tex file has 48712 bytes
dvi file has 72012 bytes
ASCII character 33!
ASCII character 34"
ASCII character 35#
numberofcharacters 48712
BODY
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\magnification=\magstep1\hoffset=0.cm
\voffset=-0.5truecm\hsize=15.8truecm\vsize=23.truecm
\baselineskip=14pt plus0.1pt minus0.1pt \parindent=12pt
\lineskip=4pt\lineskiplimit=0.1pt \parskip=0.1pt plus1pt
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%GRECO%%%%%%%%%
%
\let\a=\alpha \let\b=\beta \let\g=\gamma \let\d=\delta \let\e=\varepsilon
\let\z=\zeta \let\h=\eta \let\th=\vartheta\let\k=\kappa \let\l=\lambda
\let\m=\mu \let\n=\nu \let\x=\xi \let\p=\pi \let\r=\rho
\let\s=\sigma \let\t=\tau \let\iu=\upsilon \let\f=\varphi\let\c=\chi
\let\ps=\psi \let\o=\omega \let\y=\upsilon
\let\G=\Gamma \let\D=\Delta \let\Th=\Theta \let\L=\Lambda\let\X=\Xi
\let\P=\Pi \let\Si=\Sigma \let\F=\Phi \let\Ps=\Psi \let\O=\Omega
\let\U=\Upsilon
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%% EQUAZIONI CON NOMI SIMBOLICI
%%%
%%% per assegnare un nome simbolico ad una equazione basta
%%% scrivere \Eq(...) o, in \eqalignno, \eq(...) o,
%%% nelle appendici, \Eqa(...) o \eqa(...):
%%% dentro le parentesi e al posto dei ...
%%% si puo' scrivere qualsiasi commento; per avere i nomi
%%% simbolici segnati a sinistra delle formule si deve
%%% dichiarare il documento come bozza, iniziando il testo con
%%% \BOZZA. Sinonimi \Eq,\EQ,\EQS; \eq,\eqs; \Eqa,\Eqas;\eqa,\eqas.
%%% All' inizio di ogni paragrafo si devono definire il
%%% numero del paragrafo e della prima formula dichiarando
%%% \numsec=... \numfor=... (brevetto Eckmannn).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\global\newcount\numsec\global\newcount\numfor
\gdef\profonditastruttura{\dp\strutbox}
\def\senondefinito#1{\expandafter\ifx\csname#1\endcsname\relax}
\def\SIA #1,#2,#3 {\senondefinito{#1#2}
\expandafter\xdef\csname #1#2\endcsname{#3} \else
\write16{???? ma #1,#2 e' gia' stato definito !!!!} \fi}
\def\etichetta(#1){(\veroparagrafo.\veraformula)
\SIA e,#1,(\veroparagrafo.\veraformula)
\global\advance\numfor by 1
% \write15{\string\FU (#1){\equ(#1)}}
\write16{ EQ \equ(#1) == #1 }}
\def \FU(#1)#2{\SIA fu,#1,#2 }
\def\etichettaa(#1){(A\veroparagrafo.\veraformula)
\SIA e,#1,(A\veroparagrafo.\veraformula)
\global\advance\numfor by 1
% \write15{\string\FU (#1){\equ(#1)}}
\write16{ EQ \equ(#1) == #1 }}
\def\BOZZA{\def\alato(##1){
{\vtop to \profonditastruttura{\baselineskip
\profonditastruttura\vss
\rlap{\kern-\hsize\kern-1.2truecm{$\scriptstyle##1$}}}}}}
\def\alato(#1){}
\def\veroparagrafo{\number\numsec}\def\veraformula{\number\numfor}
\def\Eq(#1){\eqno{\etichetta(#1)\alato(#1)}}
\def\eq(#1){\etichetta(#1)\alato(#1)}
\def\Eqa(#1){\eqno{\etichettaa(#1)\alato(#1)}}
\def\eqa(#1){\etichettaa(#1)\alato(#1)}
\def\eqv(#1){\senondefinito{fu#1}$\clubsuit$#1\else\csname fu#1\endcsname\fi}
\def\equ(#1){\senondefinito{e#1}\eqv(#1)\else\csname e#1\endcsname\fi}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\let\EQS=\Eq\let\EQ=\Eq
\let\eqs=\eq
\let\Eqas=\Eqa
\let\eqas=\eqa
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\footline={\hss\tenrm\folio\hss}
\def\data{\number\day/\ifcase\month\or gennaio \or febbraio \or marzo \or
aprile \or maggio \or giugno \or luglio \or agosto \or settembre
\or ottobre \or novembre \or dicembre \fi/\number\year;\,\the\time}
%\tempo=\number\time\divide\tempo by 60}
%\newcount\tempo
\setbox200\hbox{$\scriptscriptstyle \data $}
\newcount\pgn \pgn=1
\def\foglio{\number\numsec:\number\pgn
\global\advance\pgn by 1}
\def\foglioa{A\number\numsec:\number\pgn
\global\advance\pgn by 1}
%\footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm
%\foglio\hss}
%\footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm
%\foglioa\hss}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DEFINIZIONI VARIE
%
\def\V#1{\vec#1}\let\dpr=\partial\let\ciao=\bye
\let\io=\infty\let\i=\infty
\let\ii=\int\let\ig=\int
\def\media#1{\langle{#1}\rangle}
\def\guida{\leaders\hbox to 1em{\hss.\hss}\hfill}
\def\tende#1{\vtop{\ialign{##\crcr\rightarrowfill\crcr
\noalign{\kern-1pt\nointerlineskip}
\hglue3.pt${\scriptstyle #1}$\hglue3.pt\crcr}}}
\def\otto{{\kern-1.truept\leftarrow\kern-5.truept\to\kern-1.truept}}
\def\pagina{\vfill\eject}\def\acapo{\hfill\break}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%LATINORUM
\def\etc{\hbox{\it etc}}\def\eg{\hbox{\it e.g.\ }}
\def\ap{\hbox{\it a priori\ }}\def\aps{\hbox{\it a posteriori\ }}
\def\ie{\hbox{\it i.e.\ }}
\def\fiat{{}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%DEFINIZIONI LOCALI
\def\AA{{\V A}}\def\aa{{\V\a}}\def\bv{{\V\b}}\def\dd{{\V\d}}
\def\ff{{\V\f}}\def\nn{{\V\n}}\def\oo{{\V\o}}
\def\zz{{\V z}}\def\FF{{\V F}}\def\xx{{\V x}}
\def\yy{{\V y}} \def\q{{q_0/2}}\let\lis=\overline\def\Dpr{{\V\dpr}}
\def\mm{{\V m}}
\def\ff{{\V\f}}\def\zz{{\V z}}\def\mb{{\bar\m}}
\def\UU{{\cal U}}\def\BB{{\cal B}}\def\bB{{\V\b}}
\def\DD{{\cal D}}\def\CC{{\cal C}}\def\II{{\cal I}}
\def\EE{{\cal E}}\def\MM{{\cal M}}\def\LL{{\cal L}}
\def\Sol{{\cal S}}\def\TT{{\cal T}}\def\RR{{\cal R}}
\def\WI{{W_+}}\def\WS{{W_-}}\def\sign{{\rm sign\,}}
\def\BAK{{{{\lis A}^2\over\lis K}}}
\def\thb{{{\bar \th}}}\def\fb{{{\bar \f}}}\def\psb{{{\bar \ps}}}
\def\bak{{\bar A^2\over\bar K}}
\def\={{ \; \equiv \; }}\def\su{{\uparrow}}\def\giu{{\downarrow}}
\def\mb{{\bar\m}}
\def\kb{{\bar\k}}\def\rb{{\bar\r}}\def\xb{{\bar\x}}
\def\cb{{\bar c}}\def\mb{{\bar\m}}
\def\xc{{\hat\x}}
\def\ct{{\tilde \CC}}
\def\rt{{\tilde\r}}\def\mt{{\tilde\m}}\def\kt{{\tilde\k}}
\let\ch=\chi
\def\PP{{\cal P}}
\def\bb{{\V\b}}
\def\Im{{\rm\,Im\,}}\def\Re{{\rm\,Re\,}}
\def\nn{{\V\n}}\def\lis#1{{\overline #1}}\def\q{{{q_0/2}}}
\def\atan{{\,\rm arctg\,}} \def\0{{\V0}}\def\pps{{\V\ps{\,}}}
\def\bul{{l\bar u\e^{-2}}} \def\hB{{\hat B}}
\def\*{\vskip3pt}\def\FF{{\V F}} \def\BMK{{\lis M^2\over \lis K}}
\def\arctg{{\rm arctg\,}}\def\4{{1\over4}}\def\2{{1\over2}}\def\8{{1\over8}}
\def\pd{{\h^{1/2}}}\def\md{{\h^{-1/2}}}
%%%%%%%%%%%%%%%%%% Numerazione futura
%%%% da commentare in caso sia indesiderata
%%%%%%%%%%%%%%%%%%
\openin14=houches.aux \ifeof14 \relax \else
\input houches.aux \fi
%\openout15=houches.aux
%%%%%%%%%%%%%%%%%%%%%%
%\input hhfiat
%\BOZZA
\footline={\hss\tenrm\folio\hss}
\fiat\relax
\vskip0.pt
%
{\bf Drift and diffusion in phase space. An application to celestial
mechanics.}
\vskip1.truecm\numsec=1\numfor=1
Giovanni Gallavotti\footnote{${}^1$}{Dip. di Fisica,
Universit\`a di Roma, La Sapienza, P. Moro 5, 00185 Roma, Italia.
{\it Lecture delivered at the Les Houches NATO-ASI 910820, meeting on
{\it Cellular automata and cooperative systems}, June 21- july 3, 1992:
it describes joint work (see [CG] for technical details) in
collaboration with Luigi Chierchia}, Dip. di Matematica, $II^a$
Universit\`a di Roma, via Raimondo, 00173 Roma, Italia}.
%
\vskip1.truecm
{\it Abstract: some results on the theory of Arnold's diffusion and
an application to the motion of non spherical heavenly bodies revolving
about a point in conic sections.}
\vskip1.5truecm
{\it\S1 Diffusion paths.}
\vskip1.truecm\numsec=1\numfor=1
We consider an integrable system with action angle coordinates
$(\AA,\aa)$, with $\AA$ in a sphere $V\subset R^l$, and
$\aa\in T^l$ ($l$-dimensional torus).
Let $h(\AA)$ be the hamiltonian. Let $h$ be analytic in $V$ and let
$f(\AA,\aa)$ be a perturbation analytic in $\AA,\aa$. Let $\r$ be the
amount by which the actions can be complexified and $\x$ be the amount
by which the angles can be complexified still keeping $(\AA,\aa)$ in
the holomorphy domain of $h$, $f$. The hamiltonian will be:
%
$$H(\AA,\aa)=h(\AA)+\e f(\AA,\aa)\Eq(1.1)$$
%
and the vector $\oo(\AA)\=\dpr_\AA h(\AA)$ will be called the {\it
rotation vector}. We say that $h$ is anisochronous if
$\det\dpr_{\AA^2}^2 h(\AA)\ne 0$.
Let $\LL$ be a smooth curve on the $\nn_0$-resonance:
$\RR_{\nn_0}\=\{\AA|\,\oo(\AA)\cdot\nn_0=0\}$, where $\nn_0$ is a non zero
integer components vector, and on a constant energy surface, $h(\AA)=E$,
of a anisochronous unperturbed hamiltonian.
We denote $s\to \AA(s),\, s\in [0,1]$ the parametric equations of the
curve and $D=\max_s |\oo(\AA(s))|$. Then:
\*{\it Definiton $1_1$}: {\sl The curve $\LL$, with energy $E$ and on the
resonance $\nn_0$, is a diffusion path if there exist $K,t,\t>0$ such
that:
%
$${\rm measure\ of}\{s|\,|\oo(\AA(s))\cdot\nn|^{-1}\le C|\nn|^\t\}\ge
\big(1-{K\over(DC)^{1/t}}\big)\Eq(1.2)$$
%
for all $C>0$ and for all $\nn$ not parallel to the resonance numbers
$\nn_0$.}\*
In many cases one is interested in forced systems rather than in
autonomous systems like the above \equ(1.1). Such systems are
perturbations of an integrable hamiltonian supposed to be linear in
(say) $A_1$: $h(\AA)=\o A_1+h'(\AA')$, with $\AA'=(A_2,\ldots,A_l)$,
still supposing $\det \dpr_{\AA'}^2 h'(\AA')\ne0$. The perturbation
$f$ is supposed to be $A_1$ independent (so that the variables
$A_1,\a_1$ describe the motion of a (perfect) clock). The regularity
properties on $h$ and $f$ will be the same as in the autonomous case,
and the hamiltonian is:
%
$$H(\AA,\aa)=\o A_1+h'(\AA')+\e f(\AA',\a_1,\aa')\Eq(1.3)$$
%
where $\aa'=(\a_2,\ldots,\a_l)$.
Let $\LL$ be a smooth curve in the $\nn_0\=(\n_1,\nn')$-resonance:
$\RR_{\nn_0}\{\AA'|\,\o\n_1+\oo'(\AA')\cdot\nn'_0=0\}$, where $\nn_0$
is a non zero integer components vector. The curve $\LL$ will be also
thought as a curve in $V$ by defining $A_1$ so that the energy $h(\AA)$
is a given constant $E$.
\*{\it Definition $1_2$}: {\sl The curve $\LL$, on the resonance
$\nn_0$, is a diffusion path if there exist $K,t,\t>0$ such that
\equ(1.2) holds for all $C>0$ and for all integer vectors $\nn$ not
parallel to the resonance numbers $\nn_0$.}\*
Two more classes of systems are interesting: the \ap\ unstable systems
(as opposed to the above two systems that will be called \ap\ stable),
autonomous or forced. In the new cases we shall call the canonical
variables $(\AA,I)\in V$ and $(\aa,\f)\in T^l$. In the autonomous case
the hamiltonian is:
%
$$h(\AA)+{I^2\over 2 J}+J g^2(\cos\f-1)+\e f(\AA,I,\aa,\f)\Eq(1.4)$$
%
The $h,f$ will be supposed to fulfill the above regularity and non
degeneracy conditions.
Let $\LL$ be a smooth curve on a constant energy surface $h(\AA)=E$.
Then:
\*{\it Definiton $1_3$}: The curve $\LL$, with energy $E$ is a
diffusion path if there exist $K,t,\t>0$ such that \equ(1.2) holds for
all $C>0$ and for all integer vectors $\nn$.\*
In some sense in this case the role of the resonance is plaid by the
set $(I,\f)=(0,0)$.
Finally the corresponding non autonomous case is defined
by a hamiltonian:
%
$$H(\AA,I,\aa,\f)=\o A_1+h'(\AA')+{I^2\over 2J}+Jg^2(\cos\f-1)+
\e f(\AA',I,\aa,\f)\Eq(1.5)$$
%
where $\AA=(A_1,\AA')\in V\subset R^{l-1}$,
$\aa=(\f_1,\aa')$ and $h'(\AA')$ verifying $\det\dpr_{\AA^{'2}}
h'(\AA')\ne0$. The diffusion path $\LL$ is defined to be a
curve in the space $R^{l-1}$ such that: the curve $\LL$ will be also
thought as a curve in $R\times V$ by defining $F_1$ so that the energy
$h(\AA)$ is a given constant $E$.
\*{\it Definition $1_4$}: {\sl The curve $\LL$ is a
diffusion path if there exist $K,t,\t>0$ such that \equ(1.2)
holds for all $C>0$ and for all integer vectors $\nn\ne\V0$.}\*
More generally one can consider paths $\LL$ on the surfaces of constant
$h(\AA)$ or $h'(\AA')$ without references to resonances.
\vskip1.truecm
{\it\S2 General results.}
\vskip1.truecm\numsec=2\numfor=1
The above definitions will permit us to formulate the results known in the
theory of drift and diffusion after introducing the following concept:
\*{\it Definition $2$}: {\sl A path $\LL$: $\{s\to \AA(s)\}$ is
{\it open for diffusion by a perturbation} $f$ if, given any "tube $U$
around $\LL$" (\ie an open vicinity of $\LL$), for all $\e$ small
enough, {\it but non zero}, one can determine initial data
$(\AA_\e,\aa_\e)$ (and $(I_\e,\f_\e)$ as well,
in the \ap\ unstable cases), such
that, denoting $(\AA_{\e,t},\aa_{\e,t})$ their evolution under the
hamiltonian flow with hamiltonian $H_\e$, there exists $T_\e$ and:
%
$$\eqalign{
1:\kern1.5truecm&\AA_\e\tende{\e\to0}\AA(0)\cr
2:\kern1.5truecm& \AA_{\e,t}\in U \quad{\rm for}\quad 0< t< T_\e\cr
3:\kern1.5truecm& \AA_{\e,T_\e}\tende{\e\to0}\AA(1)\cr}\Eq(2.1)$$
%
In the \ap unstable cases one adds the further requirement that:
%
$$I_{\e,0},\f_{\e,0}\tende{\e\to0}(0,0),
\qquad I_{\e,T_\e},\f_{\e,T_\e}\tende{\e\to0}(0,0)\Eq(2.2)$$ }
\*
Hence {\it a diffusion path is open under the perturbation $f$
if it is a path that can be closely followed by the actions
of a true trajectory of the perturbed system}. The latter trajectories
will be called {\it drift} trajectories, see [CG] for a motivation of
the name "diffusion".\*
It is convenient to give the definition of drift for the paths which are
not necessarily diffusion paths: in fact such a requirement would be too
restrictive.
{\bf Question}: can one give sufficient conditions for a path to be open
for diffusion?\*
A consequence of the KAM theory and of the definition of diffusion path is:
\*{\it Theorem 1: If $l=2$ no path can be open for diffusion unless it
reduces to a point}.\*
This is a trivial consequence of definition $1_3$ or $1_4$ in the cases
of \ap unstable systems; but in the case of \ap stable systems it
follows from the KAM theory.
A first positive result concerns \ap unstable systems (see [A], [CG]);
let the separatrix motion of the free pendulum, starting in $\f=\p$ at
$t=0$, be $t\to(I(t),\f(t))$, then:
\*{\it Theorem 2: Consider an a priori unstable hamiltonian (forced or
not). Suppose that the function:
%
$$M_s(\aa)\=\ig_{-\io}^\io[f(I(t),\f(t),\AA_s,\aa+\oo t)-f(0,0,\AA_s,\aa+\oo
t)]\,dt\Eq(2.3)$$
%
is such that the equation $\Dpr_\aa M(\aa)=\V0$ admits a smooth
solution $s\to\aa_s$ which is non degenerate, \ie $\d\=\det\Dpr^2
M(\aa_s)\ne0$, and suppose that $\nn\cdot\oo(\AA(s)\ne0$ for all
integres vectors $\nn$ with $|\nn|< N$ for $N$ suitably large
(depending on $\LL$). Then the perturbation $f$ opens $\LL$ for
diffusion.
Furthermore there exist three constants $T(\LL), b(\LL), c(\LL)$ such
that the time $T_\e$ can be taken for $\e$ small enough:
%
$$T_\e\le T(\LL) e^{b(\LL)\,\e^{-2}},\qquad {\rm for}\qquad
|\e|1/2$, see \S7 in [CG]. {\it However the
constant $a$ in \equ(2.6) is usually much smaller than $1/2$ and only
in very special cases it can be estimated to be $1/2$}, [N]: never
larger!
It is not impossible that, by imposing suitably many {\it ad hoc}
hypotheses on $\LL$ and $f$, one could treat some \ap stable problems. But
it seemed to us that a more interesting attitude would be to study a
particular case which admittedly has some relevance, mathematical and
physical, and which provides a natural selection of special properties
that might make the problem soluble. This is discussed in the next
section.
\vglue1.truecm
{\it\S3 Planetary precession. Existence of drift and diffusion.}
\vskip0.5truecm
\numsec=3\numfor=1
Imagine a planet $\EE$ as a homogeneous rigid body with cylindrical
symmetry. The body surface will be described in polar coordinates by
$\r=R h(\cos\th)$ for some $R$ and some $h$, $R>0,\,00$ in
his theory of lunisolar precession, finally providing a theory for
the equinoxes precession phenomenon discovered by Hipparcus
about two millennia earlier. Here we consider only the case
$k=2$: but it is very likely that what follows does not really require
neither the truncation nor neglecting the higher orders in $(R/a)^2$.
Considering such more general problems should only lead to some (minor)
modifications, except in the case $k=0$, where the result is simply
false (\ie no diffusion can take place) and the case $k=1$ which cannot
be decided by a ``lowest order'' perturbation analysis as, instead, the
cases $k\ge2$ are, (at least if the initial data are chosen as we are
going to do).
To compute the D' Alembert hamiltonian \equ(3.8) we have, of course, to
find how $\cos^2\a$ depends on the canonically conjugated variables
$(K,M,L,B,\g,\f,\ps,\l)$.
Simple spherical trigonometry arguments, (see also
[CG] appendix A8), lead to (setting $\tilde\l_T\=\l_T-\l$):
%
$$\eqalignno{
\cos\a=&\sin(\tilde\l_T-\g)\,\bigl(\cos\f\sin\th\cos\d+\sin\d\cos\th\bigr)
-\cos(\tilde\l_T-\g)
\sin\th\sin\f=\cr
=&\sin(\tilde\l_T-\g)\left((K/M)\left(1-(L/M)^2\right)^{1/2}\cos\f
+(L/M)\left(1-(K/M)^2\right)^{1/2}\right)-\cr
&-\left(1-(L/M)^2\right)^{1/2}\sin\f\cos(\tilde\l_T-\g)
\equiv s(\k\n c_\f+\m\s)-\n s_\f c&\eq(3.10)\cr}$$
%
where:
%
$$\matrix{\m \= L/M, \cr \k \= K/M, \cr} \quad \matrix{\n^2 \equiv
1-\m^2, \cr \s^2\equiv 1-\k^2, \cr} \quad \matrix{s
\equiv\sin(\tilde\l_T-\g), \cr c\equiv\cos(\tilde\l_T-\g),\cr} \quad
\matrix{s_\f \= \sin\f, \cr c_\f \= \cos\f. \cr}\Eq(3.11)$$
%
Hence we see that \equ(3.8), as well as the full \equ(3.7), does not
contain $\ps$. Therefore $L$ is a constant of motion and it will be
regarded as a parameter. It has the physical interpretation that
$\lis\n=(1-L^2/\lis M^2)^{1/2}$ is the angle between the spin axis and
the symmetry axis and in the theory of nutation it is called the {\it
eulerian nutation constant}, at the initial {\it epoch}, \ie at a
prefixed reference time, when $\lis M,\lis L,\lis
K,\lis\f,\lis\ps,\lis\g$ are the values of the canonical variables.
Therefore setting:
%
$$V\=\bigl[{(1-e\cos\l_T)^3\over(1-e^2)^3}\cos^2\a\bigr]^{[\le
2]}\=V_0+eV_1+e^2V_2\Eq(3.12)$$
%
and using:
%
$${(1-e\cos\l_T)^3\over(1-e^2)^3}=1+{3\over2}e^2-3e\cos\l+
{9\over2}e^2\cos2\l+\ldots\Eq(3.13)$$
%
one finds that:
%
$$V=\sum_{h=0}^2 e^h\sum_{r,p,j\atop r,\,p+h=even}
\lis B^h_{rpj}\cos(r\g+p\l+j\ch)\Eq(3.14)$$
%
where $\lis B^h_{rpj}$ are suitable coefficients depending on $M,K$.
For instance:
%
$$\lis B^0_{000}\=c_0\={1\over 4} [2 \s^2 \m^2 +
(1+\k^2)\n^2]\Eq(3.15)$$
Thus, setting $\lis E\= \o \BMK$ and (only for convenience of notation)
$\h_2=0$, the full (``order 2") D' Alembert hamiltonian, in the
canonical variables $(\g ,K )$,
$(\chi ,M )$, $(\l ,B )$, takes the form:
%
$$\o_TB+h_0(K,M;\h)+\h f(K,M,\g,\chi,\l;e)\Eq(3.16)$$
%
where:
%
$$h_0\=-\o_TK+{M^2\over2J}+\h\lis E\,c_0(K,M),\qquad
f\=\lis E\, [\lis V_0-c_0+e\lis V_1+e^2\lis V_2] \Eq(3.17)$$
%
with $V_i=V_i(K,M,2\g,\chi,\l)$, $c_0=c_0(K,M)$, and
$\langle\cdot\rangle$ denotes average over the angles and, more
explicitly to give an idea of the result:
%
\def\txt{\textstyle}
$$\eqalignno{\txt
\lis V_0=&\txt\sum c_j\cos j\ch +d_j \cos(2\g +j\ch)&\eq(3.18)\cr
\txt\lis V_1=&\txt\sum\Big(-3c_j\cos(\l +j\ch )
+\2d_j\big(\cos(2\g +\l +j\ch )-7\cos(2\g -\l +j\ch )\big)\Big)\cr
\txt\lis V_2=&\txt\sum\Big[
c_j\Big({3\over2}\cos j\ch +{9\over2}\cos(2\l +j\ch )\Big)+
d_j\Big({17\over2}\cos(2\g -2\l +j\ch )
-{5\over2}\cos(2\g +j\ch )\Big)\Big]\cr}$$
%
where the sums run over $j=-2,\ldots,2$ and the coefficients $\lis
B^h_{rpj}$ (cfr. \equ(3.14)) vanish unless they belong to the
following list where $|j|\le 2$:
%
$$\eqalign{
&\txt \lis B^0_{00j}\= c_j\ ,\quad \lis B^0_{20j}\= d_j,\quad
\lis B^1_{01j}\= -3 c_j\ ,\quad \lis B^1_{211} \= {d_j\over 2}\ ,
\quad \lis B^1_{2-1j}\=-{7\over 2} d_j\cr
&\txt\lis B^2_{00j}\= {3\over 2} c_j\ ,\quad \lis B^2_{02j}\={9\over 2} c_j\ ,
\lis B^2_{20j}\= -{5\over 2} d_j\ , \lis B^2_{2-2j}\={17\over 2} d_j\cr}
\Eq(3.19)$$
%
and the coefficients $c_j,d_j$ are (complicated, see [CG], appendix A14)
functions of $M,K$ and of the parameter $L$, whose explicit expression
will play no role here.
The model \equ(3.17) is a model with {\it two} parameters
$\h,e$ and it is a forced system with forcing described by the
$(B,\l)$ variables. The parameter $\h$ will be regarded as a control
parameter while $e$ wil be regarded as a perturbation parameter. We
consider a diffusion path by selecting the resonance $\nn_0=(0,2,1)$
and the line $\LL$:
%
$$s\in[0,1]\to (s\overline K+(1-s)\overline K_2,\overline M), \quad{\rm
with}\quad {\overline M\over J_3}=\o_D\Eq(3.20)$$
%
with $\lis K_2$ some constant ($<\lis M$) (recall that we are fixing the
"initial data" of "interest" to be $\lis M, \lis K$ for $M,K$).
This corresponds to $2\o_T=\o_D$, \ie to a resonance $2:1$ between the
daily rotation $\o_D$ and the annual rotation $\o_T$ ("two days in one
year").
The latter is a resonance generating, in our theory, the simplest
calculations; more interesting resonances (\eg $1:1$ or $3:2$
corresponding to $\nn_0=(0,-1,1)$ and $\nn_0=(0,3,-2)$) could also be
treated but they require more calculations and, perhaps, keeping a few
of the terms neglected in our simplified model.
One should also remark that the selected resonance is in fact a double
resonance, as also $\nn_0'=(1,-1,1)$ is a resonance vector. Hence
$\bar\LL$ is not a diffusion path in the sense of definition $1_4$ above.
Because of our selection of the resonance, the harmonic $(2\g +\chi )$
will be the angle of the pendulum (see \equ(2.5),\equ(2.6)) describing
the motion transversal to the resonance. Therefore, we perform the
following linear (canonical) change of variables,
%
$$(K ,\g ),\, (M ,\chi ),\, (B ,\l )\, \to (I ,\f ),\,
(A ,\a ),\,(B,\l )\Eq(3.21)$$
%
where $B'$ denotes here the old $B$ and $B$ becomes the corresponding
new variable defined by $B =B'-(A -a )$, with $a \=\lis K - 2 \lis M$ and:
%
$$\eqalign{
&\g = -(\a +\l +\p/2)\cr & \chi = 2(\a +\l )+\f \cr}
\qquad \eqalign{&K =2I - (A -a )+4\o_T J_3\cr&M = I +2 \o_TJ_3\cr}
\Eq(3.22)$$
%
where the shifts have been introduced so that the unstable point of the
pendulum is $\h$--close to $(I ,\f )=(0,0)$ and so that the initial
datum $(\lis K, \lis M)$ corresponds to $(\lis I ,\lis A )\=(0,0)$.
%
The hamiltonian \equ(3.17), in the canonical variables $(I ,\f ) $,
$(A ,\a )$, $(B,\l )$, takes the form (up to a neglected constant):
%
$$H=\o_TB+{I^2\over 2 J_3}% +\h J_3 g^2(\cos\f -1)
+h+\h\Big[ v_0+ ev+ e^2 v_2\Big]\Eq(3.23)$$
%
where $h=h(A ,I )$ is a suitable function.
{\it One can check that, neglecting terms of $O(\n^2)$, the integrable
part of the hamiltonian becomes: $-\h\o A-\h A^2/(8J_3)+\h O(I)+
const$ where $O(I)$ depends on $A,I$ and vanishes for $I=0$,
leading to the standard "D' Alembert equinox precession" $\dot\g'=-
\h\o$}, (see \equ(3.9) and [CG], appendix A6,A7).
For the purposes of illustration we shall further simplify the analysis
by supposing that the term with $O(I)$ is eliminated (a simplified
version still giving the D' Alembert precession as well as all the
conceptual difficulties that are treated in [CG] without introducing the
latter approximation) and that the $c_j,d_j$ coefficients are constants
and, furthermore, that most of them vanish. Namely we shall consider
the model:
%
$$H=\o_T B+\h\o A-{\h A^2\over 8J}+{I^2\over2J}+
\h(V_0+e V_1+e^2 V_2)\Eq(3.24)$$
%
where $J>0,\lis K>0$ and:
%
$$\eqalign{
V_0=&\n \lis d_1\cos(2\g+\chi)+\n c_1\cos\chi\cr
V_1=&\2\n \lis d_1\cos(2\g+\chi+\l)\cr
V_2=&\lis d_2\n^2\cos(2\g+2\l)\cr}\Eq(3.25)$$
%
with $\lis c_1,\lis d_1,\lis d_2>0$ and with $\lis\n\=\n>0$
and the angles have to be evaluated in terms of the new $\a,\f,\l$, see
\equ(3.22).
The above form of the constants reflects the form that they actually
have in the non truncated model (in particular we could absorb $\n$ into
the $c$'s and $d$'s), see [CG]: and we imagine
fixing the constants to be equal to quantities to which the
corresponding non approximated functions are close (see [CG]). For
instance $g^2=\n\lis d_1$ turns out to be: $g^2=\2\n(1+\cos i)\o_T^2$,
if $\cos i$ is $K/\lis M$ at the point of coordinate $K$ (see
\equ(3.6)).
Our simplified model becomes:
%
$$H=\o_T B+\h\o A-{\h A^2\over 8 J_3}+{I^2\over 2 J_3}+
\h g^2(\cos \f-1)+\h\big(\b v_0+ev_1+e^2v_2)\Eq(3.26)$$
%
where $\b\=1$ is introduced for later reference and:
%
$$\eqalign{
g^2=&\2\n(1+\cos i)\o_T^2\cr
v_0=&c_1\cos2(\a+\l)+\f,\qquad
v_1=2d_1\n\cos(\f+\l),\qquad
v_2={3\over4} d_2\n^2\cos2(\a+\f)\cr}\Eq(3.27)$$
%
To fix the time scale of the pendulum $(I,\f)$ in \equ(3.27) to be $\h$
independent we perform a rescaling of the action variables
$(A,I,B)=\h^\2(A',I',B')$ followed by a time rescaling by $\h^{-1}$ and
we transform the above hamiltonian into a new hamiltonian given,
abolishing the primes over the new variables to avoid introducing too
many symbols, by:
%
$$H={\o_T\over\h^\2}B+\h^\2\o A
-{\h A^2\over 8 J_3}+{I^2\over 2 J_3}+
g^2(\cos \f-1)+\big(v_0+ev_1+e^2v_2)\Eq(3.28)$$
%
where the angle variables do not change in the rescaling
transformations.
In the new variables the line $\bar\LL$ becomes simply:
%
$$\LL=\{I=0;\, -\h^{-\2}\lis A< A<\h^{-\2}\lis A\}\Eq(3.29)$$
%
and $\h^\2A$ proceeds linearly from $-\lis A$ to $\lis A$ as $s$ increases
from $0$ to $1$. If we regard the terms $v_0+e v_1+e^2v_2$ as a
perturbationn the it is immediate to check that $\LL$ is a diffusion path
and one can take the constants $K,\t,t$ to be $O(\h^{-\2}),2,7$
respectively.
The \equ(3.28) exhibits the difficulties related to the \ap stable
systems mentioned in \S1: but with a few differences due to the fact
that the system with $\h=e=0$ is quite degenerated
in the sense of \S1: the $h$ is in fact with zero hessian determinant
{\it even in the non forced variables} $K,M$: this is reflected in the
fact that the fast frequences in the \equ(3.28), which could be
considered a normal form of our hamiltonian near the resonance, are not
really fast.
In fact there are two frequences $\o_T\h^{-\2}$, the rotation speed of
the yearly motion (related by the resonance condition to that of the
daily motion), and the $\h^\2\o$, the precession speed which is of order
$\h^\2$ rather than $\h^{-\2}$. And the latter is much slower.
The theory of \S2 cannot be directly applied: for two reasons.
\item{a) }the Melnikov integral \equ(2.3) yields a critical point
$\aa_s\=\V0$, as the parameter $s$ for the $\LL$ curve varies in
$[0,1]$, at $\a=\l=0$ with a hessian matrix determinant
$\d(e,\h)=\det\Dpr_\aa^2 M(\V0)$ which has order $O(e\,
\exp{-\h^{-\2}\p/2g})$. Therefore, unless we are willing to take
$e$ extremely small compared to $\h$, the quantitative condition in
theorem 1 cannot be met.
\item{b) }even if we would be willing to take the step (very
unsatisfactory, because in realistic cases $e$ and $\h$ are not too
different) of assuming that $e$ is much smaller than
$\exp-O(\h^{-\2})$, we still could not apply the theorem 2 by the cause
that the perturbation in \equ(3.28) {\it is not} small as $e\to0$
because of the term $v_0=O(1)$.
\noindent{In} the next section we discuss how the above two
difficulties can be bypassed.
\vskip1.truecm
{\it\S4 Averaging and large homoclinic splitting.}
\numsec=4\numfor=1
\vskip0.5truecm
To use perturbation theory we must first understand why the
perturbation in \equ(3.26) can be in some sense neglected: this is
basically due to the fact that if $e=0$ the angle $\l$ rotates very
fast compared to the angle $\a$ (the first at velocity $O(\h^{-\2})$
and the second at velocity $O(\h^\2)$ relative to the resonance
oscillations which are at velocity of $O(1)$ by our rescalings). Thus
the qualitative ideas on the {\it averaging of the fast variables}
would suggest that the part $ v_0$ (and the part $e v_1$ which also
depends only on fast variables) couple to the other variables with an
{\it effective coupling of order at least as small as $O( e)$}.
We consider, in general, the hamiltonian system:
%
$$H=\md\lis\o_1 B+\pd\lis\o_2 A+\h{A^2\over 2J}+{I^2\over 2J_0}+
J_0g_0^2(\cos\f-1)+\b(F+\m f)\Eq(4.1)$$
%
with the quantities $J,J_0,g_0$ being positive constants and
with the $f,F$ being trigonometric polynomials with constant
coefficients of degree $\le N$. The function $F$ will be supposed to be
"unimodal", \ie to depend on the $\aa=(\l,\a)$, only via the quantity
$\nn_0\cdot\aa$. In other words the $F$ contains only one particular
"fast angle" and its multiples.
Then the following {\it averaging} property holds:
\*{\it Theorem 3: Suppose that $|\m|\le \h^c$, $c>10$, and fix $x>0$ and
$0<\s<1/2$. Then there is a holomorphic canonical map casting the
hamiltonian $H$ in a form (for $\h$ small):
%
$${\lis\o_1 B\over\h^{1/2}}+\h^{1/2}\lis\o_2 A+{\h A^2\over 2\hat
J}+g(\h^{1/2}A,pq)pq+\h^x\hat f(I,A,\f,\l,\a)\Eq(4.2)$$
%
with $\hat J,g,\hat f$ bounded and holomorphic in a domain
$\O(\k,\r,\x,\h^c)$ defined by:
%
$$\big\{|p|,|q|<\k,\, |\Im A|<\r\h^{-1/2},\,|\Im\a_j|<
\x,\,|\m|<\h^c\big\}\Eq(4.3)$$
%
and for $|\b|**0$ and for
$\h_0$ small enough, (one can take $\h^{1/2}_0 \log \h_0^{-1} < d\cdot
x$ for a suitable $d>0$).}\*
One tries to prove the lemma by solving recursively the equations for
the generating function of the transformation that conjugates \equ(4.1)
to \equ(4.2): the variables $p,q$ exist in the case of the free pendulum as
it is well known since Jacobi (see [CG] for an explicit construction of
the $p,q$ variables in the case of the pendulum) and in the perturbed
case they are shown to still exist and to be close to the unperturbed
ones, [CG].
The reason for the validity of \equ(4.2) is then simply that in the
free part of \equ(4.1) (\ie for $\b=0$) no strong resonances occur with
$|\nn|\le O(\h^{-1})$, and all denominators appearing in trying to
apply perturbation theory, as described, are bounded below by $O(\pd)$.
Hence we can proceed to perturbation theory of very large order,
essentially $O(\h^{-1/2})$, after taking advantage in the first step of
{\it large} denominators (of order $O(\h^{-1/2})$, \ie after taking
advantage of the averaging phenomenon) to reduce the size of $F$. At
higher order one still gains as the terms involving only $F$ still have
large divisors, while the ones involving both $f$ and $F$ are much
smaller so that a divisor as big as $\h^\2$ is not sufficient to make
them large.
The method used to deduce \equ(4.2) is very similar to the usual method
developed in the Nekhorossev resonance theory, [N], [BG], [G2]. Note
however, that in \equ(4.2) the slow modes are in the remainder term
(while at first, perhaps, one would expect them to remain of order
$\m$): see [CG] for a proof.
The above theorem can be used, in discussing \equ(3.28), to guarantee
that the line $\LL$ of two dimensional invariant tori that exist if
$\h,e=0$ and correspond to the $I=\f=0$ and $\AA(s)\in \LL$
(parameterized by $\l,\a$) still exists after the perturbation, only
slightly deformed, togheter with their stable and unstable manifolds,
or {\it whiskers}, with the exception, perhaps, of a few value of $s$
clustered in intervals, "gaps", of width of order $e^{x/t}\h^{x/t}$,
where $t$ is the constant introduced in the definition of diffusion
path. In fact the following theorem holds ([CG], \S11, lemma 5, and
remarks following it):
\*{\it Theorem 4: consider a hamiltonian which, along a diffusion
path $\LL$, has a form like \equ(4.1) with $\m$ replacing $\h^x$: then
the set of $s$ for which one cannot construct invariant two dimensional
tori, and the relative stable and unstable manifolds, which are
analytic (as functions of $e,\h$) deformations of the unperturbed
invariant tori (and of their stable and unstable manifolds)
corresponding to the line $\LL$ has measure $O(\m^{1/t})$ if
$\m<\h^c$.}\*
On the other hand the condition, in theorem 2, $\e0$ is "not too small". In fact consider the
hamiltonian \equ(4.1) near the segment $\D$ where $\dpr_A
h=\h^\2\o+\h A/8J_3$ varies, as $A\in \D$, in an interval $\pd[\lis
\o,\tilde\o]$ with $\tilde\o,\lis\o>0$. Then we can prove:
\*{\it Theorem 5: There is $c>0$ such that if $\m=\h^c$ the hamiltonian
system has invariant tori which, if $\h$ is small enough, have whiskers
with a homoclinic splitting $\d$ of $O((\h^{3/2}\m^2)^2)$ as $\h\to0$,
provided a certain sum (see [CG], \S11, lemma 4) does not vanish
accidentally. In the case of the model \equ(3.28) we find (see [CG])
up to a factor $(1+O(\n))$ and to leading order in $\h$:
%
$$\sqrt{-\d}={9\over 2}\,\h^{3/2}\,e^2\,J_3\o_T\,\sin\th\,
{\sin i\cos^2 i \,(1-\cos i)\over(1+ \cos i)}\Eq(4.4)$$
%
where $i$ denotes the inclination (\ie $\cos i=K/\lis M$ at a
generic point of $\LL$, (denoting again by $K$ the original variable of
the $z$-axis spin component); $\cos\th\=\lis L/\lis M$ is the eulerian
nutation constant. The result is uniform for $i$ in a closed interval
strictly contained in $(0,\p/2)$.}\*
Hence in the case \equ(3.24) the homoclinic splitting is a power of
$\h$, while the gaps are smaller than a prefixed power of $\h$,
provided $e=\h^c$ with $c>0$ large enough (we find that $c>92$ is
sufficient)
The above discussion shows the ideas that are behind the proof of:
\*{\it Theorem 6: Consider the precession model and the diffusion line
$\LL$ of \S3. The line $\LL$ physically describes a variation of the
inclination angle of the spin axis at fixed total angular momentum $M$
and in the resonance 2:1 between the annual motion and the daily
motion. Then $\LL$ is open for diffusion under the perturbation given
by the gravitational attraction by the centre of the keplerian orbit of
the body if $e=\h^c$ for some $c$ large enough.}\*
Note that to obtain theorems 3,4 one makes use, in [CG],
of the special structure
of the doubly resonant hamiltonian arising in the celestial problem.
Note also that one has to take $e$ to be a power of $\h$ ($e=\h^c$, with
$c>92$): the larger the power the smaller $\h$ has to be in order that
a given piece of $\LL$ be open for diffusion; on the other hand $c$
cannot be taken too small as one need it to be large to show the
averaging properties in theorem 4. The complete proof can be found in
[CG], where also the more general model \equ(3.7) is considered.
\vskip1.truecm
{\it References.}
\vskip0.5truecm
\item{[A] } Arnold, V.: {\it Instability of dynamical sistems with several
degrees of freedom}, Sov. Mathematical Dokl., 5, 581-585, 1966.
\item{[BG] } Benettin, G., Gallavotti, G.: {\it Stability of motionions near
resonances in quasi-integrable hamiltonian systems}, J. Statistical Physics,
44, 293-338, 1986.
\item{[CG] } Chierchia, L., Gallavotti, G.: {\it Drift and diffusion in
phase space}, preprint, Roma, july 1992; A plain \TeX version of the
preprint can be obtained from the Mathematical Physics Electronic
Archive of Texas University at Austin: for instructions send an E-mail
message to {\tt mp${}_-$arc@math.utexas.edu}.
\item{[DS]} Delshams, A., Seara, M.T.: {\it An asymptotic expression
for the splitting of separatrices of rapidly forced pendulum}, preprint
1991
\item{[G1] } Gallavotti, G.: {\it The elements of Mechanics}, Springer, 1983.
\item{[G2] } Gallavotti, G.: {\it Quasi integrable mechanical systems},
ed. K. Ostewalder, R. Stora, Les Houches, XLIII, 1984, "Ph\'enom\`ens
critiques, Syst\`emes aleatoires, Th\`eories de jauge, Elsevier Science,
1986, p. 539- 624.
\item{[GLT] } Gelfreich, V. G., Lazutkin, V.F., Tabanov, M.B.:{\it
Exponentially small splitting in Hamiltonian systems}, Chaos, 1 (2),
1991.
\item{[Gr] } Graff, S.M.: {\it On the conservation for hyperbolic invariant
tori for Hamiltonian systems}, J. Differential Equations 15, 1-69, 1974.
\item{[La] } Lazutkin, V.F.: {\it Separatrices splitting for standard and
semistandard mappings}, Pre\-pr\-int, 1989.
\item{[L] } de la Place, S.: {\it M\'ecanique C\'eleste}, tome II, book 5,
ch. I, 1799, english translation by Bodwitch, E., reprinted by Chelsea,
1966.
\item{[LW] } de la Llave, R., Wayne, E.: {\it Whiskered Tori},
Nonlinearity, to appear.
\item{[Me] } Melnikov, V.K.: {\it On the stability of the center for
time periodic perturbations}, Trans. Moscow Math Math. Soc., 12, 1-57,
1963.
\item{[N] } Nekhorossev, N.: {An exponential estimate of the time of
stability of nearly integrable hamiltonian systems}, Russian Mathematical
Surveys, 32, 1-65, 1975.
\item{[Nei] } Neihstad, ?.: {\it },
\item{[P] } Poincar\`e, H.: {\it Les M\'ethodes nouvelles de la m\'ecanique
c\'eleste}, 1892, reprinted by Blanchard, Paris, 1987.
\item{[Sv]} Svanidze, N.V.: {\it Small perturbations of an integrable
dynamical system with an integral invariant}, Proceed. Steklov
Institute of Math., 2, 1981
\ciao
ENDBODY
**