi)$ and
$\theta(\tau,\phi)$ of the 1-brane system.

To connect the 1-brane system to the Galileo invariant system in $(1,1)$
dimensions we follow the general investigations introduced in
Sec.~{\ref{sec:general}}. We start by rewriting $\theta(\tau,\phi)$ as
$\theta(t,x)$. Since we already have $x=x(\tau,\phi)$ in Eq.~$(\ref{sol})$,
we write $\phi=h(\tau,x)$ and change $\tau\to t$ to get to
\begin{equation}
x\equiv f_{+}(t+h(t,x))+f_{-}(t-h(t,x))
\end{equation}
in a way such that
\begin{equation}
(1+h_t)\,f'_{+}+(1-h_t)\,f'_{-}=0
\end{equation}
and
\begin{equation}
h_x\, f'_{+}-h_x \,f'_{-}=1
\end{equation}
We also get
\begin{equation}
h_t=\frac{\partial h}{\partial t}=-\,\frac{f'_{+}+f'_{-}}{f'_{+}-f'_{-}}
\end{equation}
and
\begin{equation}
h_x=\frac{\partial h}{\partial x}=\frac{1}{f'_{+}-f'_{-}}
\end{equation}
These results allow writing
\begin{equation}
\label{thetax}
\theta(t,x)=\int^{[t+h(x,t)]}\,[f'_{+}(z)]^2\,dz\,+\,
\int^{[t-h(x,t)]}\, [f'_{-}(z)]^2\,dz
\end{equation}
On the other hand, we use Eq.~{$(\ref{defg})$} to get
\begin{equation}
g=\left(\frac{\partial x}{\partial\phi}\right)^2=\frac{2\lambda}{\rho^2}
\end{equation}
and this leads to the result
\begin{equation}
\label{rho}
\rho(t,x)=\pm\,\frac{\sqrt{2\lambda}}{f'_{+}-f'_{-}}
\end{equation}
The above field configurations $\theta(t,x)$ and $\rho(t,x)$ obey
the pair of equations
\begin{eqnarray}
\label{motion1}
\frac{\partial\rho}{\partial t}+
\frac{\partial}{\partial x}\Biggl[\rho\left(\frac{\partial\theta}
{\partial x}\right)\Biggr]&=&0
\\
\label{motion2}
\frac{\partial\theta}{\partial t}+
\frac{1}{2}\left(\frac{\partial\theta}{\partial x}\right)^2&=&
\frac{\lambda}{\rho^2}
\een
which are the equations of motion $(\ref{mg1})$ and $(\ref{mg2})$ in the
$(1,1)$ dimensional case.

Furthermore, we can use Eq.~$(\ref{thetax})$ to write
\begin{equation}
\label{thetat1}
\frac{\partial\theta}{\partial t}=-2\,f'_+\,f'_-
\end{equation}
Also, the current density $J(x,t)=\rho\,(\partial\theta/\partial x)$ can be
written as
\begin{equation}
\label{current}
J(t,x)=\pm\,\sqrt{2\lambda}\,\,\frac{f'_+ + f'_-}{f'_+ -f'_-}
\end{equation}
It is now interesting to see that if we set $f_+$ or $f_-$ to zero,
that is, if we solve Eq.~$(\ref{wave})$ with only one of the two independent
solutions we get to the result that both $\theta$ and $\rho$ are
time-independent: We see that $\theta$ is time-independent directly from
Eq.~$(\ref{thetat1})$; from Eq.~$(\ref{current})$ we get that the current
density is constant, and so the continuity equation demands the density to
be time-independent. We remark that the equations of motion $(\ref{motion1})$
and $(\ref{motion2})$ impose that the current density is a specific constant
\begin{equation}
J(t,x)=\pm\,\sqrt{2\lambda}
\end{equation}
when one considers time-independent configurations. This means that the
particular solution $x(\tau,\phi)=f_+(\tau+\phi)$ or
$x(\tau,\phi)=f_-(\tau-\phi)$ to the Eq.~$(\ref{wave})$ reproduces all the
static solutions of the dynamical system governed by the action
$(\ref{actiong})$ in $(1,1)$ dimensions

For the $(1,1)$ dimensional system the energy of static configurations
$\theta=\theta(x)$ and $\rho=\rho(x)$ can be written in the
form
\begin{eqnarray}
E&=&\int dx \Biggl[\,\frac{1}{2}\,\rho\left(\frac{d\theta}{dx}
\right)^2+\frac{\lambda}{\rho}\,\Biggr]\nonumber\\
&=&\pm\,\sqrt{2\lambda}\int dx \,\frac{d\theta}{dx}+
\frac{1}{2}\int dx\, \rho\left(\frac{d\theta}{dx}\mp
\frac{\sqrt{2\lambda}}{\rho}\,\right)^2
\end{eqnarray}
and so is minimized to the value
\begin{equation}
E_M=\pm\,\sqrt{2\lambda} \,\,[\theta(x=\infty)-\theta(x=-\infty)]
\end{equation}
for field configurations that obey
\begin{equation}
\frac{d\theta}{dx}=\pm
\frac{\sqrt{2\lambda}}{\rho}
\end{equation}
which are the same solutions with constant and uniform current density
already obtained. Here we notice that the energy can also be written as
\begin{equation}
E_M=\int dx\, \rho\,\left(\frac{d\theta}{dx}\right)^2=
2\int dx\,\, \frac{\lambda}{\rho}
\end{equation}
This shows that the kinetic and potential parts of the energy contritube
equally, as usually happens with BPS solutions \cite{bps}. This means that
the system described by the Eqs.~$(\ref{motion1})$ and $(\ref{motion2})$
is the bosonic portion of some supersymmetric system, and that the 1-brane
solutions with $(\tau+\phi)$ or $(\tau-\phi)$ are the solutions that generate
all the BPS solutions of the corresponding dynamical system described by the
action $(\ref{actiong})$ in $(1,1)$ dimensions.

\subsection{Solutions for {\it d}=2}

Let us consider the $(3,1)$ dimensional problem. This is the case with
$d=2$, the membrane case where $(\phi^1,\phi^2)=(\phi,\psi)$ and
${\bf x}=(x,y)$. This problem was already considered in
\cite{hop82,bho93,hop94}. Here we have
\begin{eqnarray}
g_{11}&=&\left(\frac{\partial x}{\partial\phi}\right)^2+
\left(\frac{\partial y}{\partial\phi}\right)^2
\\
g_{12}&=&g_{21}=\frac{\partial x}{\partial\phi}\frac{\partial x}{\partial\psi}+
\frac{\partial y}{\partial\phi}\frac{\partial y}{\partial\psi}
\\
g_{22}&=&\left(\frac{\partial x}{\partial\psi}\right)^2+
\left(\frac{\partial y}{\partial\psi}\right)^2
\een
and we get
\begin{equation}
g=\left(\frac{\partial x}{\partial\phi}\frac{\partial y}{\partial\psi}-
\frac{\partial x}{\partial\psi}\frac{\partial y}{\partial\phi}\right)^2
\end{equation}
We can write $g=\{x,y\}^2$, where
\begin{equation}
\{x,y\}=\frac{\partial x}{\partial\phi}\frac{\partial y}{\partial\psi}-
\frac{\partial x}{\partial\psi}\frac{\partial y}{\partial\phi}
\end{equation}
is the Poisson bracket with respect to the membrane coordinates $(\phi,\psi)$.
We remark that such identification is only possible in the membrane case,
for $d=2$.

In Ref.~{\cite{bja98}} one finds solutions of the Galileo invariant system in
the planar ($d=2$) case, which present dilation and circular symmetries. For
this reason we choose the circular {\it Ansatz}
\begin{eqnarray}
\label{x2}
x&=&R(\tau,\phi)\,\cos(\psi)
\\
\label{y2}
y&=&R(\tau,\phi)\,\sin(\psi)
\een
In this case we get
\begin{eqnarray}
g_{11}&=&\left(\frac{\partial R}{\partial\phi}\right)^2
\\
g_{12}&=&g_{21}=0
\\
g_{22}&=&R^2
\een
which gives
\begin{equation}
\label{g2}
g=R^2\left(\frac{\partial R}{\partial\phi}\right)^2
\end{equation}
We also have
\begin{eqnarray}
g\,g^{11}&=&R^2
\\
g\,g^{12}&=&g\,g^{21}=0
\\
g\,g^{22}&=&\left(\frac{\partial R}{\partial\phi}\right)^2
\een

We use the equation of motion $(\ref{m1})$ to get from the equations for $x$
and $y$ the same equation for $R=R(\tau,\phi)$
\begin{equation}
\label{R2}
\frac{\partial^2 R}{\partial \tau^2}=R^2\,\frac{\partial^2R}{\partial\phi^2}+
R\,\left(\frac{\partial R}{\partial\phi}\right)^2
\end{equation}
This equation is solved by separating variables. Here we get
\begin{equation}
\label{R2s}
R(\tau,\phi)=2^{1/2}\,\frac{\phi}{\tau}
\end{equation}
The other Eqs.~$(\ref{c})$ and $(\ref{m2})$ give
\begin{equation}
\label{theta2s}
\theta(\tau,\phi)=-\frac{\phi^2}{\tau^3}
\end{equation}
We use Eqs.~(\ref{x2}) and (\ref{y2}) to write
\begin{eqnarray}
\frac{\partial x}{\partial\tau}\frac{\partial x}{\partial\phi}+
\frac{\partial y}{\partial\tau}\frac{\partial y}{\partial\phi}&=&
\left(\frac{\partial R}{\partial\tau}\right)
\left(\frac{\partial R}{\partial\phi}\right)
\\
\frac{\partial x}{\partial\tau}\frac{\partial x}{\partial\psi}+
\frac{\partial y}{\partial\tau}\frac{\partial y}{\partial\psi}&=&0
\een
Also, from Eq.~(\ref{R2s}) we get
\begin{eqnarray}
\frac{\partial R}{\partial\tau}&=&-2^{1/2}\,\frac{\phi}{\tau^2}
\\
\frac{\partial R}{\partial\phi}&=&2^{1/2}\,\frac{1}{\tau}
\een
and now Eq.~(\ref{theta2s}) allows checking that
\begin{eqnarray}
\frac{\partial\theta}{\partial\phi}&=&-2\,\frac{\phi}{\tau^3}=
\left(\frac{\partial R}{\partial\tau}\right)
\left(\frac{\partial R}{\partial\phi}\right)
\\
\frac{\partial\theta}{\partial\psi}&=&0
\een
in explicit agreement with the constraint Eq.~(\ref{c}), as expected.

We notice that $R^2=x^2+y^2$ and this allows getting
\begin{equation}
\phi^2=\frac{1}{2}\,\tau^2\,(x^2+y^2)
\end{equation}
and so we can write
\begin{equation}
\theta(t,x,y)=-\frac{1}{2 t}(x^2+y^2)
\end{equation}
We also have
\begin{equation}
g=\frac{2}{\tau^2}\,R^2
\end{equation}
and now Eq.~$(\ref{defg})$ allows getting
\begin{equation}
\rho(t,x,y)=\frac{\sqrt{\lambda}\,\,|t|}{\sqrt{(x^2+y^2)}}
\end{equation}
This pair of solutions identifies the circularly symmetric, dilation
invariant solutions presented in \cite{bja98} for the Galileo invariant
system in $(2,1)$ dimensions.

We remark that the above solution $(\ref{R2s})$ is a special
case of the solution found in \cite{hop94}, which gives another pair of
solution to the Galileo invariant system, a pair that presents no dilation
invariance anymore. This fact was already known to the author of
Ref.~{\cite{hop94}}.

\subsection{Solutions for {\it d}=3}

Let us now consider the 3-brane system in $(4,1)$ dimensions. We
parametrize this 3-brane with $(\phi^1,\phi^2,\phi^3)=(\phi,\chi,\psi)$ in
space ${\bf x}=(x,y,z)$. We choose the spherical {\it Ansatz}
\begin{eqnarray}
x&=&R(\tau,\phi)\,\sqrt{1-\chi^2}\,\cos(\psi)
\\
y&=&R(\tau,\phi)\,\sqrt{1-\chi^2}\,\sin(\psi)
\\
z&=&R(\tau,\phi)\,\chi
\een
In this case we use Eq.~{(\ref{matrixg})} to obtain
\begin{eqnarray}
g_{11}&=&\left(\frac{\partial R}{\partial\phi}\right)^2
\\
g_{22}&=&\frac{R^2}{1-\chi^2}
\\
g_{33}&=&R^2\,\left(1-\chi^2\right)
\een
with $g_{ij}=0$ for $i\neq j$. This implies that
\begin{equation}
\label{g3}
g=R^4\,\left(\frac{\partial R}{\partial\phi}\right)^2
\end{equation}
We then get $g\,g^{ij}=0$ for $i\neq j$, and for the diagonal elements we
have
\begin{eqnarray}
g\,g^{11}&=&R^4
\\
g\,g^{22}&=&(1-\chi^2)\,R^2\,\left(\frac{\partial R}{\partial\phi}\right)^2
\\
g\,g^{33}&=&\frac{R^2}{1-\chi^2}
\left(\frac{\partial R}{\partial\phi}\right)^2
\een
We use Eq.~{(\ref{m1})} to obtain from the equations for $x$, $y$, and $z$
the same equation for $R=R(\tau,\phi)$
\begin{equation}
\label{R3}
\frac{\partial^2 R}{\partial\tau^2}=R^4\frac{\partial^2 R}{\partial\phi^2}+
2\,R^3\,\left(\frac{\partial R}{\partial\phi}\right)^2
\end{equation}

This equation for $R(\tau,\phi)$ can also be solved by separating variables.
We see that
\begin{equation}
\label{R3d}
R(\tau,\phi)=3^{1/4}\left(\frac{\phi}{\tau}\right)^{1/2}
\end{equation}
explicitly solves Eq.~{(\ref{R3})}. This result can be used in
Eqs.~(\ref{c}) and (\ref{m2}) to give
\begin{equation}
\label{theta3s}
\theta(\tau,\phi)=-\frac{1}{4}\,3^{1/2}\,\frac{\phi}{\tau^2}
\end{equation}
We use the spherical {\it Ansatz} to write
\begin{eqnarray}
\frac{\partial x}{\partial\tau}\frac{\partial x}{\partial\phi}+
\frac{\partial y}{\partial\tau}\frac{\partial y}{\partial\phi}+
\frac{\partial z}{\partial\tau}\frac{\partial z}{\partial\phi}&=&
\left(\frac{\partial R}{\partial\tau}\right)
\left(\frac{\partial R}{\partial\phi}\right)
\\
\frac{\partial x}{\partial\tau}\frac{\partial x}{\partial\psi}+
\frac{\partial y}{\partial\tau}\frac{\partial y}{\partial\psi}+
\frac{\partial z}{\partial\tau}\frac{\partial z}{\partial\psi}&=&0
\\
\frac{\partial x}{\partial\tau}\frac{\partial x}{\partial\chi}+
\frac{\partial y}{\partial\tau}\frac{\partial y}{\partial\chi}+
\frac{\partial z}{\partial\tau}\frac{\partial z}{\partial\chi}&=&0
\een
Also, from Eq.~(\ref{R3d}) we get
\begin{eqnarray}
\frac{\partial R}{\partial\tau}&=&-\frac{1}{2}\,3^{1/4}
\left(\frac{\tau}{\phi}\right)^{1/2}\,\frac{\phi}{\tau^2}
\\
\frac{\partial R}{\partial\phi}&=&\frac{1}{2}\,3^{1/4}
\left(\frac{\tau}{\phi}\right)^{1/2}\,\frac{1}{\tau}
\een
We use Eq.~(\ref{theta3s}) to see that
\begin{equation}
\frac{\partial\theta}{\partial\phi}=-\frac{1}{4}\,3^{1/2}\,\frac{1}{\tau^2}
\end{equation}
which shows explicit agreement with the constraint Eq.~(\ref{c}).

To obtain $\theta$ in terms of the variables $(t,x,y,z)$ we notice that
the spherical {\it Ansatz} allows writing
\begin{equation}
x^2+y^2+z^2=R^2=3^{1/2}\,\frac{\phi}{\tau}
\end{equation}
We use this result in Eq.~{(\ref{theta3s})} to obtain
\begin{equation}
\label{theta3}
\theta(t,x,y,z)=-\frac{1}{4t}\left(x^2+y^2+z^2\right)
\end{equation}
The density is obtained with Eqs.~(\ref{defg}) and (\ref{g3}). It reads
\begin{equation}
\label{rho3}
\rho(t,x,y,z)=2\sqrt{\frac{2}{3}\,\lambda}\frac{|t|}{\sqrt{x^2+y^2+z^2}}
\end{equation}
and together with Eq.~(\ref{theta3}) forms a pair of solutions of the
Galileo invariant system $(\ref{actiong})$ in (3,1) dimensions.
This pair of solutions is exactly the dilation invariant solutions
found in \cite{bja98} for the Galileo invariant system in $d=3$,
with the density-dependent interaction $V(\rho)=\lambda/\rho$. They are the
solutions $(\ref{sthetad})$ and $(\ref{srhod})$ in the case $d=3$.

\section{Generalization}
\label{sec:generalization}

The results obtained in Sec.~{\ref{sec:solutions}} for the {\it d}-brane
system in $d=2$ and $3$ can be naturally extended to dimensions higher than
$d=3$. However, instead of generalizing the former spherical {\it Ansatz}
to arbitrary dimension we follow another route, which concerns generalizing
the results we have already obtained in the former Sec.~{\ref{sec:solutions}}.
We do this by first writing Eqs.~$(\ref{R2})$ and $(\ref{R3})$ together
\begin{eqnarray}
\frac{\partial^2 R}{\partial \tau^2}&=&
R^2\,\frac{\partial^2R}{\partial\phi^2}+
R\,\left(\frac{\partial R}{\partial\phi}\right)^2\hspace{1cm}(d=2)
\\
\frac{\partial^2 R}{\partial\tau^2}&=&R^4\frac{\partial^2 R}{\partial\phi^2}+
2\,R^3\,\left(\frac{\partial R}{\partial\phi}\right)^2\hspace{1cm}(d=3)
\een
These results suggest the general behavior
\begin{equation}
\label{Rg}
\frac{\partial^2 R}{\partial\tau^2}=
R^{2(d-1)}\frac{\partial^2R}{\partial\phi^2}+
(d-1)\,R^{2(d-1)-1}\left(\frac{\partial R}{\partial\phi}\right)^2
\end{equation}
This equation is nonlinear in all but the $d=1$ case,
where it reproduces the wave equation already considered in
the former Sec.~{\ref{sec:solutions}}.

Fortunately, we can solve Eq.~(\ref{Rg}) explicitly in the general $d>1$
case. We have the solution
\begin{equation}
\label{Rds}
R(\tau,\phi)=\left(d^{1/2}\,\frac{\phi}{\tau}\right)^{1/(d-1)}
\end{equation}
On the other hand, we also have the results
\begin{eqnarray}
g&=&R^2\,\left(\frac{\partial R}{\partial\phi}\right)^2\hspace{1cm}(d=2)
\\
g&=&R^4\,\left(\frac{\partial R}{\partial\phi}\right)^2\hspace{1cm}(d=3)
\een
They suggest the general behavior
\begin{equation}
\label{gg}
g=R^{2(d-1)}\,\left(\frac{\partial R}{\partial\phi}\right)^2
\end{equation}
The results for $\theta$ for $d=2$ and $3$ are given by Eqs.~(\ref{theta2s})
and (\ref{theta3s}). They can be rewritten as
\begin{eqnarray}
\theta(\tau,\phi)&=&-\,\frac{\phi^2}{\tau^3}=
-\frac{1}{2}\,\frac{1}{\tau}\,\left(2^{1/2}\,\frac{\phi}{\tau}\right)^2
\hspace{1cm}(d=2)
\\
\theta(\tau,\phi)&=&-\frac{1}{4}\,3^{1/2}\,\frac{\phi}{\tau^2}=
-\frac{1}{2}\,\frac{1}{2\tau}\,\Biggl[3^{1/4}
\left(\frac{\phi}{\tau}\right)^{1/2}\,\Biggr]^2 \hspace{1cm}(d=3)
\een
They and the result given by Eq.~{(\ref{Rds})} for $R(\tau,\phi)$
allow the following generalization
\begin{equation}
\theta(\tau,\phi)=-\frac{1}{2}\,\frac{R^2}{(d-1)\tau}
\end{equation}
which is valid under the restriction $d>1$.
For the density we use Eqs.~(\ref{defg}) and (\ref{gg}) to get
\begin{equation}
\rho(\tau,\phi)=\sqrt{\frac{2\lambda}{d}}(d-1)\frac{|\tau|}{R}
\end{equation}
These results can be rewritten as
\begin{eqnarray}
\theta(t,{\bf x})=-\frac{1}{2}\,\frac{r^2}{(d-1)t}
\\
\rho(t,{\bf x})=\sqrt{\frac{2\lambda}{d}\,}(d-1)\frac{|t|}{r}
\een
after changing $(\tau,\phi)\to (t,{\bf x})$ and using
$r=\sqrt{{\bf x}\cdot{\bf x}}$. They are the results $(\ref{sthetad})$
and $(\ref{srhod})$, found in Ref.~{\cite{bja98}} for the Galileo invariant
system $(\ref{actiong})$ in $(d,1)$ dimensions, for $d>1$.

There are at least two other interesting issues related to Eq.~(\ref{Rg}).
The first one is that it can be obtained from the Lagrangian density
\begin{equation}
{\cal L}=\frac{1}{2}\left(\frac{\partial R}{\partial\tau}\right)^2-
\frac{1}{2}\,R^{2(d-1)}\,\left(\frac{\partial R}{\partial\phi}\right)^2
\end{equation}
which can be rewritten in the form
\begin{equation}
{\cal L}=\frac{1}{2}\,{\cal G}^{\alpha\beta}\partial_{\alpha}R\,
\partial_{\beta}R
\end{equation}
where we are using that
\begin{equation}
\partial_{\alpha}=\left(\frac{\partial}{\partial\tau},
\frac{\partial}{\partial\phi}\right)
\end{equation}
and
\begin{eqnarray}
{\cal G}^{00}&=&1
\\
{\cal G}^{01}&=&{\cal G}^{10}=0
\\
{\cal G}^{11}&=&-R^{2(d-1)}
\een
in the effective $(1,1)$ dimensional space-time $(\tau,\phi)$. Here we
introduce the determinant ${\cal G}=\det({\cal G}^{\alpha\beta})=-R^{2(d-1)}$.
In general it depends on the point $(\tau,\phi)$, but in the $d=1$ case
it becomes constant, independent of both the $\phi$ and $\tau$ coordinates
of the bidimensional space-time.

The second issue related to Eq.~(\ref{Rg}) concerns the case of
considering $\tau$-independent spherical {\it Ansatz}, that is, of reducing
$R(\tau,\phi)$ to $R(\phi)$. In this case Eq.~(\ref{Rg}) becomes
\begin{equation}
\label{Rgphi}
R\,\frac{d^2R}{d\phi^2}+(d-1)\,
\left(\frac{dR}{d\phi}\right)^2=0
\end{equation}
A direct consequence of $R$ being $\tau$-independent is that
\begin{equation}
\frac{\partial g}{\partial\tau}=0
\end{equation}
Furthermore, from Eqs.~(\ref{gg}) and (\ref{Rgphi}) we obtain
\begin{equation}
\frac{\partial g}{\partial\phi}=0
\end{equation}
and so $g$ is constant. From Eq.~(\ref{m2}) we then get
\begin{equation}
\label{thetatind}
\theta=\frac{1}{2}\,g\,\tau
\end{equation}
which is independent of $\phi$ and so compatible with the constraint
Eq.~(\ref{c}). An explicit  illustration is given after solving
Eq.~(\ref{Rgphi}). Here we see that
\begin{equation}
\label{Rgenind}
R(\phi)=\phi^{1/d}
\end{equation}
solves Eq.~(\ref{Rgphi}). Also we use Eq.~(\ref{gg}) to get
\begin{equation}
g=\frac{1}{d^2}
\end{equation}
This result and Eq.~(\ref{thetatind}) allow writting
\begin{equation}
\label{thetagenind}
\theta=\frac{1}{2d^2}\,\tau
\end{equation}
Eqs.~(\ref{Rgenind}) and (\ref{thetagenind}) give another
solution for the {\it d}-brane in $d>1$ spatial dimensions. We remark that
solutions to the above second-order Eq.~(\ref{Rgphi}) can also be obtained
via the following first-order equation
\begin{equation}
\label{Rfirst}
\frac{dR}{d\phi}=\frac{1}{d}\,R^{1-d}
\end{equation}
We notice that the above solution (\ref{Rgenind}) solves the first-order
Eq.~(\ref{Rfirst}). We postpone to a future work further investigations
on this and on other related issues.

\section{Comments and conclusions}
\label{sec:comments}

In this paper we have found solutions for the relativistic {\it d}-brane
system in $(d+1,1)$ dimensions for $d=1$ and for $d>1$. The {\it d}-brane
system presents spherically symmetric solutions in $d=2, 3$
that are directly related to the spherically symmetric solutions
introduced in \cite{bja98} for the Galileo invariant system in $(2,1)$
and in $(3,1)$ dimensions. These results were generalized
to higher spatial dimensions, and so we obtained
solutions of the relativistic {\it d}-brane system in $(d+1,1)$ dimensions
that reproduce the solutions $(\ref{sthetad})$ and $(\ref{srhod})$
of the Galileo invariant system $(\ref{actiong})$ in $(d,1)$ dimensions,
for $d>1$.

These solutions are very different from the solutions one finds in $(1,1)$
dimensions. In $d=1$ the Galileo invariant system was solved exactly. The
general behavior follows as in \cite{jpo98}, which solved the equations of
motion following the route of linearization of mechanics of fluids \cite{lli}.
In Ref.~{\cite{jpo98}} one also finds
infinity sets of conserved quantities, which are proper to systems
engendering general integrability. The behavior of general integrability
in $d=1$ shows up in the present investigations via the presence of the wave
equation, which is linear and is solved exactly by standard method.
The 1-brane route to solutions to the Galileo invariant system in $(1,1)$
dimensions is very interesting since it allows obtaining the static solutions
and identifying how the BPS solutions of that dynamical system appear in the
1-brane system.

The investigations done in the present paper introduce further issues,
for instance the problem of searching for other solutions. In connection with
Ref.~{\cite{bja98}} one can ask about the possibility of transforming
solutions to new ones, using the field-dependent coordinate transformations
generated by ${\bf G}$ in Eq.~(\ref{b-}). In the hydrodynamical formulation
of quantum mechanics we can ask about the possibility of not only introducing
diffusion \cite{dgo94} but also going beyond the description of pure
states \cite{fan57}. Other interesting investigations \cite{jpo98a} have been
done, generalizing not only the way the fluid mechanical system interacts
but also how the kinematical contribution may modify its dynamical behavior.

We can also reformulate the light-cone description of the {\it d}-brane in
the way that leads to the Born-Infeld equation \cite{bin34}, as presented in
Refs.~{\cite{bho93,hop94}, for instance. As one knows, the Born-Infeld
equation arises after a nonlinear modification of standard electrodynamics,
and presents interesting properties \cite{ann89} and connections to fluid
mechanics and other issues, as the ones investigated recently in
Refs.~{\cite{cma98,gib98}}. Another interesting issue concerns the fact that
{\it d}-branes are extended objects that can be related to matrix theory and
as such may be of interest for instance in the context of holomorphic
configurations recently considered in Ref.~\cite{cta98}}.
\vskip 1cm

\begin{center}
Acknowledgments
\end{center}

The author would like to thank Roman Jackiw, from whom he has learned quite
a few interesting things related to the subject of this work. He also thanks
Jens Hoppe and Washington Taylor for interesting comments and discussions.
He is grateful to the Center for Theoretical Physics, MIT, for hospitality.

redpajama_set_name: RedPajamaArXiv


I presented a session on Social Learning at a Dutch learning conference today, ending the session by describing 'Six Week Experiments'. This notion of short, focused, and achievable, experiments is something that i'm taking from the upcoming publication of 'The Change Handbook', which tries to reconcile broad and holistic visions of change, with the grinding reality of what we should do differently next Tuesday. We need both.
The last couple of years have seen a broad and rapid shift in mindset and focus around learning: many organisations are implementing new technology, exploring collaborative communities, moving beyond purely formal approaches. But not always with aligned energy, or clear measures of success. This feels like a great area to focus on: we need learning transformation programmes, to address the big strategic questions, but we also need local, personal, and small scale experiments to inform that strategy.
Six week experiments may be about creating a shared story, a community of difference, or a short diagnostic exercise. None of these experiments will reveal one big answer, but they are a manifestation of the mechanism by which we will work out the big answer.
This entry was posted in Collaboration, Learning and tagged Collaboration, Experiments, Six Week Change, Social Learning. Bookmark the permalink.


text:
Giants' rally falls short as Nola baffles offense
Sandoval gets Giants closer in 8th; Holland allows 3 homers
PHILADELPHIA -- Maybe the Giants' 4-2 loss Tuesday night to the Phillies was a mere matter of timing, in which they were scheduled to visit Citizens Bank Park when Aaron Nola happened to throw one of the best games of his career.Or, perhaps the Giants bore witness to something more
PHILADELPHIA -- Maybe the Giants' 4-2 loss Tuesday night to the Phillies was a mere matter of timing, in which they were scheduled to visit Citizens Bank Park when Aaron Nola happened to throw one of the best games of his career.
Or, perhaps the Giants bore witness to something more imposing and electrifying: the legitimizing of a future All-Star, a peer who someday shall become peerless.
Nola dominated the Giants for seven innings, allowing one run and five hits. He struck out nearly half of the batters he faced -- 12 of 25 -- and retired at least one batter on strikes in every inning he worked but the seventh. Just as impressively, he walked none.
Nola, 24, has been around. This was his 68th career start. But with this game, he truly announced his presence to the Giants. From now on, when the Giants see the Phillies on the schedule, they'll wonder whether Nola's pitching, just as they do when they face the Dodgers and grimly await Clayton Kershaw's turn to pitch.
"He was throwing a couple of different fastballs, commanded the changeup pretty well and dropped in breaking balls," Giants catcher Buster Posey said of Nola, who recorded nine consecutive outs on strikeouts from the end of the second inning through the first two outs of the fifth. "Yeah, he was pretty good tonight."
Fabulous as Nola was, the Giants brought the potential tying run to the plate in the eighth inning. After Pablo Sandoval clobbered a pinch-hit homer off Edubray Ramos to narrow the difference to 4-2, Andrew McCutchen drew a two-out walk. Up came Posey, who was 0-for-3 but had hit safely in his last 23 games here. That streak ended, along with the Giants' hopes, when right-hander Tommy Hunter relieved Ramos and coaxed a groundout to third from Posey.
Giants starter Derek Holland continued the Giants' recent tendency to yield the long ball, allowing home runs to Aaron Altherr and Jorge Alfaro. Both came on breaking pitches, which aroused Holland's concern. He suspected that he might have tipped off the hitters to what was coming by slowing down his delivery
"As the hitter, it's easier to sit back on something like that," Holland said.
Holland (1-4) surrendered three earned runs in five innings for the third time this season. Yet manager Bruce Bochy deciphered improvement from the left-hander.
"I thought he made some really good pitches that could have gone the other way," Bochy said. "Overall, his stuff has picked up."
SOUND SMART
This marks the first time the Giants have lost back-to-back games since April 13-17 (three at San Diego, one at Arizona). But Holland insisted on remaining calm.
"We're going to bounce back," he vowed. "We've shown it."
YOU GOTTA SEE THIS
The Giants have committed 11 errors in the first five games of this trip. Tuesday, however, they tightened up their defense. Third baseman Evan Longoria made an impressive diving stop of Carlos Santana's second-inning line drive, and shortstop Brandon Crawford ranged toward the middle of the infield to grab Nola's grounder and start an inning-ending double play in the fourth inning from a difficult angle.
MITEL REPLAY OF THE DAY
The Giants generated such little offense that wasting even the slightest opportunity aroused Bochy's concern. The Phillies thought Nola picked off Gregor Blanco in the fifth inning, but replays proved inconclusive and the call stood. However, Nola legitimately picked off Blanco moments later to end the inning.
"It's something we will address," Bochy said.
Giants right-hander Chris Stratton, who starts Wednesday's 4:05 p.m. PT contest at Philadelphia, allowed two homers in a game for the first time in his career last Friday at Atlanta yet earned the decision in San Francisco's 9-4 win. Nick Pivetta will start for the Phillies.
Chris Haft has covered the Giants since 2005, and for MLB.com since 2007. Follow him on Twitter at @sfgiantsbeat and listen to his podcast.

redpajama_set_name: RedPajamaCommonCrawl


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Q: Segmentation Fault only on Ubuntu 10.04 32bit because of too small stack size I have a Java application which uses several native libraries through JNA.
The application works fine on many different systems. However, on Ubuntu 10.04 LTS - 32bit when I close the application a segmentation fault occurs.
64bit works fine and the error does not occur on other 32bit distributions. Windows works fine too.
The segmentation fault occurs in the delete function of glibc within an empty  destructor.
The problem is solved, if I increase the maximum stack size of the JVM to 32mb
(e.g. java -Xss32m ...)
Does anyone have an idea why this error only happens on Ubuntu 10.04 32bit?
I have tested several distribution within a VM.
I am using Java 8 and the default maximum stack size is 4MB.
I dont know if this is relevant to the JVM, but ulimit -s is 8MB.

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