Plain TeX must run twice
BODY
\def\mela{\relax}
\magnification1200
\hsize 14cm
%file macrodario.tex
\catcode`@=11
%
%------------------------- comandi riservati ---------------------------
%
\def\b@lank{ }
\newif\if@simboli
\newif\if@riferimenti
\newif\if@bozze
\newif\if@data
\def\bozze{\@bozzetrue
\immediate\write16{!!! INSERISCE NOME EQUAZIONI !!!}}
\newwrite\file@simboli
\def\simboli{
\immediate\write16{ !!! Genera il file \jobname.SMB }
\@simbolitrue\immediate\openout\file@simboli=\jobname.smb
\immediate\write\file@simboli{Simboli di \jobname}}
\newwrite\file@ausiliario
\def\riferimentifuturi{
\immediate\write16{ !!! Genera il file \jobname.aux }
\@riferimentitrue\openin1 \jobname.aux
\ifeof1\relax\else\closein1\relax\input\jobname.aux\fi
\immediate\openout\file@ausiliario=\jobname.aux}
\newcount\eq@num\global\eq@num=0
\newcount\sect@num\global\sect@num=0
\newcount\para@num\global\para@num=0
\newcount\const@num\global\const@num=0
\newcount\lemm@num\global\lemm@num=0
\newif\if@ndoppia
\def\numerazionedoppia{\@ndoppiatrue\gdef\la@sezionecorrente{\the\sect@num}}
\def\se@indefinito#1{\expandafter\ifx\csname#1\endcsname\relax}
\def\spo@glia#1>{} % si applica a \meaning\xxxxx; butta via tutto quello
% che produce \meaning fino al carattere >
% (v. manuale TeX, pag. 382, \strip#1>{}).
\newif\if@primasezione
\@primasezionetrue
\def\s@ection#1\par{\immediate
\write16{#1}\if@primasezione\global\@primasezionefalse\else\goodbreak
\vskip\spaziosoprasez\fi\noindent
{\bf#1}\nobreak\vskip\spaziosottosez\nobreak\noindent}
%--------------------------- Indice -------------------------------
\newif\if@indice
\newif\if@ceindice
\newwrite\file@indice
\def\indice{
\immediate\write16{Genera il file \jobname.ind}
\@indicetrue
\immediate\openin2 \jobname.ind
\ifeof2\relax\else
\closein2\relax
\@ceindicetrue\fi
\if@ceindice\relax\else
\immediate\openout\file@indice=\jobname.ind
\immediate\write
\file@indice{\string\vskip5pt
\string{ \string\bf \string\centerline\string{ Indice
\string}\string}\string\par}
\fi
}
\def\quiindice{\if@ceindice\vfill\eject\input\jobname.ind\else\vfill\eject
\immediate\write\file@indice{\string{\string\bf\string~
Indice\string}\string\hfill\folio}
\null\vfill\eject\null\vfill\eject\relax\fi}
%
%------------------------------ a disp. dell'utente: sezioni -------------
\def\sezpreset#1{\global\sect@num=#1
\immediate\write16{ !!! sez-preset = #1 } }
\def\spaziosoprasez{50pt plus 60pt}
\def\spaziosottosez{15pt}
\def\sref#1{\se@indefinito{@s@#1}\immediate\write16{ ??? \string\sref{#1}
non definita !!!}
\expandafter\xdef\csname @s@#1\endcsname{??}\fi\csname @s@#1\endcsname}
\def\autosez#1#2\par{
\global\advance\sect@num by 1\if@ndoppia\global\eq@num=0\fi
\global\lemm@num=0
\global\para@num=0
\xdef\la@sezionecorrente{\the\sect@num}
\def\usa@getta{1}\se@indefinito{@s@#1}\def\usa@getta{2}\fi
\expandafter\ifx\csname @s@#1\endcsname\la@sezionecorrente\def
\usa@getta{2}\fi
\ifodd\usa@getta\immediate\write16
{ ??? possibili riferimenti errati a \string\sref{#1} !!!}\fi
\expandafter\xdef\csname @s@#1\endcsname{\la@sezionecorrente}
\immediate\write16{\la@sezionecorrente. #2}
\if@simboli
\immediate\write\file@simboli{ }\immediate\write\file@simboli{ }
\immediate\write\file@simboli{ Sezione
\la@sezionecorrente : sref. #1}
\immediate\write\file@simboli{ } \fi
\if@riferimenti
\immediate\write\file@ausiliario{\string\expandafter\string\edef
\string\csname\b@lank @s@#1\string\endcsname{\la@sezionecorrente}}\fi
\goodbreak\vskip 48pt plus 60pt
\noindent{\bf\the\sect@num.\quad #2}
\if@bozze
{\tt #1}\fi
\par\nobreak\vskip 15pt
\nobreak}
\def\blankii{\blank\blank}
\def\destra#1{{\hfill#1}}
\font\titfnt=cmssbx10 scaled \magstep2
\font\capfnt=cmss17 scaled \magstep4
\def\blank{\vskip 12pt}
\def\capitolo#1#2\par{
\global\advance\sect@num by 1\if@ndoppia\global\eq@num=0\fi
\global\lemm@num=0
\global\para@num=0
\xdef\la@sezionecorrente{\the\sect@num}
\def\usa@getta{1}\se@indefinito{@s@#1}\def\usa@getta{2}\fi
\expandafter\ifx\csname @s@#1\endcsname\la@sezionecorrente\def
\usa@getta{2}\fi
\ifodd\usa@getta\immediate\write16
{ ??? possibili riferimenti errati a \string\sref{#1} !!!}\fi
\expandafter\xdef\csname @s@#1\endcsname{\la@sezionecorrente}
\immediate\write16{\la@sezionecorrente. #2}
\if@simboli
\immediate\write\file@simboli{ }\immediate\write\file@simboli{ }
\immediate\write\file@simboli{ Sezione
\la@sezionecorrente : sref. #1}
\immediate\write\file@simboli{ } \fi
\if@riferimenti
\immediate\write\file@ausiliario{\string\expandafter\string\edef
\string\csname\b@lank @s@#1\string\endcsname{\la@sezionecorrente}}\fi
\par\vfill\eject
\destra{\capfnt {\la@sezionecorrente}\hbox to 10pt{\hfil}}
\blankii\noindent{\titfnt\baselineskip=20pt
\hfill\uppercase{#2}}\blankii
\if@indice
\if@ceindice\relax\else\immediate\write
\file@indice{\string\vskip5pt\string{\string\bf
\la@sezionecorrente.#2\string}\string\hfill\folio\string\par}\fi\fi
\if@bozze
{\tt #1}\par\fi\nobreak}
\def\semiautosez#1#2\par{
\gdef\la@sezionecorrente{#1}\if@ndoppia\global\eq@num=0
\fi
\global\lemm@num=0
\global\para@num=0
\if@simboli
\immediate\write\file@simboli{ }\immediate\write\file@simboli{ }
\immediate\write\file@simboli{ Sezione ** : sref.
\expandafter\spo@glia\meaning\la@sezionecorrente}
\immediate\write\file@simboli{ }\fi
\s@ection#2\par}
%------------------paragrafi----------------------------------------
\def\pararef#1{\se@indefinito{@ap@#1}
\immediate\write16{??? \string\pararef{#1} non definito !!!}
\expandafter\xdef\csname @ap@#1\endcsname {#1}
\fi\csname @ap@#1\endcsname}
\def\autopara#1#2\par{
\global\advance\para@num by 1
\xdef\il@paragrafo{\la@sezionecorrente.\the\para@num}
\vskip10pt
\noindent {\bf \il@paragrafo\ #2}
\def\usa@getta{1}\se@indefinito{@ap@#1}\def\usa@getta{2}\fi
\expandafter\ifx\csname
@ap@#1\endcsname\il@paragrafo\def\usa@getta{2}\fi
\ifodd\usa@getta\immediate\write16
{??? possibili riferimenti errati a \string\pararef{#1} !!!}\fi
\expandafter\xdef\csname @ap@#1\endcsname{\il@paragrafo}
\def\usa@getta{\expandafter\spo@glia\meaning
\la@sezionecorrente.\the\para@num}
\if@simboli
\immediate\write\file@simboli{ }\immediate\write\file@simboli{ }
\immediate\write\file@simboli{ paragrafo
\il@paragrafo : pararef. #1}
\immediate\write\file@simboli{ } \fi
\if@riferimenti
\immediate\write\file@ausiliario{\string\expandafter\string\edef
\string\csname\b@lank @ap@#1\string\endcsname{\il@paragrafo}}\fi
\if@indice
\if@ceindice\relax\else\immediate\write
\file@indice{\string\noindent\string\item\string{
\il@paragrafo.\string}#2\string\dotfill\folio\string\par}\fi\fi
\if@bozze
{\tt #1}\fi\par\nobreak\vskip .3 cm \nobreak}
%------------------------------ a disp. dell'utente: equazioni -----------
\def\eqpreset#1{\global\eq@num=#1
\immediate\write16{ !!! eq-preset = #1 } }
\def\eqlabel#1{\global\advance\eq@num by 1
\if@ndoppia\xdef\il@numero{\la@sezionecorrente.\the\eq@num}
\else\xdef\il@numero{\the\eq@num}\fi
\def\usa@getta{1}\se@indefinito{@eq@#1}\def\usa@getta{2}\fi
\expandafter\ifx\csname @eq@#1\endcsname\il@numero\def\usa@getta{2}\fi
\ifodd\usa@getta\immediate\write16
{ ??? possibili riferimenti errati a \string\eqref{#1} !!!}\fi
\expandafter\xdef\csname @eq@#1\endcsname{\il@numero}
\if@ndoppia
\def\usa@getta{\expandafter\spo@glia\meaning
\il@numero}
\else\def\usa@getta{\il@numero}\fi
\if@simboli
\immediate\write\file@simboli{ Equazione
\usa@getta : eqref. #1}\fi
\if@riferimenti
\immediate\write\file@ausiliario{\string\expandafter\string\edef
\string\csname\b@lank @eq@#1\string\endcsname{\usa@getta}}\fi}
\def\eqsref#1{\se@indefinito{@eq@#1}
\immediate\write16{ ??? \string\eqref{#1} non definita !!!}
\if@riferimenti\relax
\else\eqlabel{#1} ???\fi
\fi\csname @eq@#1\endcsname }
\def\autoeqno#1{\eqlabel{#1}\eqno(\csname @eq@#1\endcsname)\if@bozze
{\tt #1}\else\relax\fi}
\def\autoleqno#1{\eqlabel{#1}\leqno(\csname @eq@#1\endcsname)}
\def\eqref#1{(\eqsref{#1})}
%----------- Lemmi automatici: a disposizione dell'utente ----------------
\def\lemmalabel#1{\global\advance\lemm@num by 1
\xdef\il@lemma{\la@sezionecorrente.\the\lemm@num}
\def\usa@getta{1}\se@indefinito{@lm@#1}\def\usa@getta{2}\fi
\expandafter\ifx\csname @lm@#1\endcsname\il@lemma\def\usa@getta{2}\fi
\ifodd\usa@getta\immediate\write16
{ ??? possibili riferimenti errati a \string\lemmaref{#1} !!!}\fi
\expandafter\xdef\csname @lm@#1\endcsname{\il@lemma}
\def\usa@getta{\expandafter\spo@glia\meaning
\la@sezionecorrente.\the\lemm@num}
\if@simboli
\immediate\write\file@simboli{ Lemma
\usa@getta : lemmaref #1}\fi
\if@riferimenti
\immediate\write\file@ausiliario{\string\expandafter\string\edef
\string\csname\b@lank @lm@#1\string\endcsname{\usa@getta}}\fi}
\def\autolemma#1{\lemmalabel{#1}\csname @lm@#1\endcsname\if@bozze
{\tt #1}\else\relax\fi}
\def\lemmaref#1{\se@indefinito{@lm@#1}
\immediate\write16{ ??? \string\lemmaref{#1} non definita !!!}
\if@riferimenti\else
\lemmalabel{#1}???\fi
\fi\csname @lm@#1\endcsname}
%--------------- bibliografia automatica: riservati ----------------------
\newcount\cit@num\global\cit@num=0
\newwrite\file@bibliografia
\newif\if@bibliografia
\@bibliografiafalse
\newif\if@corsivo
\@corsivofalse
\def\title#1{{\it #1}}
\def\rivista#1{#1}
\def\lp@cite{[}
\def\rp@cite{]}
\def\trap@cite#1{\lp@cite #1\rp@cite}
\def\lp@bibl{[}
\def\rp@bibl{]}
\def\trap@bibl#1{\lp@bibl #1\rp@bibl}
\def\refe@renza#1{\if@bibliografia\immediate % scrive su .BIB
\write\file@bibliografia{
\string\item{\trap@bibl{\cref{#1}}}\string
\bibl@ref{#1}\string\bibl@skip}\fi}
\def\ref@ridefinita#1{\if@bibliografia\immediate\write\file@bibliografia{
\string\item{?? \trap@bibl{\cref{#1}}} ??? tentativo di ridefinire la
citazione #1 !!! \string\bibl@skip}\fi}
\def\bibl@ref#1{\se@indefinito{@ref@#1}\immediate
\write16{ ??? biblitem #1 indefinito !!!}\expandafter\xdef
\csname @ref@#1\endcsname{ ??}\fi\csname @ref@#1\endcsname}
\def\c@label#1{\global\advance\cit@num by 1\xdef % assegna il numero
\la@citazione{\the\cit@num}\expandafter
\xdef\csname @c@#1\endcsname{\la@citazione}}
\def\bibl@skip{\vskip 5truept}
%------------------------ bibl. automatica: a disp. dell'utente ------------
\def\stileincite#1#2{\global\def\lp@cite{#1}\global
\def\rp@cite{#2}}
\def\stileinbibl#1#2{\global\def\lp@bibl{#1}\global
\def\rp@bibl{#2}}
\def\corsivo{\global\@corsivotrue}
\def\citpreset#1{\global\cit@num=#1
\immediate\write16{ !!! cit-preset = #1 } }
\def\autobibliografia{\global\@bibliografiatrue\immediate
\write16{ !!! Genera il file \jobname.BIB}\immediate
\openout\file@bibliografia=\jobname.bib}
\def\cref#1{\se@indefinito % se indefinito definisce
{@c@#1}\c@label{#1}\refe@renza{#1}\fi\csname @c@#1\endcsname}
\def\upcref#1{\null$^{\,\cref{#1}}$}
\def\cite#1{\trap@cite{\cref{#1}}} % [5]
\def\ccite#1#2{\trap@cite{\cref{#1},\cref{#2}}} % [5,6]
\def\ncite#1#2{\trap@cite{\cref{#1}--\cref{#2}}} % [5-8] senza definire
\def\upcite#1{$^{\,\trap@cite{\cref{#1}}}$} % ^[5]
\def\upccite#1#2{$^{\,\trap@cite{\cref{#1},\cref{#2}}}$} % ^[5,6]
\def\upncite#1#2{$^{\,\trap@cite{\cref{#1}-\cref{#2}}}$} % ^[5-8] senza def.
\def\clabel#1{\se@indefinito{@c@#1}\c@label % sola definizione
{#1}\refe@renza{#1}\else\c@label{#1}\ref@ridefinita{#1}\fi}
\def\cclabel#1#2{\clabel{#1}\clabel{#2}} % def. doppia
\def\ccclabel#1#2#3{\clabel{#1}\clabel{#2}\clabel{#3}} % def. tripla
\def\biblskip#1{\def\bibl@skip{\vskip #1}} % spaziatura nella bibl.
\def\insertbibliografia{\if@bibliografia % scrive la bibliografia
\immediate\write\file@bibliografia{ }
\immediate\closeout\file@bibliografia
\if@indice
\if@ceindice\relax\else\immediate\write
\file@indice{\string\vskip5pt\string{\string\bf\string~
Bibliografia\string}\string\hfill\folio\string\par}\fi\fi
\catcode`@=11\input\jobname.bib\catcode`@=12\fi
}
%--------- per comporre il file con la bibliografia --------------
\def\commento#1{\relax}
\def\biblitem#1#2\par{\expandafter\xdef\csname @ref@#1\endcsname{#2}}
% ricordare: una lista in chiaro della bibliografia si
% ottiene eseguendo $ TEX BIBLIST
%---------------- titolo in cima alla pagina, data.-----------------
\def\data{\number\day.\number\month.\number\year}
\def\datasotto{\@datatrue
\footline={\hfil{\rm \data}\hfil}}
\def\titoli#1{\if@data\relax\else\footline={\hfil}\fi
\xdef\prima@riga{#1}\voffset+20pt
\headline={\ifnum\pageno=1
{\hfil}\else\hfil{\sl \prima@riga}\hfil\folio\fi}}
\def\duetitoli#1#2{\if@data\relax\else\footline={\hfil}\fi
\voffset=+20pt
\headline={\ifnum\pageno=1
{\hfil}\else{\ifodd\pageno\hfil{\sl #2}\hfil\folio
\else\folio\hfil{\sl #1}\hfil\fi} \fi} }
\def\la@sezionecorrente{0}
% ------------------COSTANTI ---------------------------------
\def\const@label#1{\global\advance\const@num by 1\xdef
\la@costante{\the\const@num}\expandafter
\xdef\csname @const@#1\endcsname{\la@costante}}
\def\cconlabel#1{\se@indefinito{@const@#1}
\const@label{#1}\fi}
\def\constnum#1{\se@indefinito{@const@#1}
\const@label{#1}\fi\csname @const@#1\endcsname}
\def\ccon#1{C_{\constnum{#1}}}
\catcode`@=12
%------------------ FORMATI TEOREMI E GENERALI --------------------
\def\abstract{
\vskip48pt plus 60pt
\noindent
{\bf Abstract.}\quad}
\def\summary{
\centerline{{\bf Summary.}}\par}
\def\firma{\noindent
\centerline{Dario BAMBUSI}\par\noindent
\centerline{Dipartimento di Matematica dell'Universit\`a,}\par\noindent
\centerline{Via Saldini 50, 20133 Milano, Italy.}\par}
\def\theorem#1#2{\par\vskip4pt
\noindent {\bf Theorem \autolemma{#1}.}{\sl \ #2}
\par\vskip10pt}
\def\semitheorem#1{\par\vskip4pt
\noindent {\bf Theorem.}{\sl \ #1}
\par\vskip10pt}
\def\lemma#1#2{\par\vskip4pt
\noindent {\bf Lemma \autolemma{#1}.}{\sl \ #2}
\par\vskip4pt}
\def\proof{\par\noindent{\bf Proof.}\ }
\def\proposition#1#2{\par\vskip4pt
\noindent {\bf Proposition \autolemma{#1}.}{\sl \ #2}
\par\vskip10pt}
\def\corollary#1#2{\par\vskip4pt
\noindent {\bf Corollary \autolemma{#1}.}{\sl \ #2}
\par\vskip10pt}
\def\remark#1#2{\par\vskip4pt
\noindent {\bf Remark \autolemma{#1}.}{\sl \ #2}
\par\vskip4pt}
\def\definition#1{\par\vskip2pt
\noindent {\bf Definition.}{\sl \ #1}
\par\vskip2pt}
%------------------- ROUTINE DI USO GENERALE -----------------------
\def\norma#1{\left\Vert#1\right\Vert}
\def\perogni{\forall\hskip1pt}
\def\meno{\hskip1pt\backslash}
\def\frac#1#2{{#1\over #2}}
\def\fraz#1#2{{#1\over #2}}
\def\interno{\vbox{\hbox{\vbox to .3 truecm{\vfill\hbox to .2 truecm
{\hfill\hfill}\vfill}\vrule}\hrule}\hskip 2pt}
\def\quadratino{
\hfill\vbox{\hrule\hbox{\vrule\vbox to 7 pt {\vfill\hbox to
7 pt {\hfill\hfill}\vfill}\vrule}\hrule}\par}
\font\strana=cmti10
\def\lie{\hbox{\strana \char'44}}
\def\ponesotto#1\su#2{\mathrel{\mathop{\kern0pt #1}\limits_{#2}}}
\def\Sup{\mathop{{\rm Sup}}}
%\def\Sup#1{\hskip2pt\ponesotto{{\rm Sup}}\su{#1}}
\def\tdot#1{\hskip2pt\ddot{\null}\hskip2.5pt \dot{\null}\kern -5pt {#1}}
\def\diff#1#2{\frac{\partial #1}{\partial #2}}
\def\base#1#2{\frac{\partial}{\partial#1^{#2}}}
\def\charslash#1{\setbox2=\hbox{$#1$}
\dimen2=\wd2
\setbox1=\hbox{/}\dimen1=\wd1
\ifdim\dimen2>\dimen1
\rlap{\hbox to \dimen2{\hfil /\hfil}}
#1
\else
\rlap{\hbox to \dimen1{\hfil$#1$\hfil}}
/
\hfil\fi}
\def\Re{{\rm \kern 0.4ex I \kern -0.4 ex R}}
\def\Sh{{\rm Sh}\hskip1pt}
\def\Ch{{\rm Ch}\hskip1pt}
\def\poisson#1#2{\left\{#1 ,#2\right\} }
\def\toro{{\bf T}}
\def\Na{{\bf N}}
\def\Ra{{\bf Z}}
\def\id{{\bf 1}}
\def\Cm{{\bf C}}
\def\reale{{\rm Re}\hskip2pt}
\def\imma{{\rm Im}\hskip2pt}
\def\rin{{\bf Z}}
\def\pmb#1{\setbox0=\hbox{#1}\ignorespaces
\hbox{\kern-.02em\copy0\kern-\wd0\ignorespaces
\kern.05em\copy0\kern-\wd0\ignorespaces
\kern-.02em\raise.02em\box0 }}
\def\vett#1{\pmb{$#1$}}
\def\A{{\cal A}}
\def\B{{\cal B}}
\def\C{{\cal C}}
\def\D{{\cal D}}
\def\E{{\cal E}}
\def\F{{\cal F}}
\def\G{{\cal G}}
\def\H{{\cal H}}
\def\I{{\cal I}}
\def\L{{\cal L}}
\def\M{{\cal M}}
\def\N{{\cal N}}
\def\O{{\cal O}}
\def\P{{\cal P}}
\def\Q{{\cal Q}}
\def\R{{\cal R}}
\def\S{{\cal S}}
\def\T{{\cal T}}
\def\U{{\cal U}}
\def\V{{\cal V}}
\def\W{{\cal W}}
\def\Z{{\cal Z}}
\def\sym{\nabla^\Omega}
\def\uno{{\kern+.3em {\rm 1} \kern -.22em {\rm l}}}
\def\unpo{\vskip3pt}
\def\unp{\vskip6pt}
\def\a{\`a\ }
\def\o{\`o\ }
\def\e{\`e\ }
%------------------------------ F I N E ---------------------------------
\def\cmp{Commun. Math. Phys.}
\def\cmf{Commun. Math. Phys.}
\def\gio{A. Giorgilli}
\def\gal{L. Galgani}
\def\bgg{G. Benettin, \gal, \gio}
\def\zamp{J. Appl. Math. Phys. (ZAMP)}
\def\catene{bam93a}
\def\giro{bam91b}
\def\breather{bam95?}
\def\adiabat{bam95b}
\def\dipolouno{bam94??}
\def\lia{bam91a}
\def\dipolo{bam94?}
\biblitem{Lor09}H.A. Lorentz: The theory of Electrons, Dover, (New York
1952); first edition 1909.
\biblitem{dir38}P.A.M. Dirac: \title{Classical Theory of Radiating
Electrons. }\rivista{Proc. Roy. Soc.} {\bf A167}, 148-168 (1938).
\biblitem{Mor}G.~Morpurgo: Introduzione alla fisica delle particelle.
Zanichelli (Bologna, 1987)
\biblitem{boh48}D. Bohm, M. Weinstein: \title{The Self Oscillations of a
Charged Particle.} \rivista{Phys. Rev.} {\bf 74}, 1789-1798 (1948).
\biblitem{gra82}W.T. Grandy, Ali Aghazadeh: \title{Radiative Corrections
for Extended Charged Particles in Classical Elecrodynamics.}
\rivista{Ann. Phys.} {\bf 142}, 284-298 (1982).
\biblitem{Yag92}A.D.~Yaghjian: Relativistic Dynamics of a Charged
Sphere. Lect.~Notes.~Phys. m.11, Springer Verlag (Berlin Heidelberg, 1992).
\biblitem{bam94?}D.~Bambusi, D.~Noja:
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print.
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electrodynamics of point particles in the dipole approximation and proof
of the Abraham--Lorentz--Dirac equation}. Quaderno n$^0$24/94
Dipartimento di Matematica ``F.Enriques'', University of Milano (1994).
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G.R.G. {\bf 26}, 167--201 (1994)
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\rivista{Ann. der Phys.} {\bf 10}, 105-179 (1903).
\biblitem{kra48}H.A.~Kramers: \title{Nonrelativistic Quantum
Electrodynamics and the Correspondence Principle}, Rapports et
discussion du huitieme Conseil de Physique Solvay. Published by
R.~Stoops (Brussels 1950).
\biblitem{vankamp}N.G.~Van Kampen:\title{ Contribution to the quantum
theory of light scattering} Kgl. Danske Videnskab. Selskab, Mat.
Fys. Medd. {\bf 26} No. 15 (1951).
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lectures in physics, Mainly Electromagnetis and Matter, Vol. II.
Addison--Wesley (Redwood City, 1963)
\biblitem{gal89a}L. Galgani, C. Angaroni, L. Forti, A. Giorgilli,
F. Guerra:\title{
Classical Electrodynamics as a Nonlinear Dynamical System.}
\rivista{Phys. Lett. A,} {\bf 139}, 221-230, (1989).
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the Pauli-Fierz Model of Classical Electrodynamics.} \rivista{ Ann.
Inst. H. Poincar\'e, } Physique th\'eorique {\bf 58}, 155-171 (1993).
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Field Theories} \rivista{Phys. Rev.} {\bf 125}, 1422-1428 (1962).
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on Euclidean Spaces. Princeton University Press (Princeton, New Jersey
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\biblitem{gal94}A.~Carati, P.~Delzanno,~L.~Galgani, J.~Sassarini:
\title{Nonuniqueness properties of solutions of the Lorentz Dirac
equation}
\rivista{Nonlinearity}. {\bf 8} 65--79 (1995).
\def\dati{{\tt mettimi}}
%------------- definizioni --------------------------
\def\trivial#1{\relax}
\def\mem{m_{{\rm em}}}
\def\rot{{\rm rot}\hskip1pt}
\def\mtot{m_{{\rm tot}}}
\def\ald{Abraham--Lorentz--Dirac}
\def\ml{Maxwell--Lorentz\ }
\def\bold#1{{\bf #1}}
\def\intre{\int_{\Re^3}}
\def\sob#1{H_*^{\{#1\}}}
\def\aomega{\bold A^{l}_{\omega}}
\def\arun{\bold A^l_r}
\def\Ao{\bold{A}^{o,l}_{\omega}}
\def\Ae{\bold{A}^{e,l}_{\omega}}
\def\ao#1#2{\bold{A}^{o,#1}_{#2}}
\def\ae#1#2{\bold{A}^{e,#1}_{#2}}
\def\rhoadiomega{\left|\hat\rho_a\left(\frac{\omega}{c} \right)\right|^2}
\def\rhodiomega{\left|\hat\rho\left(\frac{\omega}{c} \right)\right|^2}
\def\rhidiomega{\left|\hat\D\left(\frac{a\omega}{c} \right)\right|^2}
\def\rhodik{\left|\hat\rho(k)\right|^2}
\def\conf{\Q}
\def\scal#1#2{\left\langle #1;#2\right\rangle_{\conf}}
\def\vk{{\bf k}}
\def\ve{\bold e_j(\vk)}
\def\sldue#1#2{\left\langle #1;#2\right\rangle_{L^2}}
\def\mdi#1{\mtot (#1){#1}^2-\alpha}
\def\massa{(\mdi{\omega})^2+\gamma\omega^6}
\def\ottoterzi#1{\frac83\pi\frac{e}{c^4}{#1}^2\alpha\hat
\rho\left(\frac{#1}{c} \right) }
\def\intzero{\int_0^\infty}
\def\interre{\int_{-\infty}^{\infty}}
\def\piu#1#2{\frac{#1}{#1+#2}}
\def\menom#1#2{\frac{#1}{#1-#2}}
\def\effe{f(\nu)\left|\hat\rho(k)\right|^2}
\def\ba{{\bf A}}
\def\bw{\bold w}
\def\bv#1{\bold u_{#1}}
\def\gr#1{\bold #1}
\def\lea{\A}
%----------- fine definizioni --------------------------
\duetitoli{D.~Bambusi}{Rigorous deduction of Lorentz--Dirac equation}
\numerazionedoppia
\riferimentifuturi
\autobibliografia
\def\lia{bam91a}
\def\dip{\cite{\dipolo}}
\def\vzero{\dot \bold q_0}
\def\qzero{\bold q_0}
\def\azero{\ddot \bold q_0}
\def\ald{Lorentz--Dirac}
{\bf
\centerline{A PROOF OF THE LORENTZ--DIRAC EQUATION}
\centerline{FOR CHARGED POINT PARTICLES}
}
\vskip48pt
\centerline{Dario BAMBUSI}
\centerline{Dipartimento di Matematica dell'Universit\`a,}
\centerline{ Via Saldini 50, 20133 MILANO, Italy.}
\centerline{email: bambusi@mat.unimi.it}
\par
\abstract {\it A rigorous deduction of the nonrelativistic
Lorentz--Dirac equation from the Maxwell--Lorentz system in the dipole
approximation is given. Previously the validity of the equation had been
checked, using the explicit solution of the complete system in terms of
normal modes, only for very special initial data for the complete
system; here instead the proof, valid for all initial data, is given by
making rigorous the standard heuristic deduction, and this is
obtained by overcoming a
serious problem there involved. Finally a formula is given, which
enables one to calculate, from the initial field, the initial
acceleration to be used as an initial datum for the Lorentz--Dirac
equation.}
\autosez{1}Introduction
The dynamics of a classical point--like charged particle (with
renormalized mass $m$
charge $e$ and position vector $\bold q$) is usually studied, in the
non--relativistic approximation, taking as a mathematical model the so
called Lorentz--Dirac (LD) equation
$$
m\tau\tdot {\bold q}=m\ddot {\bold q}-\bold F(\bold q,t)\
,
$$
where $\bold F(\bold q,t)$ is a mechanical force and $\tau=2e^2/3mc^3$
is a parameter having the dimension of a time. However, the fundamental
model that should be studied is the Maxwell--Lorentz system (ML), namely
Newton's law for the particle with Lorentz force due to the
electromagnetic field, and Maxwell equations for the field, with a
current due to the particle's motion. Now, the LD equation for the
particle was deduced from the Maxwell--Lorentz system, in the non
relativistic approximation\upcite{Lor09}, already at the beginning of the
century, but such a deduction can only be considered as a heuristic one
which leaves open some delicate and serious problems. In fact, even
Dirac, when he introduced his relativistic version of the
equation\upcite{dir38}, could not deduce it in a clear way, and was
obliged to state explicitly that he was actually postulating it. So, it
might be that the LD equation is not a rigorous consequence of classical
electrodynamics, and indeed there is a quite diffuse skepticism about
its validity. In the present paper we prove (for a particular class of
mechanical forces) equivalence of the linearized Maxwell--Lorentz system
(the so called dipole approximation) with the \ald\ equation.
The proof is obtained by overcoming a serious difficulty contained in
the classical heuristic deductions of the LD
equation\upccite{Mor}{boh48} (see also \cite{gra82}). Such a difficulty
was already pointed out, at a heuristic level, by
Yaghjian\upcite{Yag92}, who concluded that one should either avoid to
study point particles or modify the model. On the contrary, in \dip\
(see also \cite{bam94??}) it was rigorously proved that the particle's
motion corresponding to some particular initial data for the field
(essentially, vanishing ``free field'') satisfies the LD equation. This
was obtained by direct check using the explicit solution of the complete
system in terms of normal modes. Here instead, we go back to the
classical heuristic deduction, solve the difficulty pointed out by
Yaghjian and obtain a simple and direct proof of the LD equation, which
turns out to be valid for essentially all (smooth) initial data for the
field (see theorem~\lemmaref{dip} below for a more precise statement).
Moreover, we deduce a formula enabling one to calculate the initial
acceleration to be inserted in the LD equation in order to obtain a
particle's motion in agreement with the ML system in the dipole
approximation. A similar formula was already contained in a paper by
Kijowski\upcite{kij94a}, although with a completely different
interpretation (see sect.~\sref{4} below).
In what follows we shall consider only the Maxwell--Lorentz system {\it
in the dipole approximation }(see eq. \eqref{ml} below), and for brevity
the qualification ``in the dipole approximation'' will be understood.
To come to a precise statement we first recall that the dynamics of a
point particle in the ML system is not well defined (indeed one has to
calculate the electric field at the particle's position, where it is
typically infinite); the usual way to overcome this problem
(see e.g. [\cref{Lor09},
\cref{abr03}, \cref{kra48}, \cref{vankamp}, \cref{FeyII}]) consists in
regularizing the model by taking a rigid extended particle (or
equivalently a cutoff on the Fourier modes) and to study the limit of
the interesting quantities as the particle's ``radius'' $R$ tends to
zero.
In order to study the behaviour of the particle, following \cite{boh48}
and many other authors, we first deduce a closed equation for its motion
in the regularized system, and
then study its limit as $R\to0$. It is well known, heuristically, that
such an equation converges to the LD equation {\it provided one neglects
some terms} which are very small after a very short time. First we show
that these terms, although very concentrated in time, produce very large
effects. Indeed, we will show that the solution of the equation in
which the ``transient terms'' are taken into account does not have a
limit as $R\to0$. This is a formalization of the fact, understood by
Yaghjian, that such transient terms are very important. Then, following
\cite{\dipolo} we overcome the above difficulty by imposing a certain
relation
between the initial particle's velocity and the initial datum for the
field. Precisely, motivated by the fact that a particle in uniform
motion is followed by a
field$^{[\cref{abr03},\cref{gal89a},\cref{\lia}]}$, which in the point
limit is singular at the particle's position, we consider the class of
initial fields differing from such an ``adapted field'' only by regular
terms. It turns out that the contribution of the adapted field to the
equation of motion exactly cancels the dangerous ``transient terms''.
In such a way the essential difficulty is eliminated and the problem of
taking the point limit is just reduced to a purely technical one, which
can be dealt with by standard methods; we will in fact use Laplace
transform. We thus obtain that the limit motion exists and satisfies the
LD equation
$$
m\ddot {\bold q}=m\tau\tdot{\bold q}+\bold F_L(t,\bold A_0,\dot{\bold
A}_0)+\bold F(\bold q,t)\ ,\autoeqno{ald.2}
$$
where $\bold F_L(t,\bold A_0,\dot{\bold A}_0)$ is a function of time
depending parametrically on the initial data for the field (it
is in fact
the linearization of the Lorentz force due to the free evolution of
the initial fields), while $\bold F(\bold q,t)$ is a mechanical force that
is here assumed to be the sum of a linear part and a part depending
only on time, namely $\bold F(\bold q,t)=-\alpha\bold q+\bold f(t)$,
with a constant $\alpha$, and a given $\bold f$.
\unpo
Our study also leads to solve the ``problem of the initial
acceleration''. Let us recall it briefly. The solution of the Cauchy
problem for eq.~\eqref{ald.2} requires knowledge of the initial
particle's acceleration in addition to the mechanical state, namely
position and velocity. Thus, since in the regularized Maxwell--Lorentz
system the motion is completely determined by the initial mechanical
state and the initial fields, one should determine the initial
particle's acceleration from the initial data for the complete system,
and use such a ``correct acceleration'' as an initial datum in the LD
equation. We will prove that the correct initial particle's acceleration
is given by
$$
\ddot {\bold q}_0=\frac32\frac ce \lim_{r\to0}r\frac1{4\pi r^2}
\int_{S_r}\dot {\bold A}_0(\bold x)d\sigma\ ,\autoeqno{4.11}
$$
where $S_r$ is the sphere of radius $r$ centered at the origin (i.e. the
point where the interaction of the particle with the field is localized
in the dipole approximation) and $d\sigma$ its surface element. As a
corollary, we obtain that each solution of the Cauchy problem for
\eqref{ald.2} is the point limit of the particle's motion corresponding
to a definite solution of the ML system in the dipole approximation.
\unp
As anticipated above the technical tool used here in studying the point
limit is the Laplace transform. The limit $R\to0$ is obtained by
exchanging the inverse Laplace transform and the point--limit. This
requires an estimate which is non trivial.
The paper is organized as follows. In sect.~\sref{2} we revisit the
classical deduction of the LD equation in order to point out the
difficulties related to it. In sect.~\sref{3} we solve such
difficulties, study the point--limit of the particle's motion and deduce
the condition on the initial acceleration. Sect.~\sref{4} contains a
brief discussion. The appendix contains the estimates needed in order to
exchange the point limit and the inverse Laplace transform.
\noindent
{\it Acknowledgement} This work is part of a project for the study of
classical electrodynamics as a dynamical system initiated by Luigi
Galgani, involving Andrea Carati, Diego Noja, and Andrea Posilicano (and
me). I thank all of them for many discussions and suggestions. This work
has been developed with the support of the grant EC contract
ERBCHRXCT940460 for the project ``Stability and universality in
classical mechanics".
\autosez{2}On the classical deduction of the LD equation
Consider the regularized ML system in the dipole approximation. In the
Coulomb gauge it has the form$^{[\cref{vankamp},\cref{col62}]}$
$$
\frac1{c^2}\ddot {\bold A}-\bigtriangleup \bold A=
\frac{4\pi e}c \bold j_{tr}\ ,\quad
m_0\ddot {\bold q} =-\intre \frac{e}c \rho_R(x)\dot {\bold A}
(x)d^3x+\bold F(\bold q,t)\ .
\autoeqno{ml}
$$
where $\bold F(\bold q,t)$ represents a mechanical force,
$m_0$ is the particle's bare mass, $\bold A$ the vector potential
(which is subjected to the condition $div\bold A\equiv 0$), $\bold
j_{tr}$ is the transversal part (i.e. the divergence free part)
of the current $\bold j(\bold x) :=
e\dot\bold q \rho_R(\bold x)$. Finally $\rho_R$ is the charge
distribution of the particle. We assume (as in \cite{\dipolo})
that $\rho_R$ is of the form
$$
\rho_{R}(x):=\frac1{R^3}\D\left(\frac xR\right)\ , \autoeqno{pa.1}
$$
where $\D$ is a fixed positive, normalized (in $L^1$), spherically
symmetric $C^\infty$ function tending to zero at infinity faster than
any power, $R$ is a positive parameter playing the role of particle's
radius, and $x=|\bold x|$. We will also need the technical assumption
that the Fourier transform of $\D$ is monotonically decreasing.
If $\bold F(\bold q,t)$ is regular, then for $R>0$ and in a suitable
function space, system \eqref{ml} is well posed (see \cite{\lia});
in particular one has
that $\bold q(.)$ is $C^2$.
We perform now the Fourier transform of the form factor (choosing the
normalization convention which makes it an $L^2$ isometry) and of the
fields
$$
\bold A(\bold x)=\sum_{j=1,2}\frac1{(2\pi)^{3/2}}\intre \hat
A_j(\vk)\ve e^{i\bold k\bold x}d^3x\ ,
$$
where, as usual $\ve$, $j=1,2$ are unit polarization vectors,
constituting with $\bold k$ an orthonormal basis
of $\Re^3$. Then, using the parameter variation formula,
we can write the solution of the first of equations \eqref{ml} in
terms of the initial data for the field and of the particle's motion,
namely
$$
\hat A_{j}(\bold k,t)=\hat A_{0,j}(\bold k)\cos (\omega_k t)+\frac{\dot
{\hat A}_{0,j}(\bold k)}{\omega_k}\sin(\omega_kt)+\int_0^t \frac{4\pi
ec}{\omega_k}\left[\dot {\bold q}(s)\cdot \ve\right] \hat\rho_R(k)\sin
[\omega_k(t-s)]ds\ ,\autoeqno{ar}
$$
where $\hat A_{0,j}$ and $\dot {\hat A}_{0,j}$ are the Fourier
transforms of the initial fields $\bold A_{0}$, $\bold {\dot A}_{0}$,
while $\hat \rho_R(k)=\hat \D(kR)$ is the Fourier transform of
$\rho_R$, and $\omega_k:=ck$. Substituting \eqref{ar} in the Newton
equation for the particle (i.e.~in the second of \eqref{ml}) and
performing the angular integrations one obtains a closed
integro-differential equation for the particle's motion, namely
$$
m_0\ddot \bold q(t)=
\int_0^t\dot{ \bold q}(s) \ddot M_R(t-s)ds +\bold F(\bold q,t)+
\bold F^R_L(t,\bold A_0,\dot {\bold
A}_0)\
,\autoeqno{3.21}
$$
where
$$
\eqalign{
M_R(t)&:=\frac{32}3 \pi^2
\frac{e^2}{c^2}
\int_0^\infty |\hat \rho_R(k)|^2\cos(ckt)dk
\cr
&
= \frac1R\frac{32}3 \pi^2
\frac{e^2}{c^2}
\int_0^\infty \left|\hat \D(s)\right|^2\cos\left(cs\frac tR \right)ds
\ ,
\cr
\bold F^R_L(t,A_0,\dot
A_0)
&:=\sum_j\intre e\hat \rho_R^*(k)k\hat
A_{0,j}(\bold k)\sin (ckt)\ve d^3k
\cr
&- \sum_j\intre e\hat\rho_R^*(k)\frac{1}{c} \dot
{\hat A}_{0,j}(\bold k)\cos (ckt)\ve d^3k
\ .} \autoeqno{l}
$$
Remark that $\bold F_L$ is the linearization of the Lorentz force due to
the free evolution of the initial data for the fields.
>From \eqref{3.21} it is
easy to prove more regularity of $\bold q(.)$. In particular if $\bold
F(\bold q,t)$ is $C^1$, and the initial data data for the field are
such that $\bold F_L^R$ is $C^1$ as a function of time, then one has
that $\bold q(.)$ is $C^3$. So it is possible to perform two integrations
by parts obtaining
$$
(m_0+M_R(0))\ddot \bold q(t)=\dot
\bold q_0 \dot M_R(t)+\ddot \bold q_0M_R(t)+
\int_0^t\tdot{ \bold q}(s) M_R(t-s)ds +\bold F(\bold q,t)+
\bold F^R_L(t,A_0,\dot
A_0)\
.\autoeqno{3}
$$
At this step {\it the standard deduction of the LD equation neglects
the two terms $\vzero \dot
M_R$ and $\azero M_R$ as pertaining to a very short transient}
(one has
$M_R(t)=N(t/R)/R$, for a regular $N$ which goes to zero faster than any
power as its argument tends to infinity, see \eqref{l}).
Then, taking $m_0$ as a function of $R$ in such a way that
$m=m_0+M_R(0)$
is fixed, namely
$$
m_0(R):=m-M_R(0)\ , \autoeqno{m}
$$
where $m$ is identified with the physical mass, one is
reduced to the equation
$$
m\ddot {\bold q}=\int _0^\infty \tdot{\bold q}(s)M_{R}(t-s)ds+\bold
F(\bold q,t)+
\bold F^R_L(t,A_0,\dot
A_0)\ \
.\autoeqno{1.1}
$$
It is easy, at a heuristic level, to take the limit of \eqref{1.1} and
to show that it gives the LD equation.
We claim that, notwithstanding the fact that they are very concentrated
in time, the terms $\vzero \dot M_R$ and $\azero M_R$ produce very
important effects, so that they cannot be neglected. In particular we
are going to prove that, due to their existence, {\it the solution of
the Cauchy problem for \eqref{3} (and therefore for \eqref{ml}) does not
have a limit as $R\to0$}.
For simplicity we consider the case of vanishing initial field, and
vanishing mechanical force, and take the Laplace transform of the
original
eq.~\eqref{3}. Denoting by $\bar {\bold q}=\bar {\bold q}(p)$ the
Laplace transform of $\bold q$, and by $\bar M_R(p)$ the Laplace
transform of $M_R$, we obtain
$$
m(\bar {\bold q} p^2-{\bold q}_0p-\dot {\bold q}_0)=\bar M_R(p)
(\bar {\bold q}
p^3-{\bold q}_0p^2-\dot {\bold q}_0p-\ddot
{\bold q}_0)+\dot {\bold q}_0[p\bar M_R(p)-M_R(0)]+\bar M_R(p)
\ddot {\bold q}_0\
.\autoeqno{4.4}
$$
A first remark is that the last term at r.h.s., namely $\bar M_R\ddot
{\bold q}_0$, exactly cancels the other term proportional to $\ddot
{\bold q}_0$, namely the one coming from the Laplace transform of $\tdot
{\bold q}$. As a consequence, the solution of the Cauchy problem for
eq.~\eqref{3} requires knowledge of ${\bold q}_0$, and $\dot {\bold
q}_0$ only. So, the effect of the ``transient term'' $M\ddot {\bold
q}_0$ is the following: {\it the solution of the Cauchy problem for
equation \eqref{3} does not require knowing the initial particle's
acceleration $\ddot{\bold q}_0$}.
It is easy to see (at least heuristically) that $\bar M(p)\to m\tau$ as
$R\to0$ (see \eqref{p.3}), while $M_R(0)\to+\infty$, so that all terms
in \eqref{4.4} have a well defined limit as $R\to0$, except the {term
proportional to $M_R(0)$}, which {causes the limit of $\bar {\bold q}$
not to exist.} This term comes from the Laplace transform of $\vzero
\dot M(t)$; so, we conclude that {\it the term $\dot M\vzero$ causes the
point limit of the particle's motion not to exist}, at least when $\dot
{\bold q}_0\not=0$. Such a result was already proved, with a different
method, in \cite{\dipolo}. We conclude that a correct deduction of the
LD equation cannot be obtained by neglecting the ``transient terms'' of
equation \eqref{3}. We recall that Yaghjian in \cite{Yag92} made some
heuristic considerations in order to show that the transient terms we
considered here are important. This led him to conclude that equation
\eqref{3} should be modified in order to deal with small particles, and
he suggested a possible modification. Here, we take a different point of
view: we will retain the model and show that, provided initial data are
chosen in a good way, the point limit dynamics exists.
\autosez{3}Point-limit of the particle's motion and \ald\ equation
We recall that in the ref.~\cite{\dipolo} it was proved that the point
limit of the particle's motion in the ML exists also when the initial
particle's velocity is different from zero, {\it provided} one considers
initial data for the fields ``adapted'' to the initial particle's
velocity. Precisely, in \cite{\dipolo}, for each $\bold v\in\Re^3$ a
field $\bold X^R_{{\bold v}} (\bold x)=\bold X^R(\bold v,\bold x)$ was
considered, whose components are given in Fourier transform by
$$
\hat X^R_{\bold v,j}(\bold k)=\hat X^R_{j}(\bold v,\bold k)
:=\frac{4\pi e}{c}\hat \rho_R(k)\frac{\bold
v\cdot \ve}{k^2}\ ,\quad j=1,2\ ,\autoeqno{7}
$$
and then it was proved that the particle's motion corresponding to
initial data of the form $(\bold q_0,\dot \bold q_0, \bold
A_0, \dot \bold A_0)=(\bold q_0,\dot \bold q_0, \bold X^R_{\dot {\bold
q}_0}+\bold
A'_0, \dot \bold A_0)$, where $ (\bold A'_0, \dot \bold A_0)$ are
regular fields, has a limit as the radius $R$ tends to zero.
Motivated by the above considerations we study initial data of the form
$$
(\bold q_0,\dot \bold q_0, \bold
A_0, \dot \bold A_0)=(\bold q_0,\dot \bold q_0, \bold X^R_{\dot {\bold
q}_0}+\bold
A'_0,
\bold X^R_{\bold a}+\dot\bold A'_0)\ ,\autoeqno{8}
$$
where $\bold a\in\Re^3$ is a parameter, and
$(\bold A'_0, \dot\bold A'_0)$, are regular (say $C^\infty_c$, where $c$
stands for ``compactly supported'') fields.
By a simple calculation one has
$$
\bold F^R_L(t,\bold X^R_{\dot {\bold q}_0},\bold X^R_{\bold a})=-\dot
{\bold q}_0\dot M_R(t)- \bold
a M_R(t)\ ,\autoeqno{fx}
$$
so that, corresponding to the above kind of initial data, eq.~\eqref{3}
takes the form
$$
m\ddot \bold q=(\ddot \bold q_0-\bold a)M_R(t)+ \int_0^\infty\tdot {\bold
q}(t-s)M_R(s)ds+\bold F^R_L(t,\bold A'_0,\dot\bold A'_0
)+\bold F(\bold q,t)
\ .\autoeqno{8.1}
$$
In order to study rigorously the limit $R\to0$ of its solution we will
use the Laplace transform, for this reason we will consider only the
linear case in which the mechanical force reduces to
$$
\bold F(\bold q,t)=-\alpha\bold q+\bold f(t)\ ,\autoeqno{f}
$$
with a given $\bold f$.
\theorem{dip}{Let $\bold a\in \Re^3$ be fixed; consider the solution of
the Cauchy problem for system \eqref{ml} with force given by \eqref{f},
with $\bold f\in C^1([0,+\infty])\cap L^1([0,+\infty])$
and $\alpha\in\Re$. Take initial
data of the form
$$
(\bold q_0,\dot \bold q_0, \bold
A_0, \dot \bold A_0)=(\bold q_0,\dot \bold q_0, \bold X^R_{\dot {\bold
q}_0}+\bold
A'_0,
\bold X^R_{\bold a}+\dot\bold A'_0)\ ,\autoeqno{e8}
$$
with $ (\bold A'_0, \dot\bold
A'_0)\in C^\infty_c\times C^\infty_c$, and $\bold X^R$ defined by
\eqref{7}.
For each positive $R$ let $\bold
q^R(t)$ be the corresponding particle's motion. Then, as $R$ tends to
zero, $\bold q^R(.)$ converges in the sense of distributions to
the solution of the Cauchy problem
$$
\eqalign{
m\ddot\bold q&=m \tau\tdot{\bold q}+\bold F(\bold q,t)+\bold F^{R=0}_L
(t,\bold A'_0,
\dot\bold A'_0)
\cr
\qzero&=\qzero\ ,\quad\vzero=\vzero\ ,\quad
\azero=\bold a\ ,
}\autoeqno{1.2}
$$
with $\bold F^{R=0}_L(t,\bold A'_0,
\dot\bold A'_0)$ defined by \eqref{l}.}
The idea of the proof is to solve \eqref{8.1} by Laplace transform, take
the limit of the Laplace transform of the
solution as $R\to0$ and compare it with the Laplace
transform of the solution of \eqref{1.2}. This can be made rigorous
through some technical estimates which are deferred to the
appendix.
The above theorem has a double content. Indeed, on the one hand it
ensures that the point limit of the particle's motion satisfies the LD
equation with a Lorentz force due to the free evolution of a part of the
initial fields, which might be called ``free fields''. On the other hand
it gives a relation between the initial acceleration and the ``non
free'' part of the initial field. We will show below that it is also
possible to deduce the initial particle's acceleration from the initial
field.
\remark{co}{In ref.~\cite{\dipolo} it was proved, with a different
method, that, at least when $\bold F(\bold q,t)=-\alpha\bold q$ with
$\alpha>0$, the convergence of $\bold q^R$ is actually in
$H^2([-T,T],\Re^3)$, so that the particle's position and velocity
converge uniformly as the radius tends to zero.}
\remark{2}{Theorem \lemmaref{dip} also holds if the fields $(\bold A'_0,
\dot\bold A'_0)$, are much more irregular than assumed in the statement.
For example it is enough to assume that such fields give rise to a
Lorentz force $\bold F_L^{0}(.,\bold A'_0, \dot\bold A'_0)$ which is
$C^1([0,+\infty])\cap L^1([0,+\infty])$.
}
>From theorem \lemmaref{dip} we also have the following important
\corollary{c}{Let $\bold q(t)$ be a solution of the Cauchy problem
\eqref{1.2} for the LD equation; then $\bold q(.)$ is the limit of the
particle's motion corresponding to the solution of the ML system
\eqref{ml} with initial data ($\bold q_0$,$\dot{\bold q}_0$,$\bold
X^R_{\dot {\bold q}_0}+ \bold A'_0$,$\bold X^R_{\ddot {\bold
q}_0}+\dot\bold A'_0$).}
Concerning the initial acceleration to
be inserted in the LD equation we have the following
\proposition{p.i}{For a field of the form $\bold A=\bold X^0_{\bold
a}+\bold A'$, with $\bold a\in\Re^3$ and
$\bold A'\in C^\infty_c$, one has
$$
\bold a=\frac32\frac ce \lim_{r\to0}r\frac1{4\pi r^2}\int_{S_r}
\bold A(\bold x)d\sigma\ ,\autoeqno{d.11}
$$
where, as above $S_r$ is the sphere of radius $r$ centered at the
origin, and $d\sigma$ its
surface element.}
\proof It is just a calculation. First notice that $\bold A'$ does not
contribute to the limit in \eqref{d.11}. So denote by $\bold Z(r)$
the average of $\bold X_{\bold a}^0$
over a sphere of radius $r$. We calculate
it. Recall that the Fourier transform commutes with the
spherical average\upcite{SteWei},
so that by calculating the angular
integration in Fourier transform (see \eqref{7}) one has
$$
\hat {\bold Z}(k)=4\pi \frac ec \frac23\frac{\bold
a}{k^2}\frac1{(2\pi)^{3/2}}\ ,
$$
which gives $\bold Z(r)=2e\bold a/(3cr)$. From this \eqref{d.11}
immediately follows. \quadratino
\remark{ia}{Equation \eqref{d.11} enables one to explicitly calculate the
initial acceleration in terms of the initial field; the corresponding
formula is given
$$
\ddot {\bold q}_0=\frac32\frac ce \lim_{r\to0}r\frac1{4\pi r^2}
\int_{S_r}\dot {\bold A}_0(\bold x)d\sigma\ .\eqno{\eqref{4.11}}
$$
}
So, the initial acceleration is proportional to the
coefficient of $r^{-1}$ in the power expansion of the spherical average
of the electric field.
\remark{r+1}{
The initial acceleration to be inserted in the LD equation
is not free: in order to obtain a particle's motion which is coherent
with the Maxwell--Lorentz system one {\it must} use the initial
acceleration
given by \eqref{4.11}.}
\remark{r+2}{For the class of initial fields allowed by theorem
\lemmaref{dip} one has, in the point limit, a formula similar to
\eqref{4.11} relating the initial particle's velocity and the initial
value of the vector potential $\bold A_0$, namely
$$
\dot {\bold q}_0=\frac32\frac ce \lim_{r\to0}r\frac1{4\pi r^2}
\int_{S_r}{\bold A}_0(\bold x)d\sigma\ .\autoeqno{4.12}
$$}
Concerning the behaviour of the singularity of the fields for $t>0$ we
have
\remark{r+3}{The solutions of the limit field equation
$$
\frac1{c^2}
\ddot {\bold A}-\bigtriangleup\bold A=\frac{4\pi}{c}\left( \bold {\dot
q}\delta\right)_{{\rm tr}}\ \autoeqno{4.10}
$$
(where the index ``tr'' denotes, as above, the divergence free part)
satisfy
$$
\ddot {\bold q}(t)=\frac32\frac ce \lim_{r\to0}r\frac1{4\pi r^2}
\int_{S_r}\dot {\bold A}(\bold x,t)d\sigma\ ,
\quad \dot {\bold q}(t)=\frac32\frac ce \lim_{r\to0}r\frac1{4\pi r^2}
\int_{S_r}{\bold A}(\bold x,t)d\sigma\ \autoeqno{4.13}
$$
(to prove \eqref{4.13} use retarded potentials).}
\remark{r+4}{We did not prove that, as $R\to0$, the field converges in a
suitable sense to a solution of \eqref{4.10}. In the case $\alpha>0$,
$\bold f(t)\equiv 0$, this could be obtained easily by exploiting the
strong convergence result for the particle's motion given in
\cite{\dipolo} (see remark \lemmaref{co}).}
A general study of the behaviour of the field in the point limit is
beyond the aim of this note and is deferred to future work.
\autosez{4}Discussion
We have thus proved that the \ald\ equation is essentially equivalent to
the Maxwell--Lorentz system in the dipole approximation. In this
connection we remark that, since (due to mass renormalization), the
original Newton's equation for the particle, which involves the bare
mass, is not defined in the point--limit, one can say that the
Lorentz--Dirac equation is the point--like version of Newton
equation with Lorentz force when the self interaction is taken into
account. However, such an interpretation is possible only if a
prescription is given for the initial particle's acceleration, and this
is afforded by \eqref{4.11},
which allows the practical use of the LD equation. We also remark
that the generic runaway behaviour, which is well known to be present in
the \ald\ equation, turns out as a consequence
to be present (and generic) also in the
Maxwell--Lorentz system in dipole approximation. Moreover, some
interesting phenomena recently found in the LD equation\upcite{gal94}
turn out to pertain also to the Maxwell--Lorentz system (in the dipole
approximation).
\unpo
As recalled in the introduction, in \cite{\dipolo} it was already
proved, in the case $\bold F(\bold q,t)=-\alpha\bold q$ with $\alpha>0$,
that the limit particle's motion corresponding to adapted initial data
(namely of the form $(\bold q_0,\dot \bold q_0, \bold A_0, \dot \bold
A_0)=(\bold q_0,\dot \bold q_0, \bold X^0_{\dot {\bold q}_0},0)$)
satisfy the LD equation. So the present paper (which uses a different
and simpler technique) gives an extension of such a result. Moreover, we
give here a recipe for the calculation of the correct initial
acceleration.
On the other hand we think that the analysis of ref.~\cite{\dipolo}
(which is very close to that of van Kampen\upcite{vankamp}) gives a
geometrical insight in the dynamics, which could be very useful for
further developments of the theory of point particles, e.g. for the
study of non--runaway motions, and in investigating the possible
existence of a renormalized Hamiltonian. Moreover, the technique of
paper \cite{\dipolo} gives a much stronger convergence for the
particle's motion (see remark \lemmaref{co} above).
\unpo
Concerning formula \eqref{4.11} for the initial acceleration, we already
recalled that it is essentially contained in a paper by Kijowski. Indeed
Kijowski realized that the field calculated via the retarded potentials
has the asymptotic behaviour \eqref{4.13} on the particle's position,
and he interpreted eqs.~\eqref{4.13} as boundary conditions that have to
be imposed on the fields, and subsequently he upgraded the first of
\eqref{4.13} to the status of dynamical equation for the particle,
introducing a new boundary condition at the particle's position. Our
result is that the first of \eqref{4.13} has to be explicitly imposed at
$t=0$, if the point--limit dynamics has to exist. On the contrary, the
astonishing fact happens that the particle's acceleration immediately
adapts itself to the initial electric field, so that the second of
\eqref{4.13} is automatically satisfied at $t=0$. No boundary conditions
are needed for positive times (except those at infinity).
\autosez{app}Appendix: Proof of theorem \lemmaref{dip}.
For simplicity we will deal explicitly only with the case
$(\bold A'_0, \dot\bold A'_0)=0$ and $\bold F(\bold q,t)\equiv0$. The
proof in the general case requires only minor changes; in particular,
when $\alpha\not=0$ the present technique gives a convergence slightly
weaker than that ensured by lemma \lemmaref{l.2} below.
We split the proof into two lemmas.
\lemma{l.1}{There exists a positive $r_*$, and a positive $w_*$ such
that, for $0w_*$, the solution of \eqref{8.1} with
$\bold A'_0=\dot\bold A'_0=0$ and $\bold (\bold q,t)\equiv0$ is given by
$$
\bold q^R(t)=
\frac1{2\pi}
\int_{\Re}\bar{\bold q}^{R}(w+is)e^{(w+is)t}ds\ ,\autoeqno{p.9}
$$
where
$$
\bar {\bold q}^R(p)=\frac{{\bold q}_0}{p}+\frac{\dot {\bold q}_0}{p^2}+
\frac{\bold a\bar M_{R}(p)}{
(m-p\bar M_R(p))p^2}\ .\autoeqno{l.1}
$$
}
\proof By the standard theory of Laplace transform
the candidate to be the Laplace transform of the solution of \eqref{8.1}
is \eqref{l.1}. We will prove that \eqref{l.1} is analytic in the region
${\rm Re}(p)\geq w_*$, and moreover tends to zero like $|p|^{-1}$ as $|p|\to
\infty$, so that it is the Laplace transform of the function $\bold q^R$
given by \eqref{p.9}.
The only nontrivial term to be studied is the last one in \eqref{l.1}.
So, we consider only the case $\qzero=\vzero=0$. First notice that $\bar
M_R$ is bounded and analytic for ${\rm Re}(p)\geq\epsilon>0$. We give
now some useful formulas which follow directly form the definition of
$\bar M_R$, namely
$$
m-p\bar M_{R}(p)=
m-\frac{32}3\pi^2\frac{e^2}{c^2}
p^2\int_0^\infty\frac{\left|\hat\rho_R(k)\right|^2}
{p^2+(ck)^2}dk=m_0+\frac{32}3\pi^2e^2
\int_{0}^{\infty} \frac{k^2\left|\hat\rho_R
(k)\right|^2}{p^2+(ck)^2}dk
\ ,\autoeqno{p.3}
$$
$$
\bar
M_{R}(p)=\frac{16}{3i}\pi^2\frac{e^2}{c^2}
\int_{-\infty}^{\infty}\frac{\left|\hat\rho_R(k)
\right|^2}{ck-ip}dk\ ,
\autoeqno{h.1}
$$
where the function $\hat\rho_R(k)$ has been extended to the whole real
axis by symmetry.
Fix $R>0$. From the last of \eqref{p.3} one has that $m-p\bar
M_{R}(p)\to m_0(R)$ as $|p|\to \infty$, and therefore the last term of
\eqref{l.1} behaves as $|p|^{-2}$ when $|p|\to\infty$. We prove now that
the function \eqref{p.3} does not vanish for ${\rm Re}(p)\geq w_*$ with a
suitable $w_*$ independent of $R$. Denoting $p^2=\nu+i\omega$, one has
that the imaginary part of $m-p\bar M_R$ is proportional through
inessential coefficients to (use \eqref{p.3})
$$
\omega\int_0^\infty\frac{\left|\hat\rho_R(k)\right|^2
(ck)^2}{(\nu+(ck)^2)^2+
\omega^2}dk\ ,\autoeqno{p.2}
$$
which vanishes only for $\omega=0$. So $m-p\bar M_R$ can vanish only on
the real axis (i.e. when $\omega=0$).
We are led to look for
{\it real} solutions of the equation
$m-p\bar M_R(p)=0$.
Taking the derivative with respect to $p$ one checks that the l.h.s.~is
monotonically decreasing (for positive $p$), and therefore such an
equation has at most one positive solution. Moreover, the l.h.s.~is a
regular function of $R$ and $p$, and therefore, by the implicit function
theorem, the solution depends continuously on $R$. So, for small $R$ the
only solution of our equation is close to the solution of the limit
equation $ m-\tau mp=0 $. Therefore there exists $w_*$ such that
\eqref{l.1} is analytic in the region ${\rm Re}(p)\geq w_*$ (it is the sum of
functions which are the inverse of nonvanishing analytic functions).
\quadratino
\lemma{l.2}{As $R\to0$ the function defined by \eqref{p.9}, \eqref{l.1}
converges, in the dual to $H^{1}([-T,T])$, to
$$
\frac1{2\pi}\int_{{\rm Re}(p)=w\geq w_*}\left( \frac{\qzero}p+\frac{\vzero}
{p}+\frac{m\tau\bold a}{(m-\tau mp)p^2} \right)e^{pt} d{\rm Im}(p)
\ .\autoeqno{55}
$$
}
\proof Denote $p=w+is$. We will prove that $\bold
q^R(t)e^{-wt}$ (which is obviously extended to the whole $\Re$ by zero)
converges in the dual to $H^{1}(\Re)$ as $R\to0$ to the function
\eqref{55}. To this end we have to
prove that for any $f\in H^1$ the integral of $f$ multiplied by $\bold
q^R(t)e^{-wt}$ converges as $R\to0$. In practice we will prove that
$\bar{\bold q}^R(w+is)$ is uniformly bounded, so that Lebesgue theorem
gives the result.
Again we will consider only the case $\qzero=\vzero=0$. Notice that
$\bar M$ is globally bounded uniformly in $R$. Now, we are going to
estimate (from below) uniformly in $R$ the denominator of \eqref{l.1}.
We first consider the case of large $s$, and estimate $m-p\bar M_R$ by
the modulus of its imaginary part. Then we consider the case of finite
$s$.
So, consider the imaginary part of $m-p\bar M$,
which is given by
\eqref{p.2}.
Calculating the derivative of \eqref{p.2} with respect to $R$ it is easy
to see that \eqref{p.2} is a function of $R$ which is monotonically
decreasing for $\omega>0$. Therefore, for $R\leq R_*$ one has
$$
|{\rm Im}(p\bar M_{R}(p))|\geq |{\rm Im}(p\bar M_{R_*}(p))|\geq Cs^{-2}\
,\quad \perogni |s|\geq S\ ,
$$
with $S$ and $C$ {\it independent of $R$}.
We study now $|s|\leq S$. By the
standard theory of the Hilbert transform\upcite{SteWei}
(based on the idea of
taking the boundary value of functions analytic on a half plane) using
$|\hat \D|^2\in H^1(\Re)$ one has, as $R\to0$,
$$
\frac1{2i}\int_{-\infty}^{\infty}\frac{|\hat \D(b)|^2}{cb-iRp}db\to
\frac1{2i}{P}\int_{-\infty}^{\infty}\frac{|\hat
\D(b)|^2}{cb}db+\frac\pi{2c}
\left|\hat \D(0)\right|^2=\frac\pi{2c}
\left|\hat \D(0)\right|^2=
\frac1{16\pi^2c}\ ,
$$
where ${P}\int_{-\infty}^{\infty}$ denotes the principal value
(recall that $p$ is fixed and
${\rm Re}(p)>0$).
It
follows that, for $|s|