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FM 01-13
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Lagrange, Vibrating String, 
Lagrange Inversion, Implicit Functions, Kepler Equation, Quantum Field 
Theory, Combinatorial KAM
---------------1305141520958
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%italian=48
\documentclass[11pt]{article}
%\documentclass[pra,preprint]{revtex4}
\newcount\biblio
\title{Aspects of Lagrange's Mechanics and their Legacy}
\author{\small\textsc{Giovanni Gallavotti}
\\
\small Accademia dei Lincei \& INFN-Roma1,
Italia}%

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%

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\begin{document}
\maketitle

\begin{abstract}\0From the ``vibrating string'' and ``Kepler's equation''
 theories to relativistic quantum fields, perturbation theory, (divergent)
 series resummations, KAM theory.\end{abstract} \*

\0{\small {\bf Key words:} \it Vibrating string, Quantum fields,
  Perturbation theory, Divergent series resummation, KAM }%
\footnote[1]{{\small\texttt giovanni.gallavotti@roma1.infn.it}  \small
  extended version of a talk at the ``{\it Il segno di Lagrange nella Matematica
  contemporanea}'', Torino, Politecnico, 10 April 2013.}
%
\*

\def\SEC{Continuity and the vibrating string}
\section{\SEC}\label{sec1}
\iniz 

In 1759 an important open problem was how general was the solution
by Euler of the wave equation for a string 
starting from an initial configuration in which it is given a
shape $u_0(x)=\f(x)$ and no initial velocity:

\be\eqalign{
\dpr_t^2 u=&c^2 \dpr^2_x u, \qquad x\in[0,a],\ u(a)=u(b)=0\cr
u(x,t)=& \f(x-ct)+\f(x+ct)
\cr}\Eq{e1.1}\ee
%
According to D'Alembert's arguments it should have been necessary that
$\f(x)$ be at least a {\it smooth} (twice differentiable in present
notations or, in D'Alembert's language, subject to the {``loi de
  continut\'e}) function of $x\in R$ which is {\it periodic with period
  $2a$} and odd around $0$ and $a$. In his notations $\f$ had to satisfy
conditions at $0,a$ allowing its continuation to a $2a$--periodic function
of class $\CC_2$, odd around $0$ and $a$.

Euler claimed that Eq.\equ{e2.1} would be a solution
$u(x,t)=\f(x-ct)+\f(x+ct)$ if $\f(x)$ was simply defined outside the
interval $[0,a]$ just continuing it as a periodic function of period $2a$
odd around $0$ and $a$. He insisted that the ($\CC_1$) smoothness of
$\f(x)$ is sufficient: this means that as time $t$ becomes $>0$ the form
of the string appears (in general) to keep points with discontinuous
curvature: {\it i.e.} points in which the second derivative is not
defined. This led D'Alembert to think that when the continuation of $\f$
was not smooth the solutions did not make sense.

Neither was able to produce a rigorous argument. Other theories were due
to Taylor who had proposed that a general solution of the string motion
(starting from an initial configuration in which the string is given a
shape $u_0(x)=\f(x)$) is a sum $u(x)=\sum_{n} \a_n \sin(\frac{2\p}{2a} n
x)\cos(\frac{2\p}{2a} n c t)$: a view supported by D. Bernoulli (according
to Lagrange). This is criticized by Euler who objects that
such expressions only represent D'Alembert's solutions (which is true {\it if
the series converges} in class $\CC_2$).

All solutions had been obtained by arguments relying on unproved properties
based on intuition or experience and this is the reason identified by
Lagrange as the source of the controversies. 

Lagrange's idea is that the problem must first be (in modern locution) {\it
  regularized}: this means imagining the string to consist of an indefinite
number $m$ of small particles aligned (in the rest state) and elastically
interacting with the nearest neighbors possibly moving orthogonally to the
rest line and with extremes $y_0=y_m=0$ fixed.

In this way the problem becomes a clearly posed mechanical problem and the
equations of motion are readily derived: the only question is therefore
finding the properties of their solutions and of the limit in which the
mass of the particles tends to zero while their number and the strength of
the elastic force tend to infinity so that a continuum string motion
emerges, \cite[T.I, p.71]{La867}: 
\* 
%
\0{\it Il resulte de tout cet expos\'e que l'Analyse que nous avons
  propos\'ee dans le Chapitre pr\'ec\'edent est peut-\^etre, la seule qui
  puisse jeter sur ces mati\`eres obscures une lumi\`ere suffisante \`a
  \'eclaircir les doutes qu'on forme de part et d'autre.}
%cit1 p.71 (gv -> 35)
\*

The analysis is today well known: he notices that the problem is (in modern
language) the diagonalization of a $(m-1)\times(m-1)$ tridiagonal symmetric
matrix. The solution is perhaps the first example of the diagonalization
procedure of a large matrix. The result is the representation of the general
motion of the chain with general initial data for positions {\it and
  velocities}, \cite[T.I, p.97]{La867}:

\*
\0{\it .. je ne crois pas qu'on ait jamais donn\'e pour cela une formule
  g\'en\'erale, telle que nous venons de la trouver.}
%cit2 p.97 (gv -> 64)
\*

Lagrange's very remarkable three memories, the first two constitute a
veritable monograph, consist in showing that the general motion has, for
suitable $\o_h$, the form (translated in modern language)
%
\be  y^{(\d)}(\x,t)=\sum_{h=1}^{m-1}
\Big\{
\wt A_h\sqrt{\frac2m}\sin\frac{\p\,h}a\x\,\cdot\cos\o_h t+
\wt B_h\sqrt{\frac2m}\sin\frac{\p\,h}a\x\,\cdot\sin\o_h
t\Big\},\Eq{e1.2}\ee
%
where, denoting by $\d$ (called $dx$ by Lagrange) the mesh of the discretized
positions so that the number of small masses, located at points $\x=i\d$,
is $\frac{a}\d$ and $ \wt A_h,\wt B_h,\o_h$ are derived from the initial
profiles of positions $Z(\x)$ and velocities $U(\x)$, \cite[T.I.,
  p.163]{La867}: via the expressions

\be \kern-3mm\eqalign{
&\sqrt{\frac2m}\wt A_h= \frac2m \sum_{i=1}^{m-1} \big(\sin\frac{\p h}a\x\big)
Z(\x)\tende{\d\to0}\frac2a\int_0^a Z(x) \big( \sin\frac{\p h}a
x\big)\,dx\cr
&\sqrt{\frac2m}\wt B_h= \frac2{\o_h\,m}
\sum_{i=1}^{m-1}\big(\sin\frac{\p h}a\x\big)
U(\x)\tende{\d\to0}
\frac2{\lis\o_h\,a}\int_0^a U(x) \big( \sin\frac{\p h}a
x\big)\,dx,\cr
&\o_h=c\sqrt{2\frac{1-\cos(\frac{\p h\d}{a})}{\d^2}}\tende{\d\to0}
\lis\o_h=c\frac{\p h}a}\Eq{e1.3}\ee
%
where $c$ is $\sqrt{\frac\t\m}$ where $\t$ is the tension and $\m$ is the
density of the string. The formulae do not require smoothness assumptions
on the data $Z,U$ other than integrability as stated in answer to
D'Alembert's critique, \cite[T.I, p.324]{La867}:\*
%
\0{\it Mais je le prie de faire attention que, dans ma solution, la
  d\'etermination de la figure de la corde \`a chaque instant d\'epend
  uniquement des quantit\'es $Z$ et $U$, lesquelles n'entrent point dans
  l'op\'eration dont il s'agit. Je conviens que la formule \`a laquelle
  j'applique la m\'ethode de M. Bernoulli est assujettie \`a la loi de
  continuit\'e; mais il ne me parait pas s'ensuivre que les quantit\'es $Z$
  et $U$, qui constituent le coefficient de cette formule, le soient aussi,
  comme M. d'Alembert le pr\'etend.}
%
\*

Hence, setting $\x=x$, the general solution:
%
\be \eqalign{
u(x,t)=&\sum_{h=0}^\infty\sin\frac{\p h}a
x \,\Big\{ (\frac2a \int_0^a Z(x') \sin\frac{\p h}a
x'\, dx')\cos\lis\o(h) t\cr
&+ (\frac2a \int_0^a U(x') \sin\frac{\p h}a
x'\,dx') \frac{\sin\lis\o(h) t}{\lis\o(h)}\Big\}\cr}\Eq{e1.4}\ee
%
is found: and it is far more general than
the previous solutions because it permits initial data with $U\ne0$, {\it
  i.e.} with initial data in which the string is already in motion. In the
case $U=0$ it coincides with Euler's as it can be written
$\f(x+ct)+\f(x-ct)$ with $\f$ odd around $0$ and $a$ and $2a$-periodic
aside from the obvious convergence and exchange of sum and limits problems.

Check of Eq.\equ{e1.3} is based on some trigonometric properties: basically
on ``Cote's formula'', \cite[T.I, p.75]{La867},
$|a^m-b^m|=\prod_{h=0}^{m-1} (a^2-2ab \cos\frac{2\p}mh+b^2)^{\frac12}$
(derivable from $a^m-b^m= \prod_{h=0}^{m-1} (a e^{i\frac{2\p}m h}-b)$)
which is used instead of the modern $\sum_{h=0}^{m-1} e^{i\frac{2\p}m
  h(p-q)}=m \d_{p,q}$).

The continuum limit Eq.\equ{e1.4} is identified with Euler's result: it is
remarkably found together with interesting considerations and attempted
justifications of resummations of divergent series like, \cite[T.I,
  p.111]{La867},

\be \cos x+\cos 2x+\cos 3x +\ldots=-\frac12\Eq{e1.5}\ee
%
an incorrect statement immediately criticized by D'Alembert but whose use
Lagrange defended correctly, \cite[T.I, p.322]{La867}.\footnote{\small
  Lagrange uses this formula to infer the value of the difference
  $\D(x,\th)=\frac12\sum_{N=1}^\infty \cos
  n(x-\th)-\frac12\sum_{N=1}^\infty\cos n(x+\th)=0$ which ``today'' would
  instead be $(-\frac12 +\p\d(x-\th)
  +\frac12-\p\d(x+\th))=\p(\d(x-\th)-\d(x+\th)) \ne0$ unless $\th=0$. His
  argument is to consider first Eq.\equ{e1.5}, which equals
  $-\frac12+\p\d(x)$, and erroneously sets it $-\frac12$, and the claim
  that $\D(x,0)=0$ would be correct. Looking at the consequence of this
  error on Lagrange's theory it appears, as he correctly states in
  \cite[T.I, p.111]{La867}, that what is really used is the truncated version of
  the above relations and the Eq.\equ{e1.5} was only an alternative
  proposition: {\it il ne sera pas hors de propos de d\'emontrer encore la
    m\^eme proposition d'une autre mani\`ere}, \cite[T.I, p.109]{La867}. }
%cit3 p.111 (gv -> 75)

The analysis might appear not rigorous in today sense: not really because of
the statement Eq.\equ{e1.5} nor because the statement that $\sin
\frac{2\p}2 m(\frac{x}a\pm\frac{H t}T)=0$ as consequence of $m=\infty$,
\cite[T.I, p.102]{La867} which may make the eyebrows frown, as of course
D'Alembert's did, and for which Lagrange almost ``apologized'': 
\*

\0``{\it Je conviens que je ne me suis pas exprim\'e assex exactment ..}'',

\*\0\cite[T.I, p.322]{La867}, while making it more plausible
(it seems to be an early version of what is today the ``Lebesgue theorem''
for Fourier series).

Also several other objections by D'Alembert and D. Bernoulli do not appear
to have been answered very convincingly by the present standards in the
memory in defense of the theory in \cite[T.I, p.319-332]{La867}. The short
note is still of great interest as it shows the Lagrange struggles and gets
very close to the modern notion of ``weak solution'' of a PDE, see also
\cite[T.I, p.177]{La867}, and to a formalization of a theory of resummation
of divergent series.

Today a mathematician will probably consider it not completely rigorous
only because it applies to piecewise differentiable initial data $Z,U$
for the positions and speeds of the string elements: but a proof of the
exchanges of limits (and their existence) needed to ``pass to the continuum
limit'' is not even mentioned.

Its need did not occur to Lagrange, in his 23-d year of age, as well as to
Euler himself: the easy proof is in all textbooks, {\it e.g.}
\cite[Ch.4.5]{Ga008}, and it is rightly considered that, {\it de facto},
Lagrange solved at the same time the controversies on the proof of
completeness of the basis $\sin \frac{\p}{a}nx$ for the functions on
$[0,a]$ vanishing at $0,a$, {\it i.e} showed the convergence of Fourier's
series for functions piecewise continuously differentiable.

The theory of the vibrating strings, in Lagrange, was motivated and became
part of a long and detailed study of the propagation of sound in two
exhaustive Memoires (followed by a short one to answer criticism): in which
sound propagation, \Ie the wave equation, is analyzed in the one
dimensional case and for three dimensional spherical waves (whose equation
is taken from Euler and which he reduces to the one dimensional theory via
the remarkable change of variables $u(x)=x^{-2}{\int^x z(x)x dx}$).

In the second Memory the ($\CC_2$) continuity problem is also examined from a
new viewpoint: namely to study the motion of the {\it internal
  points} of a string with extremes fixed. It is proved that the points
move following the solution found with the discretization method. This is
interesting also because the method presented will be recognized to be a
precursor of the modern notion of {\it weak solution} of a PDE, \cite[T.I,
  p.177]{La867}: \*

\0{\it Les transformations dont je fais usage dans cette occasion sont celles
qu'on appelle int\'egrations par parties, et qui se d\'emontrent 
ordinairement par les principes du calcul diff\'erentiel; mais il n'est pas
difficile de voir qu'elles ont leur fondement dans le calcul g\'en\'eral des
sommes et des diff\'erences; d'o\`u il suit qu'on n'a point \`a craindre
d'introduire par l\`a dans notre calcul aucune loi de continuit\'e entre les
diff\'erentes valeurs de $z$.}
\*

The vibrating string theory and the continuum as a limit of a
microscopically discrete reality was developed by Lagrange at a time when
the atomistic conceptions were being established. The approach he adopted
is a key legacy at the basis of methods currently employed in the most
diverse fields, see for instance \cite{Pr009}: a further example will be
discussed in the next section.
 
%cit5 (158 -> 122 ok)
%cit6 (158 -> 122 ok)
%cit7 (322 --> 286 ok) obiez 1
%cit8     ``          obiez 2
%cit9 (323 ``  287 ok)        obiez 3

\def\SEC{QFT: quantum elastic string}
\section{\SEC}
\label{sec2}\iniz

The key idea in the vibrating string has been that it is a continuous
system which should be regarded as, and behaves as, a limit case of a
system of infinitely many adjacent particles whose motion should be described
via the ordinary equations {\it without} requiring new principles.
%
%cit10 (45 --> 9 ok)
%cit11 (55 --> 19 ok)

Interestingly this problem has reappeared essentially for the same reasons
in recent times. The theory of elementary particles requires at the same
time quantum mechanics and (at least) special relativity: it became soon
clear that it could be appropriately formulated as a theory of quantized
fields which met immediately impressive successes in the description of
electromagnetic interactions of photons and electrons and of weak
interactions. Particles were naturally represented by particular states of
a field which could describe waves as well, \cite[Sec.I]{Ga985b}.

The simplest example is a ``scalar field'', $\f(x)$, in space time
dimension $2$ corresponding classically to the Lagrangian
%
\be\eqalign{
{\cal L}=&\frac\m2\int_\a^\b \Big(\dot\f(x)^2- c^2(\frac{d\f}{d
    x}(x))^2
%+\cr&
-(\frac{m_0 c^2}{\hbar})^2\f(x)^2\,-\, I(\f(x))\Big)\,dx
\cr}\Eq{e2.1}\ee
%
where $I(\f)$ is some function of $\f$. If $I=0$ this is a vibrating string with
density $\m$, tension $\t=\m c^2$  and an elastic pinning force $\m
(\frac{m_0 c^2}\hbar)^2$. 

\eqfig{300}{65}{
\ins{-15}{0}{$O$}
\ins{260}{0}{$L$}
}{Fig1}{Fig.1}

\kern4mm
\0{\small Figure 4.1: chain of oscillators elastically bound by nearest
  neighbors and to centers aligned on an axis orthogonal to their vibrations
and Dirichlet's boundary condition.}
\*

The nonlinearity of the resulting wave equation produces the result
that when two or more wave packets collide they emerge out of the
collision quite modified and do not just go through each other as in
the case of the linear string, so that their interaction is
nontrivial. 

Naively the quantum states will be, by the ``natural extension of the usual
quantization rules'', functions $F(\f)$ of the function $\f$ describing the
configurational shape of the elastic deformations. The Hamiltonian operator
acts on the wave function $F$ as
%
\be\eqalign{
({\cal H} F)(\f)=&
\int_{0}^L
\left(-\frac{\hbar^2}{2\m}
\frac{\d^2 F}{\d\,\f(x)^2}(\f)+
\frac\m2\Big(
c^2
(\frac{\dpr\f}{\dpr\V x}(\V x))^2+\right.\cr
&\left.+(\frac{m_0 c^2}{\hbar})^2 
\f(x)^2+ I(\f(\V x))\Big)\, F(\f)\right)
\,d^D\V x
\cr}\Eq{e2.2}\ee
%
where $\frac\d{\d\f(x)}$ is the functional derivative operator (a notion
also due to Lagrange and to his calculus of variations) 
and it should be defined in the space $L_2(``d\f'')$, where the scalar
product ought to be $(F,G)=\int \overline{F(\f)}\,G(\f)\,''d\f''$ and
$''d\f''=\prod_{\V x\in[0,L]} d\f(\V x)$.

Even though by now the mathematical meaning that one should try to attach
to expressions like the above as ``infinite dimensional elliptic
operators'' and ``functional integrals'' is quite well understood,
particularly when $I\equiv0$, formulae like the above are still quite
shocking for conservative mathematicians, even more so because they turn
out to be very useful and deep.

One possible way to give meaning to \equ{e2.2} is to go back to first
principles and recall the classical interpretation of the vibrating string
as a system of finitely many oscillators, following Lagrange's brilliant
theory of the discretized wave equation and of the related Fourier series
summarized in Sec.\ref{sec1}.

Suppose, for simplicity, that the string has periodic boundary conditions
instead of the Dirichlet's conditions studied by Lagrange; replace it with
a lattice $ Z_\d$ with mesh $\d>0$ and such that $L/\d$ is an integer. In
every point $n \d$ of $ Z_\d$ put an oscillator with mass $\m\,\d$,
described by a coordinate $\f_{n\d}$ giving the elongation of the
oscillator over its equilibrium position, and subject to elastic pinning
force with potential energy $\frac12 \m \d\,(\frac{m_0 c^2}{\hbar})^2
\f^2_{n\d}$, to a nonlinear pinning force with potential energy $\frac12\m
\d I(\f_{n\e})$. and finally to a linear elastic tension, coupling nearest
neighbors at positions $n\d,(n+1)\d$, with potential energy $\frac12\m
\d^{-1}c^2 (\f_{n\d}-\f_{(n+1)\d})^2$.

Therefore the Lagrangian of the system, in the more general case of space
dimension $D\ge1$ ($D=1$ is the vibrating string, $D=2$ is the vibrating
film, {\it etc}, \Ie space time dimension $d=D+1$), is
%
\be\kern-3mm\eqalign{
{\cal L}=&\frac\m2 \d^D\sum_{n \d\in \L_0}\Big(
\dot\f_{n\d}^2-c^2 \sum_{j=1}^D \frac{(\f_{n \d+ e_j \d}-
\f_{n\d})^2}{\d^2}
-(\frac{m_0 c^2}{\hbar})^2\f^2_{n\d}- I(\f_{n\d})\Big)
\cr}
\Eq{e2.3}\ee
%
where $e_j$ is a unit vector oriented as the directions of the
lattice; if $n \d+e_j \d$ is not in $\L_0$ but $n \d$ is in
$\L_0$ then the $j$-th coordinate equals $L$ and $n
\d+e_j \d$ has to be interpreted as the point whose $j$-th coordinate
is replaced by $1$; {\it i.e.} is interpreted with periodic
boundary conditions with coordinates identified modulo $L$.

It should be remarked that for $I=0$ and $m_0=0$ Eq.\equ{e2.3} is, for
$D'=1$, the Lagrangian of the vibrating string introduced by Lagrange in
his theory of sound.

Of course there is no conceptual problem in quantizing the system: it will
correspond to the familiarly elliptic operator on 
$L_2(\prod_{n \e} d\f_{n
  \e})=L_2(R^{\frac{L}\d})$:
%
\be\eqalign{
\HH_{\d}=&
-\frac{\hbar^2}{2\m \d}\sum_{n\d\in\L_o}\frac{\dpr^2}{\dpr \f_{n
\d}^2}+\frac{\m \d}{2}
\sum_{n \d\in\L_o}\cr
&\Big( c^2\frac{(\f_{n \d+e
    \d}-\f_{n \d})^2}{\d^2}+
\big(\frac{m_0 c^2}{\hbar}\big)^2 \f_{n \d}^2+ I(\f_{n \d})\Big)
\cr}\Eq{e2.4}\ee
%
with $\CC^\infty_0(R^{\frac{L}\d})$ as domain (of essential
self-adjoint\-ness) provided $I(\f)$ is assumed bounded below, as it should
always be: 
\eg in the case of the so called {\it $\l \f^4$ theory} with
$I(\f)=\int_0^L (\l \f(x)^4+\m\f(x)^2+\n)dx$ with $\l>0$.

At this point it could be claimed, with Lagrange \cite[T.I, p.55]{La867},
\*

\0{\it Ces \'equations, comme il est ais\'e de le voir, sont en m\^eme
  nombre que les particules dont on cherche les mouvements; c'est pourqoi,
  le probl\`eme \'etant d\'ej\`a absolument determin\'e par leur moyen, on
  est oblig\'e de s'en tenir l\`a, de sorte que toute condition
  \'etrang\`ere ne peut pas manquer de rendre la solution insuffisante et
  m\^eme fautive.}
\*

This can be appreciated by recalling that in developing the theory, {\it
  even in the simplest case of the $\l\f^4$ field, just mentioned},
difficulties arise which have given rise to many discussions.

In particular the attempts to study the properties of the operator $\cal H$
via expansions in $\l,\m,\n$ immediately lead to nonsensical results
(infinities or indeterminate expressions). At the beginning of QFT the
results were corrected by adding ``{\it counterterms}'' amounting, in the
case of the string {\it i.e.} of space-time of dimension $2$, to make
$\m,\n$ infinite.

The subtraction prescriptions, known as ``{\it renormalization}'') were not
really arbitrary: the remark was that {\it all results} were given by many
integrals which were divergent but in which a divergent part could be
naturally isolated by bounding first the integration domains and by
determining $\m,\n$ as functions of the parameters defining the domains
boundaries so that the infinities disappear when the boundaries of the
domains of integration are removed.

{\it Reassuringly} the choice of the divergent ``counterterms'' $\m,\n$
turned out to be essentially independent of the particular result that was
being computed.

Even though the {\it renormalization} procedure was unambiguous (at least
it was claimed to be such) it was not clear (at least not to many) that
some new rule, \Ie a new physical assumption, was not implicitly introduced
in the process. This is analogous to the controversy on the vibrating
string solved by Lagrange with his theory of sound via discretization of
the string, solution of its motion and remotion of the regularization, \Ie
taking the particles mass to $0$, the tension to $\infty$ at proper rates
as $\d\to0$ so that all results remained well defined and converging to
limits.

In the 1960's reducibility of renormalization theory to the rigorous study
of the properties of the operator in Eq.\equ{e2.4} was started and at least
in the cases of the quantum string and of the quantum film (\Ie the cases
of space time dimension $2$ and $3$, respectively) it was fully understood
in a few (very few) cases via the work of Nelson, Glimm-Jaffe, Wilson who
showed clearly that no infinities really arise if the problem is correctly
studied, \cite{Ga985b}: \Ie taking seriously the discretized Hamiltonian
computing physically relevant quantities and passing to the continuum
limit, just as Lagrange did in his theory of sound.

This has been a major success of Physics and Analysis. The problem remains
open in the case of $d=D+1=4$ space-time dimensions and the reason is precisely
that the nature of the regularization becomes essential in higher
dimension: it seems that the naive discretization on a square lattice
described here implies assumptions on the Physics at very small scale so
that it might be impossible to perform the continuum limit $\d\to0$ unless
the regularization of the formal functional derivatives in the Hamiltonian
is chosen conveniently. It is conjectured that a naive discretization like
the above Eq.\equ{e2.4} cannot lead to a nontrivial result if $d=4$ and
$I(\f)$ is a fourth order polynomial.

Therefore a challenge remains to explain why the renormalized quantum
electrodynamics in $4$-space time dimensions, only defined as a formal
power expansion in the elementary electric charge, gives amazingly precise
results perfectly agreeing with experiments although the known
regularizations are all conjectured to yield trivial result if studied
under full mathematical rigor. In recent times the regularization choice
appears to have been related to very new conceptions of the nature at very
small space-time scale (QCD and string theory): even in this respect
Lagrange's method of attacking problems avoiding the introduction of new
principles or of understanding their necessity is still fertile.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\SEC{Perturbations in field theory}
\section{\SEC}
\label{sec3}\iniz
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The problems of QFT are often studied by perturbation theory and a key role
is played by {\Ie scale invariant theories}. For instance consider the
elliptic operator $\HH_\d$ in Eq.\equ{e2.4} for a quantum string, written
as $H^0_\d+I_\d$ separating the interaction $I_\d(\f)=\l\f^4+\m\f^2+\n$.

Perturbation theory as developed by Lagrange might seem at first
sight quite far from its use in QFT: yet it is quite close as it will be
discussed in this section after giving some details on the form in which it
arises in renormalization theory and how it appears as an implicit
functions problem.

Restricting the discussion to the scalar $\f^4$-systems, Eq.\equ{e2.4}, the
breakthrough has been Wilson's theory which shows that the values of
physical observables can be can be constructed, for the $\f^4$ system in
dimensions $d=2,3$, as power series in parameters $
\L_k\defi(\l_k,\m_k,\n_k)$, $k=0,$ $1,2,\ldots$, called {\it running
  coupling constants}, which are related by

\be
\L_k= M \L_{k+1}+ B(\{\L_r\}_{r=k+1}^{\infty}),\qquad k=0,1,\ldots\Eq{e3.1}\ee
%
where $M$ is a diagonal matrix with elements $m_1,m_2,m_3$. The physical
meaning of the running constants is that they control the physical
phenomena that occur on length scale $2^{-k} ell_0$ here $\ell_0$ is a
natural length scale associated with the system, \eg
$\ell_0=\frac{\hbar}{m_0 c^2}$.

The series for the observable quantities values are {\it well defined and
  no infinities appear provided} the sequence $\L_k$ consists of uniformly
bounded elements.

Hence the physical observables values can be expressed in terms of a well
defined power series in few parameters, the $\L_k$'s, which are in turn
functions of three independent parameters (the $\L_0$, for instance) whose
values define the theory. However the values of physical observables are
very singular functions and their expansion in powers of the physical
parameters (\Ie $\L_0$, for instance) would be meaningless. In other words
the infinities appearing in the heuristic theory are due to singularities
in the $\L_k$ as functions of a single one among them: they are due to
overexpanding the solution.

This leaves however open the problems:

\0(1) existence of a bounded sequence of ``running couplings'' $\L_k$

\0(2) convergence of the series for the observables values

The second question has a negative answer because it is immediate that
$\l_k<0$ cannot be expected to be allowed: in the $\f^4$-systems they can
be shown to be asymptotic series in space-time dimensions $2,3$ .

The first question has an answer in space-time dimensions $2,3$ (ultimately
due to $m_1<1$ and $m_2,m_3>1$) while in $4$ dimensions space-time one more
expansion parameter $\a_k$ is needed and $m_1,m_4=1$ but it seems
impossible to have a bounded sequence with $\l_k>0$ (the {\it triviality
  conjecture} proposes actual impossibility but it is a delicate and open
problem).

The relation with Lagrange's work arises from Eq.\equ{e3.1} considered as
an equation for $\L\defi \{\L_k\}_{k=0}^\infty$ of the form
$A=\L-\MM(\L)-\BB(\L)$ with $A=0$. In the development of perturbation
theory the request of a bounded solution of the latter implicit function
problem arises in the form that is naturally identified with the
boundedness requirement on the solution that Lagrange's theory of
``litteral equations'' would yield (see below) for Eq.\equ{e3.1}.

In \cite[T.III, p.25]{La867} the following formula is derived
%
\be\a=x-\f(x),\quad\otto\quad
\ps(x)=\ps(\a)+ \sum_{k=1}^\infty \frac1{k!}
\dpr^{k-1}_\a(\f(\a)^k\dpr_\a\ps(\a))
\Eq{e3.2}\ee
%
{\it for all} $\ps$. In particular for
$\ps(\a)\equiv\a$ 
%
\be x(\a)=\a+\sum_{k=1}^\infty \frac1{k!}
\dpr^{k-1}_\a(\f(\a)^k),\qquad x(0)=\sum_{k=1}^\infty \frac1{k!}
\dpr^{k-1}_\a(\f(\a)^k)\Big|_{\a=0}\Eq{e3.3}\ee
%
are expressions for the inverse function and for a solution of
$x=\f(x)$. For the remarkable formula to works it is necessary that the
series converges: it does not always select among the roots the closest to
$\a$ and this is pointed out in \cite{Ch846}, criticizing a related
statement by Lagrange; it provides an interesting discussion of the
properties of the root obtained by replacing $\f$ by $t \f$ and studying
which among the roots is associated with the sum of the series if the
latter converges up to $t=1$, theorem 3, p.18.

The formulae can be generalized to $\a,x$ in $R^n,\, n\ge1$ and are
used in the derivation and on the detailed analysis of
Eq.\equ{e3.1} via the remark that the fixed point equation of
Lagrange is equivalent to the following {\it tree expansion} of the $k-th$
term in Eq.\equ{e3.2} (when $\ps(x)=x$):
%
\be \frac1{k!} \dpr^{k-1}_\a(\f(\a)^k)=\sum_\th {\rm Val}(\th)\Eq{e3.4}\ee
%
where $\th$ is a tree graph, \Ie
%
it is a ``decorated'' tree with 
\\
(1) $k$ branches $\l$ of equal length oriented towards the
``root'' $r$,
\\
(2) each node as well as the root (not considered a node) 
carries a label $j_v\in\{1,\ldots,n\}$ 
\\
(3) at each node $v$ enter $k_v$ branches
$\l_1\equiv v v_1,\ldots,\l_{k_v}\equiv v v_{k_v}$ where
$v_1,\ldots,v_{k_v}$ are the $k_v$ nodes preceding $v$ (it is $\sum_{v<r}k_v=k$).
and
\\
(4) the node $v$ symbolizes the tensor
$\dpr^{k_v}_{x_{j_{v_1}},x_{j_{v_2}},\ldots x_{j_{v_{k_v}}} 
}\f_{j_v}(\a)$

\eqfig{300}{145}{
\ins{-5}{71}{$r$}
\ins{14}{71}{$j$}%\n\kern-2pt=\kern-2pt\n_{\l_{0}}$}
%\ins{-5}{91}{$j$}
%\ins{14}{71}{$k$}%\kern-2pt=\kern-2pt n_{\l_{0}}$}
\ins{24}{91}{$\l_{0}$}
\ins{50}{71}{$v_0$}
%\ins{46}{91}{$\n_{v_0}$}
\ins{127}{100.}{$v_1$}
%\ins{121.}{125.}{$\n_{v_1}$}
\ins{92.}{41.6}{$v_2$}
\ins{158.}{83.}{$v_3$}
\ins{191.7}{133.3}{$v_5$}
\ins{191.7}{100.}{$v_6$}
\ins{191.7}{71.}{$v_7$}
\ins{191.7}{-6.3}{$v_{11}$}
\ins{141.7}{20}{$v_{12}$}
\ins{191.7}{18.6}{$v_{10}$}
\ins{166.6}{42.}{$v_4$}
\ins{191.7}{54.2}{$v_8$}
\ins{191.7}{37.5}{$v_9$}
}{Fig2}{Fig.2}

\kern.7cm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\0Fig.2:\ {\small A decorated tree. Labels $j_{v}, v<r$
not marked and intended as
contracted.}
\*
%
\0(5) Two such decorated trees are considered equivalent if 
superposable by pivoting the branches around the nodes without
permitting overlapping.
\\
%
\0(6) The {\it value}, ${\rm Val}\,(\th)$ is defined as

\be {\rm Val}\,(\th)=\prod_{v\in\th} \Big(\frac1{k_v!}
\dpr^{k_v}_{j_{v_1},\ldots,j_{v_{k_v}}}\f_{j_v}(\a)\Big)\Eq{e3.5} \ee
%
 with sum over node labels, {\it except the root label $j=j_{r}$},
is understood.
\*

It can be said that also the determination and theory of the functions $B$
follows a path that can be traced to Lagrange's theory of ``litteral
equations''.

The analysis in the $\f^4$-field theory context of the just described
method of getting to the running couplings expansion can be found in
\cite{Ga985b}.  Lagrange made use of Eq.\equ{e3.4} in his works on
celestial mechanics, relying on his study of Kepler's equation giving
eccentric anomaly $\x$ or true anomaly $u$ in terms of average anomaly
$\ell$:

\be \ell=\x+e\sin\x,\qquad \x=\ell+\sum_{k=1}^\infty
\frac{e^k}{k!}\dpr^{k-1}_\ell(\sin^k\ell)
\Eq{e3.6}\ee
%
The coefficients can be readily evaluated if $\sin^k\x$ is expressed in
terms of angles multiples of $\ell$ and since $\frac{
  du}{d\x}=\frac{(1-e^2)^{\frac12}}{1+e\cos\x}$ the expansion of the true
anomaly is also computed applying Eq.\equ{e3.2} with $\f(\ell)=e \sin\ell$
and $\ps(\ell)=\frac{\sqrt{1-e^2}}{1+e\cos\ell}$, \cite[T.III,
  p.114]{La867}.

Several applications of perturbation theory have been considered by
Lagrange often leading, as a byproduct, to key discoveries like the
determination of the five Lagrangian points, \cite[T.VI, p.280]{La867},
found while attempting to find approximate solutions to the three body
problem. And the many works on the secular variations of the planetary
nodes, the determination of the orbit of comets, the Moon librations.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\SEC{Overview on  Mechanics}
\section{\SEC}
\iniz\label{sec4}\iniz
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Lagrange's contribution to the formulation and application of the
principles of Mechanics is so well known and modern that it is continuously
used today. The 1788 {\it Trait\'e de M\'ecanique Analytique}, whose second
edition is in \cite[T.XI]{La867} and \cite[T.XII]{La867}, is entirely based
on a consistent reduction to first principles of every problem considered.

This means constantly returning to applying the least action principle in
the form of the combination the the principle of virtual works with the
principle of D'Alembert, \cite[p.51]{Da743}: 
%
\*
%
\0{\it D\'ecompos\'es les Mouvemens $a,b,c,\&tc$ imprim\'es \`a chaque
  Corps, chacun en deux autres $\a,\a';\b,\b';\g,\g'\&tc$ qui soient tels,
  que si l'on n'e\^ut imprim\'e aux Corps que les Mouvemens $\a,\b,\g,\&tc$
  ils eussent p\^u conserver ces Mouvemens sans se nuire r\'eciproquement;
  et que si on ne leur e\^ut imprim\'e que les Mouvemens $\a',\b',\g',\&tc$
  le syst\`eme fut demeur\'e en repos; il est clair que $\a,\b,\g,\&tc$
  seront les Mouvemens que ces Corps prendront en vertu de leur action. Ce
  Q.F. trouver.}%%%%
\footnote{The last words, ``What was to be found'', are there because the
  principle is ``derived'' as an answer to a problem posed in the
  previous page. Here $a,b,c,\&tc$ are the impressed forces $\V f_i$, and
  $\a,\b,\g,\&tc$ are $m_i \V a_i$ while $\a',\b',\g',\&tc$ are the
  constraint reactions $-\V R_i$: \Ie $\V f_i=m_i\V a_i-\V R_i$; in modern
  form, \cite[\S3.18,Vol.2]{LA927}, the $\V f_i-m_i\V a_i$ may be called
  the ``lost forces'' and the principle is {\it During the motion of a
    system of point masses, however constrained and subject to forces, at
    each instant the lost forces are equilibrated by virtue of the
    constraints}. } \*

Lagrange's addition is clearly explicitly stated in the second work on the
Moon librations where he says, about his method: \*

\0 {\it... Elle n'est autre chose que le principe de Dynamique de
  M. d'Alembert, r\'eduit en formule au moyen du principe de l'\'equilibre
  appel\'e commun\'ement loi des vitesses virtuelles. Mais la combinaison
  de ces deux principes est un pas qui n'avait pas \'et\'e fait, et c'est
  peut-\^etre le seul degr\'e de perfection qui, apr\`es la d\'ecouverte de
  M. d'Alembert, manquait encore \`a la Th\'eorie de la Dynamique.},
\cite[V.,p11]{La867}.\footnote{\Ie $\sum_i (\V f_i-m_i\V a_i)\cdot\d\V
  x_i=0$.} \*

The equvalent action principle is used in the form \*

\0{\it Dans le mouvement d'un syst\`eme quelconque de corps anim\'es par
  des forces mutuelles d'attraction, ou tendantes \`a des centres fixes, et
  prortionnelles \`a des fonctions quelconques des distances, les courbes
  d\'ecrites par les diff\'erents corps, et leurs vitesses, sont
  n\'ecessairement telles que la somme des produits de chaque masse par
  l'int\'egrale de la vitesse multipli\'ee par l'\'el\`ement de la courbe
  est un maximum ou un minimum, pourvu que l'on regarde les premiers et les
  derniers points de chaque courbe comme donn\'es, en sorte que les
  variations des coordonn\'ees r\'epondantes à ces points soient nulles.

C'est le th\'eor\`eme dont nous avons parl\'e \`a la fin de la premi\`ere
Section, sous le nom de Principe de la moindre action}, 
\cite[T.XI, p. 318]{La867}.  \*

He arrives at the {\it Trait\'e de M\'ecanique Analytique}, which ``brought
Rational Mechanics to a state of perfection'', \cite[p.221]{Li967}, as the
completion of a long series of remarkable applications. Namely

\0(1) the
theory of sound (\Ie the vibrating string) and the theory of the librations
of the Moon, 1763, \cite[T.VI, p.8]{La867} (revisited in \cite[T.V, p.5]{La867},
1781), which is perhaps the first example of development and systematic
application of pertubation theory based on analytical mechanics, \Ie on the
action principle and the Euler-Lagrange equations.

\0(2) the theory of rigid body, which he undertakes curiously insisting
in avoiding starting from the proper axes of rotation, \cite[T.III,
  p.579]{La867}:
\*

\0{\it ... ce qui exige la r\'esolution d'une \'equation
  cubique. Cependant, \`a consid\'erer le Probl\`eme en lui-même, il semble
  qu'on devrait pouvoir le r\'esoudre directement et ind\'ependamment des
  propri\'et\'es des axes de rotation, propri\'et\'es dont la
  d\'emonstration est assez difficile, et qui devraient d'ailleurs \^etre
  plut\^ot des cons\'equences de la solution m\^eme que les fondements de
  cette solution}, \*

\0he rederives the theory in an original way, certainly very involved for
today eyes (used to Euler's equations as beginning point
\cite[p. 402]{Eu853}, \cite{FGG013}), obtaining, as a byproduct, the
theorem about the reality of the eigenvalues of a $3\times3$ symmetric
matrix, \cite[T.III, p. 605]{La867}, and (later, in the {\it Trait\'e}) the
motion of the ``Lagrange's top'', \cite[T.XII, p. 253]{La867} \*

\0{\it Ce cas est celui o\`u l'axe des coordonn\'ees $c$, c'est-à-dire la
  droite qui passe par le point de suspension et par le centre de
  gravit\'e, est un axe naturel de rotation, et o\`u les moments d'inertie
  autour des deux autres axes sont \'egaux (art. 32), ce qui a lieu en
  g\'en\'eral dans tous les solides de r\'evolution, lorsque le point fixe
  est pris dans l'axe de r\'evolution.  La solution de ce cas est facile,
  d'après les trois int\'egrales qu'on vient de trouver}, \*

\0a remarkable integrability property that Poisson rediscovered (in fact he
does not quote Lagrange, see the footnote by J. Bertrand at the {\it
  loc.cit.}).

\0(3) the theory of the secular variations of the planetary elements
and several other celestial questions (the fifth volume of the collected
paper is entirely dedicated to celestial mechanics, \cite[T.V]{La867})
opening the way to Laplace's celestial mechanics and slightly preceding the
completion and publication of the M\'ecanique. This was the first time
analytical mechanics was used to attempt {\it long time} orbital predictions.

\0(4) the contributions are not only directed towards the principles but
concentrate on concrete problems for instance the integration by separation
of variables, \Ie by quadratures, of several dynamical systems: \eg the two
centers of gravitational attraction in the three dimensional case,
\cite[T.II, p.67]{La867}, the mentioned symmetric top.

\0(5) The {\it Trait\'e} also reflects the attitude of Lagrange regarding
matter as constituted by particles: \Ie his consistently kept atomistic
view, see for instance \cite[T.XI, p.189]{La867}: \*

\0{\it Quoique nous ignorions la constitution int\'erieure des fluides,
  nous ne pouvons douter que les particules qui les composent ne soient
  mat\'erielles, et que par cette raison les lois g\'en\'erales de
  l'\'equilibre ne leur conviennent comme aux corps solides}, \*

\0his conception of Mechanics as based on a variational principle was
adopted universally by the physicists that developed further the atomistic
theories: in particular Clausius and Boltzmann make constant use of the
action principle using also Lagrange's notations, \cite[p. 25]{Bo866},
\cite{Cl871}, and of the well known commutation rule $\d d=d\d$ introduced by
Lagrange and become universal, \cite[T.I, p.337]{La867}.

\0(6) Very often the ``Lagrange multipliers'', that he introduced to
simplify in various cases calculus of variations problems, are incorporated
in the variational equation of Euler-Lagran\-ge, \cite[T.XI, p.340]{La867}.

\0(7) It is important to comment also on the style of all his papers which
continues to inform the {\it Trait\'e}: at a time when quoting other works
was not very usual all papers of Lagrange start with a dense summary of the
previous works on the subject from which he draws the foundations of his
contributions and gives an important help to the historians of science.

\0(8) The terminology used by Lagrange has become the adopted terminology
in most cases: at times, however, the reader might be confused by the use
of terms which no longer have the same meaning and sorting things out
requires very careful reading: an example is in \cite[T.XI, p.244]{La867}
where the footnote added by J.  Bertrand provides an essential help to the
reader who has not read enough in detail the earlier pages and tomes.

\0(9) In the {\it Trait\'e} the theory of the small oscillations in systems
of several degrees of freedom is treated as a perturbation theory problem
and applied to the rigid mody motions and to celestial mechanics: it was
employed extensively in the 1800-ths, starting with Laplace, until Poincar\'e
made clear the need of substantial improvements on the method by pointing
out the existence of motions that could not be reduced to simple ``quasi
periodic'' ones. New ideas came much later with the work of Siegel,
Kolmogorov, Arnold, Moser, Eliasson: Lagrange's imprint on the problem
however remains not only through the secular perturbations treatment but
also through another of his contributions, namely the theory of the implict
functions, Eq.\equ{e3.2}, as outlined in the next section.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\SEC{KAM}\section{\SEC}
\label{sec5}\iniz
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The problem of stability of quasi periodic motions is closely related to
Lagrange's inversion formula Eq.\equ{e3.2} and to its combinatorial version
\equ{e3.4}. It can be formulated as an implicit functions problem for a
function $\V h(\mathbf\a)$ defined on the torus $T^\ell$ satisfying, in the
simplest non trivial cases, the {\it Hamilton--Jacobi} equation
%
\be {\cal K}\V h(\Ba)\defi \V h(\Ba)+(\Bo\cdot\BDpr)^{-2}\Big(\e\BDpr
f(\Ba+\V h(\Ba))\Big)=\V 0\Eq{e5.1}\ee
%
where $\Bo\in R^\ell$ is a vector with Diophantine property:
$|\Bo\cdot\Bn|\ge C_0^{-1}|\n|^{-\t}$ for some $C_0,\t>0$ and $\BDpr f$ is
the gradient of an analytic even (for simplicity) function on $T^\ell$,
$(\Bo\cdot\BDpr)^{-2}$ is the linear (pseudo)differential operator on the
functions analytic and odd on $T^\ell$ applied to the function of $\Ba$
$\Ba\to \e\BDpr f(\Ba+\V h(\Ba))$ and $\V h(\Ba)$ is to be determined, odd
in $\Ba\in T^\ell$.

This is leads to consider an infinite dimensional version of Lagrange's
inversion: it can be solved in exactly the same way writing $\V A=\V
h+{\cal K}\V h$ and solving for $\V h$ when $\V A=\V 0$, as in
Eq.\equ{e3.2}.  This is more easily done if the Eq.\equ{e5.1} is considered
in Fourier's transfom. Writing $\V h(\Ba)=\sum_{k=1}^\infty \e^k \V
h^{[k]}(\Ba)$ and denoting $\V h_{\Bn,j}^{(k)}$ the Fourier transform of 
$\V h^{(k)}_j,\,\Bn\in Z^\ell$ 
an expression for $j$-th component $h^{[k]}_{\Bn,j}$ is given
via trees with $k$ root-oriented branches as:

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\eqfig{300}{140}{
\ins{-5}{71}{$r$}
\ins{-5}{91}{$j$}
\ins{14}{71}{$\Bn\kern-2pt=\kern-2pt\Bn_{\l_{0}}$}
\ins{24}{91}{$\l_{0}$}
\ins{50}{71}{$v_0$}
\ins{46}{91}{$\Bn_{v_0}$}
\ins{127}{100.}{$v_1$}
\ins{121.}{125.}{$\Bn_{v_1}$}
\ins{92.}{41.6}{$v_2$}
\ins{158.}{83.}{$v_3$}
\ins{191.7}{133.3}{$v_5$}
\ins{191.7}{100.}{$v_6$}
\ins{191.7}{71.}{$v_7$}
\ins{210}{5}{$v_{11}$}
\ins{141.7}{20}{$v_{12}$}
\ins{191.7}{18.6}{$v_{10}$}
\ins{166.6}{42.}{$v_4$}
\ins{191.7}{54.2}{$v_8$}
\ins{191.7}{37.5}{$v_9$}
}{Fig3}{Fig.3}
\*
\0Fig.3: {\small A tree $\th$ con
$k_{v_0}=2,k_{v_1}=2,\ldots$ and
$k=13$, with a few decorations\vfil}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this case on each node there is an extra label $\Bn_v$ (marked in Fig.3
only on $v_0$ and $v_1$) and on each line there is an extra label
$\Bn(\l)=\sum_{w\le v} \Bn_w$; also in this case the labels $j_v,
j_{v_1},\ldots,j_{v_{k_v}}$ associated with the nodes have to be contracted
when appearing twice (\Ie unless $v=r$ as $j_r=j$ appears only once). The
trees will be identified if reducible to each other by pivoting as in the
simple scalar case of Sec.\ref{sec3}. After some algebra it appears that

\be\eqalign{\V h^{[k]}_\Bn=&\sum_\th {\rm Val}(\th),\qquad
%\cr
{\rm Val}(\th)=\Big(\prod_v \frac{f_{\Bn_v}}{k_v!}\Big)\Big(\prod_\l
\frac{\Bn_v\cdot\Bn_{v'}}{(\Bo\cdot\Bn(\l))^2}\Big)\cr}\Eq{e5.2}
\ee
%
where the sum is over all trees with $k$ branches.

Eq.\equ{e5.2} was developed in the context of celestial mechanics by
Lindstedt and Newcomb. It turns the proof of the KAM theorem into a simple
algebraic check in which the main difficulty of the {\it small divisors},
which appears because a naive estimate of $\V h^{[k]}_\Bn$ has size
$O(k!^\t)$ (making the formula illusory, because apparently divergent), can
be solved by checking that the values of the trees which are too large have
competing signs and almost cancel between themselves leading to an estimate 
 $|\V h^{[k]}_\Bn|\le c^k e^{-\k |\Bn|}$ for suitable $c,\k>0$.
 
The tree representation is particularly apt to exhibit the cancellations
that occur: the consequent proof of the KAM theorem, \cite{Ga994b}, is not
the classical one and it is often considered too complicated. In this
respect a comment of Lagrange is relevant: \*

\0{\it D'ailleurs mes recherches n'ont rien de commun avec le
  leurs que le probl\`eme qui en fait l'object; et c'est toujours
  contribuer \`a l'avancement des Math\'e\-matiques que de montrer comment
  on peut r\'esoudre les m\^eme questions et parvenir au m\^eme resultats
  par des voies tr\`es-diff\'erentes; les m\'ethodes se pr\`etent par ce
  moyen un jour mutuel et en acqui\`erent souvent un plus grand degr\'e
  d'\'evidence et de g\'en\'eralit\'e.} 
\*

\0\cite[T.6, p.280]{La867}.

Going back to Kepler's problem the work of Carlini and of Levi-Civita
(independent, later) made clear that Lagrange's series, Eq.\equ{e3.6}, can
be resummed into a power series in the parameter $\h=\frac{
  e\,\exp\sqrt{1-e^2}}{1+\sqrt{1-e^2}}$ with radius of convergence $1$,
\cite[Appendice, p.44]{Ca818},\cite{LC904}, thus redetermining the
D'Alembert's radius of convergence $r^*=0.6627434...$ of the power series
in $e$ (called ``Laplace's limit'')\footnote{This is not the only
  resummation of Lagrange's series: the most famous is perhaps Carlini's
  resummation in terms of the Bessel functions $J_n(z)$, namely $\x=\ell
  +\sum_{n=1}^\infty \frac2n J_n(n e) \sin n\ell$, found at the same time
  by Bessel, \cite{Ca817}, \cite{Ca818}, \cite{Be818}, \cite{Ja850},
  \cite{Co992}.} as the closest point to $0$ of the curve
$|\h|=1$. Furthermore for $e$ real and $e<1$ it is $\h<1$ and an expansion
for the eccentric anomaly is obtained by a series convergent for all
eccentricities $e<1$.

Recently the same formula has been used to study resonant quasi periodic
motions in integrable systems subject to a perturbing potential $\e V$ with
$\e$ small: the resulting tree expansion allowed the study of the series in
cases in which it is likely to be not convergent: in spite of this it has
been shown that a resummation is possible and leads to a representation of
the invariant torus which is analytic in $\e$ in a region of the form,
\cite{GG005e},

\eqfig{250}{110}{ \ins{75}{95}{$ \rm complex \atop \e\rm-plane$}
}{Fig4}{Fig.4} \*\* 

\0where the contact points of the holomorphy region and
the negative real axis have a Lebesgue density point at $0$; and in other
cases to a fractional power series representation, \cite{GGG006}, or also
to a Borel summability in a region with $\e=0$ on its boundary,
\cite{CGGG006}.

The resummation leading to the above result is a typical resummation that 
appears in the eighteenth century analysis and is precisely the one that is
used against the objections of D'Alembert to the vibrating string solution:

\* \0{\it Or je demande si, toutes les fois que dans une formule
  alg\'ebrique il se trouvera par exemple une s\'erie g\'eom\'etrique
  infinie, telle que $1-x+x^2-x^3+\ldots$, on ne sera pas en droit d'y
  substituer $\frac1{1+x}$ quoique cette quantit\'e ne soit r\'eellement
  \'egale \`a la somme de la s\'erie propos\'ee qu'en supposant le dernier
  terme $x^\infty$ nul. Il me semble qu'on ne saurait contester
  l'exactitude d'une telle substitution sans renverser les principes les
  plus communs de l'analyse.}  \* 
%
\0\cite[T.I, p.323]{La867}: the modernity
of this defying viewpoint will not escape the readers' attention.  \*\*

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\SEC{Comments}
\section{\SEC}
\label{sec6}\iniz
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\0(1) The work of Lagrange has raised a large number of comments and deep
critical analysis, here I mention a few: \cite{Fr983}, \cite{Fr985},
\cite{Pa003}, \cite{AFG004}.
%
\\
% 
(2) Although born and raised in Torino there are very few notes written in
Italian: all of them seem to be in mail exchanges; a remarkable one has
been inserted in the Tome VII of the collected works, \cite[T.VII,
  p. 583]{La867}. It is a very deferent letter in which he explains a
somewhat unusual algorithm to evaluate derivatives and integrals. Consider
the series

\be [xy]^m\defi \sum_{k=0}^\infty{m\choose k} [x]^{m-k}[y]^k\Eq{e6.1}\ee
%
(which the 
Lagrange writes without the brackets, added here for clarity). The
positive powers are interpreted as derivatives (or more properly as
infinitesimal increments with respect to the variation of an {\it unspecified}
variable) and the negative as integrals provided at least $x$ or $y$ is an
infinitesimal increment. Thus for $m>0$ and integer the expression is a
finite sum and yields the Leibnitz differentiation rule
$d(xy)^m=\sum_{k=0}^\infty{m\choose k} (dx)^{m-k}(dy)^k$. If $m=-1-k<0$
then $[dx]^{-1-k}$ is interpreted as $k+1$ iteration of indefinite
integration over an infinitesimal interval $dx$; this means $dx^{-1}=x,
dx^{-2}=\frac{x^2}{2dx},\ldots, [dx]^{-1-k} =\frac{x^{k+1}}{(k+1)!\,
  dx^k}$.
 
For $m=-1$ Lagrange gives the example $\int y dx$: from Eq. \equ{e6.1}

\be \eqalign{ &[dx\, y]^{-1}=\sum_{k=0}^\infty (-1)^k [dx]^{-1-k} d^ky\cr
  &\int y dx= \sum_{k=0}^\infty \frac{(-1)^k x^{k+1}}{(k+1)! dx^k} d^ky=
\sum_{k=0}^\infty \frac{(-1)^k x^{k+1}}{(k+1)!}\frac{d^ky}{dx^k}
\Eq{e6.2}\cr} \ee
%
(a relation that the 18 years old Lagrange attributes to ``Giovanni
Bernoullio'', 1694). A further example is worked out:

\be [dx dy]^{-2}=\sum_{k=0}^\infty {-2\choose k} [dx]^{-2-k} d^{k+1}y
=\sum_{k=0}^\infty(-1)^k\frac{k+1}{(k+2)!} x^{k+2}\frac{d^{k+1}y}{dx^{k+1}}
\Eq{e6.3}\ee
%
which therefore yields the indefinite integral $\int\int dy\,dx$ which is
shown to be identical to $\int y dx$ simply by differentiating the {\it
  r.h.s.} of \equ{e6.3} and checking its identity with the {\it r.h.s.} of
\equ{e6.2} (Lagrange suggests, equivalently, to differentiate both
equations twice).

The letter is signed {\it Luigi De La Grange}, and
addressed to ``Illustrissimo Signor Da Fagnano'', well known mathematician,
who would soon help Lagrange to get his first paper published.
%
\\ 
%
(3) Consideration of friction is not frequent in the works of Lagrange, it
is mentioned in a remark on the vibrating strings theory, \cite[T.I, p.109,
  241]{La867}, and the analysis on the tautochrone curves in T.II,III. 

It is also considered in the astronomical problems to examine the consequences,
on the variations of the planetary elements, of a small rarefied medium
filling the solar system (if any) in the Tomes VI and VII.
Or in the
influence of friction on the oscillations of a pendulum in the Trait\'e
(XII) and in fluid mechanics problems. 

Although his attention to applications has been constant (for instance
studying the best shape to give to a column to strengthen it) friction
enters only very marginally in the remarkable theory of the anchor
escapement, \cite[T.IV, p.341]{La867}: this is surprising because it is an
essential feature controlling the precision of the clocks and the very
possibility of building them, \cite[Ch.1,Sec.2.17]{Ga985b}.
%
\\
%
(4) Among other applications discussed by Lagrange are problems in Optics,
again referring to the variational properties of light paths, and in
Probability theory.  
\\
% 
(5) Infinitesimals, in the sense of Leibnitz, are pervasive in his work (a
nice example is the 1754 letter to G.C. Fagnano quoted above which shows
that Lagrange learnt very early their use and their formidable power of
easing the task of long algebraic steps (at the time there were already
some objections to their use): this makes reading his papers easy and
pleasant; at the time there were several objections to their use which
eventually lead to the rigorous refoundation of analysis.

However Lagrange himself seems to have realized that something ought to be
done in systematizing the foundations of analysis: the tellingly long title
of his lecture notes {\it Th\'eorie des Fonctions analytiques, contenants
  les principes du calcul differentiel d\'egag\'e de toutes
  consid\'erations d'infiniment petits ou d'evanouissants, de limites ou de
  fluxions et r\'eduits \`a l'analyse alg\'ebrique de quantit\'es finies},
\cite{La797}, and the very first page of the {\it Le\c{c}ons sur le calcul
  des fonctions}, \cite{La806}, are a clear sign of his hidden qualms on
the matter.

The fact that they developed at a late stage of his life, while he
wholeheartedly adopted the Leibnitz methods in his youth, show that the
problem of mathematical rigor had grown, even for the great scientists, to
a point that it was necessary to work more on it.
\\
%
(6) An informative history of his life can be found in the ``Notices sur la
vie et les ouvrages'' in the preface by M. Delambre of the Tome I of the
collected works, \cite{La867} and in the commemoration by P. Cossali at the
University of Padova, \cite{Co813}. If abstraction is made of the bombastic
rethoric celebrating, in the first (quite a) few pages, Eugene Napoleon
and, in the last (quite a) few pages, Napoleon I himself the remaining
about 130 pages of Cossali's {\it Elogio} contain a useful and detailed
summary and evaluation of all the works of Lagrange (also exposed in a
circumvoluted rethorical style).
%\vfill\eject
\def\SEC{References}

\small
%\bibliography{0Bib}
\bibliographystyle{unsrt}

\begin{thebibliography}{10}

\bibitem{La867}
J.L. Lagrange.
\newblock {\em Oeuvres}.
\newblock Gauthiers-Villars, Paris, 1867-1892.

\bibitem{Ga008}
G.~Gallavotti.
\newblock {\em The Elements of Mechanics (II edition);}.
\newblock http://ipparco.roma1.\-infn.it, Roma, 2008 [I edition was Springer
  1984].

\bibitem{Pr009}
E.~Presutti.
\newblock {\em Scaling limits in Statistical Mechanics and Microstructures in
  Continuum Mechanics}.
\newblock Springer, Berlin, 2009.

\bibitem{Ga985b}
G.~Gallavotti.
\newblock Renormalization theory and ultraviolet stability for scalar fields
  via renormalization group methods.
\newblock {\em Reviews of Modern Physics}, 57:471--562, 1985.

\bibitem{Ch846}
{F. Chi\`o}.
\newblock {\em {Ricerche sopra la serie di Lagrangia, Memoria I}}.
\newblock {Giuseppe Pomba e C. Editori}, Torino, 1846.

\bibitem{Da743}
J.B. D'Alembert.
\newblock {\em {Trait\'e de Dynamique}}.
\newblock David l'ain\'e, Paris, 1743.

\bibitem{LA927}
T.~Levi-Civita and U.~Amaldi.
\newblock {\em Lezioni di Meccanica Razionale}.
\newblock Zanichelli, Bologna, 1927, ristampe 1974, 2013.

\bibitem{Li967}
A.~Lichnerowicz.
\newblock {La M\'ecanique \`a l'Acad\'emie}.
\newblock In Academie des Sciences, editor, {\em {Troisi\`eme Centenaire}},
  pages 213--238. Gauthier-Villars, Paris, 1967.

\bibitem{Eu853}
L.~Euler.
\newblock {\em {Mechanik}}.
\newblock (ed. J.Ph.Wolfers), C. A. Koch's Verlagshandlung, Greifswald, 1853.

\bibitem{FGG013}
J.P. Fran{\c{c}}oise, P.~Garrido, and G.~Gallavotti.
\newblock Rigid motions: action-angles, relative cohomology and polynomials
  with roots on the unit circle.
\newblock {\em Journal of Mathematical Physics}, 54:032901, 2013.

\bibitem{Bo866}
L.~Boltzmann.
\newblock {\em {\"U}ber die mechanische {B}edeutung des zweiten {H}auptsatzes
  der {W\"a}rme\-theorie}, volume 1, p.9 of {\em {W}is\-sen\-schaft\-li\-che
  {A}bhandlungen, ed. {F}. {H}asen{\"o}hrl}.
\newblock Chel\-sea, New York, 1968.

\bibitem{Cl871}
R.~Clausius.
\newblock {Ueber die Zur{\"u}ckf{\"u}hrung des zweites Hauptsatzes der
  mechanischen W{\"a}rmetheorie und allgemeine mechanische Prinzipien}.
\newblock {\em Annalen der Physik}, 142:433--461, 1871.

\bibitem{Ga994b}
G.~Gallavotti.
\newblock {Twistless KAM tori}.
\newblock {\em Communications in Mathematical Physics}, 164:145--156, 1994.

\bibitem{Ca818}
F.~Carlini.
\newblock {Ricerche sulla convergenza della serie che serve alla soluzione del
  problema di Keplero}.
\newblock {\em Effemeridi Astronomiche di Milano, German translation by Jacobi
  as DOI: 10.1002/asna.18500301502, p.213-228}, 10:2--48, 1818.

\bibitem{LC904}
T.~Levi-Civita.
\newblock {Sopra l'equazione di Kepler}.
\newblock {\em Rendiconti dell'Accademia dei Lincei}, {XIII}, also in {Opere,
  T.2}:260--268, 1904.

\bibitem{Ca817}
F.~Carlini.
\newblock {Ricerche sulla convergenza della serie che serve alla soluzione del
  problema di Keplero}.
\newblock {\em Giornale di Fisica, Chimica e Storia Naturale German translation
  by Jacobi as DOI: 10.1002/asna.18500301402, p.197-212}, 10:458--460, 1817.

\bibitem{Be818}
F.W. Bessel.
\newblock {\em Briefwechsel zwischen W. Olbers und F.W. Bessel}, volume T.2.
\newblock Avenarius und Mendelsohnn, 1818.

\bibitem{Ja850}
C.G.J. Jacobi.
\newblock {Untersuchungen die Convergenz der Reihe durch welche das Kepler'sche
  Problem gelost wird}.
\newblock {\em {Astronomische Nachrichten}}, 50:197--244, 1850.

\bibitem{Co992}
P.~Colwell.
\newblock {Bessel Functions and Kepler's Equation}.
\newblock {\em The Anerican Mathematical Monthly}, pages 45--48, 1992.

\bibitem{GG005e}
G.~Gallavotti and G.~Gentile.
\newblock Degenerate elliptic resonances.
\newblock {\em Communications in Mathematical Physics}, 257:319--362, 2005.

\bibitem{GGG006}
G.~Gallavotti, G.~Gentile, and A.~Giuliani.
\newblock {Fractional Lindstedt series}.
\newblock {\em Journal of Mathematical Physics}, 47:012702 (+33), 2006.

\bibitem{CGGG006}
O.~Costin, G.~Gallavotti, G.~Giuliani, and G.~Gentile.
\newblock {Borel summability and Lindstedt series}.
\newblock {\em Communications in Mathematical Physics}, 269:175--193, 2006.

\bibitem{Fr983}
C.~Frazer.
\newblock {J. L. Lagrange's Early Contributions to the Principles and Methods
  of Mechanics}.
\newblock {\em Archive for History of Exact Sciences}, 28:197--241, 1983.

\bibitem{Fr985}
C.~Frazer.
\newblock {J. L. Lagrange's Changing Approach to the Foundations of the
  Calculus of Variations}.
\newblock {\em Archive for History of Exact Sciences}, 32:151--191, 1985.

\bibitem{Pa003}
M.~Panza.
\newblock The origins of analytic mechanics in the 18th century.
\newblock {\em in {A history of analysis}, ed. H.N. Jahnke, AMS}, pages
  137--153, 2003.

\bibitem{AFG004}
T.~Archibald, C.~Frazer, and I.~Grattan-Guinness.
\newblock {\em The History of Differential Equations, 1670–-1950}, volume
  51/2004.
\newblock {Reports of the Mathematisches Forschungsinstitut Oberwolfach}, 2004.

\bibitem{La797}
J.L. Lagrange.
\newblock {\em {Th\'eorie des Fonctions analytiques}}.
\newblock {Journal de l'\'Ecole Polytechnique}, Imprimerie de la Republique,
  Paris, 1797 (V).

\bibitem{La806}
J.L. Lagrange.
\newblock {\em {Le\c{c}ons sur le calcul des fonctions}}.
\newblock Courcier, Paris, 1806.

\bibitem{Co813}
P.~Cossali.
\newblock {\em Elogio di Luigi Lagrange}.
\newblock Tipografia Bettoni, Padova, 1813.

\end{thebibliography}


\*
\0{\tt http://ipparco.roma1.infn.it\\
giovanni.gallavotti@roma1.infn.it}

\end{document}

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0 90 punto 70 90 punto 120 60 punto 180 30 punto
160 130 punto 200 110 punto 240 170 punto 
240 130 punto 240 90 punto 240 0 punto 240 30 punto 200 60 punto 
240 70 punto 240 50 punto 70 90 0 90 frecciac
120 60 70 90 frecciac
160 130 70 90 frecciac
200 110 160 130 frecciac
240 170 160 130 frecciac 240 130 200 110 frecciac
240 90 200 110 frecciac
180 30 120 60 frecciac 240 0 180 30 frecciac %240 0 120 60 frecciac
240 30 120 60 frecciac 200 60 120 60 frecciac
240 70 200 60 frecciac 240 50 200 60 frecciac
grestore


---------------1305141520958
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Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="Fig4.eps"

/origine1assexper2pilacon|P_2-P_1| { 
4 2 roll 2 copy translate exch 4 1 roll sub 
3 1 roll exch sub 2 copy atan rotate 2 copy 
exch 4 1 roll mul 3 1 roll mul add sqrt } def 
 
/punta0 {0 0 moveto dup dup 0 exch 2 div lineto 0 
lineto 0 exch 2 div neg lineto 0 0 lineto fill 
stroke } def 

/dirpunta{% x1 y1 x2 y2 
gsave origine1assexper2pilacon|P_2-P_1| 
0 translate 7 punta0 grestore} def 

/puntatore {4 copy 4 copy 4 copy 4 copy 
dirpunta 4 2 roll moveto lineto stroke} def 

/punto { % x y punto
gsave 3 0 360 newpath arc fill stroke grestore} def


/R {100} def
/h {1} def
/p1 {20} def
/p2 {23} def
/p3 {30} def
/p4 {34} def
/p5 {40} def
/p6 {45} def
/p7 {60} def
/p8 {66} def
/p9 {80} def
/p10 {88} def


0.5 0.5 scale
0 R translate

R 0 moveto
0 h p1 {dup dup mul exch neg R add exch R div lineto} for 
R p1 p2 add 2 div sub 0 lineto
p2 R sub neg 
p2 dup mul R div lineto 
p2 h p3 {dup dup mul exch neg R add exch R div lineto} for 
R p3 p4 add 2 div sub 0 lineto
p4 R sub neg 
p4 dup mul R div lineto 
p4 h p5 {dup dup mul exch neg R add exch R div lineto} for 
R p5 p6 add 2 div sub 0 lineto
p6 R sub neg 
p6 dup mul R div lineto 
p6 h p7 {dup dup mul exch neg R add exch R div lineto} for 
R p7 p8 add 2 div sub 0 lineto
p8 R sub neg 
p8 dup mul R div lineto 
p8 h p9 {dup dup mul exch neg R add exch R div lineto} for 
R p9 p10 add 2 div sub 0 lineto
p10 R sub neg 
p10 dup mul R div lineto 
p10 h R {dup dup mul exch neg R add exch R div lineto} for 
0 100 moveto
105 85 
180 50
180 0
curveto
R 0 
lineto
0.7 setgray fill
stroke

1 -1 scale
R 0 moveto
0 h p1 {dup dup mul exch neg R add exch R div lineto} for 
R p1 p2 add 2 div sub 0 lineto
p2 R sub neg 
p2 dup mul R div lineto 
p2 h p3 {dup dup mul exch neg R add exch R div lineto} for 
R p3 p4 add 2 div sub 0 lineto
p4 R sub neg 
p4 dup mul R div lineto 
p4 h p5 {dup dup mul exch neg R add exch R div lineto} for 
R p5 p6 add 2 div sub 0 lineto
p6 R sub neg 
p6 dup mul R div lineto 
p6 h p7 {dup dup mul exch neg R add exch R div lineto} for 
R p7 p8 add 2 div sub 0 lineto
p8 R sub neg 
p8 dup mul R div lineto 
p8 h p9 {dup dup mul exch neg R add exch R div lineto} for 
R p9 p10 add 2 div sub 0 lineto
p10 R sub neg 
p10 dup mul R div lineto 
p10 h R {dup dup mul exch neg R add exch R div lineto} for 
0 100 moveto
105 85 
180 50
180 0
curveto
R 0 
lineto
1 -1 scale
0.7 setgray fill
stroke

0 setgray

R 0 moveto
0 h p1 {dup dup mul exch neg R add exch R div lineto} for 
R p1 p2 add 2 div sub 0 lineto
p2 R sub neg 
p2 dup mul R div lineto 
p2 h p3 {dup dup mul exch neg R add exch R div lineto} for 
R p3 p4 add 2 div sub 0 lineto
p4 R sub neg 
p4 dup mul R div lineto 
p4 h p5 {dup dup mul exch neg R add exch R div lineto} for 
R p5 p6 add 2 div sub 0 lineto
p6 R sub neg 
p6 dup mul R div lineto 
p6 h p7 {dup dup mul exch neg R add exch R div lineto} for 
R p7 p8 add 2 div sub 0 lineto
p8 R sub neg 
p8 dup mul R div lineto 
p8 h p9 {dup dup mul exch neg R add exch R div lineto} for 
R p9 p10 add 2 div sub 0 lineto
p10 R sub neg 
p10 dup mul R div lineto 
p10 h R {dup dup mul exch neg R add exch R div lineto} for 
0 100 moveto
105 85 
180 50
180 0
curveto
R 0 
lineto
stroke

1 -1 scale
R 0 moveto
0 h p1 {dup dup mul exch neg R add exch R div lineto} for 
R p1 p2 add 2 div sub 0 lineto
p2 R sub neg 
p2 dup mul R div lineto 
p2 h p3 {dup dup mul exch neg R add exch R div lineto} for 
R p3 p4 add 2 div sub 0 lineto
p4 R sub neg 
p4 dup mul R div lineto 
p4 h p5 {dup dup mul exch neg R add exch R div lineto} for 
R p5 p6 add 2 div sub 0 lineto
p6 R sub neg 
p6 dup mul R div lineto 
p6 h p7 {dup dup mul exch neg R add exch R div lineto} for 
R p7 p8 add 2 div sub 0 lineto
p8 R sub neg 
p8 dup mul R div lineto 
p8 h p9 {dup dup mul exch neg R add exch R div lineto} for 
R p9 p10 add 2 div sub 0 lineto
p10 R sub neg 
p10 dup mul R div lineto 
p10 h R {dup dup mul exch neg R add exch R div lineto} for 
0 100 moveto
105 85 
180 50
180 0
curveto
R 0 
lineto
1 -1 scale
stroke

R R neg R R puntatore
0 0 R 2 mul 0 puntatore

---------------1305141520958--
