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Jacobi.tex
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\documentclass[11pt]{article}
\usepackage{url}
\usepackage{graphicx}
\usepackage{color}
\begin{document}
\section*{Integrable systems yield porisms} % : resonant tori}
It is common knowledge that a $n$-degrees of freedom Hamiltonian system is {\it integrable} if one can find, around any point, local coordinates (local, but not so much) $(I_1, \cdots, I_n, \theta_1, \cdots \theta_n), \, \theta_i \equiv \theta_i + 2\pi$, called {\it action-angle}, in which the Hamiltonian becomes
only a function of the $I_j$, which are then constants of motion. \\
Trajectories wind invariant tori (called Lagrangian) with frequencies $\omega_j = \partial H/\partial I_j$.
In two degrees of freedom, when the ratio $\omega_1/\omega_2 = p/q $ is rational,
all the orbits in such a `resonant torus' are periodic, and they all have the same period. Taking $(p,q)=1$, the fundamental period $T = 2\pi p/\omega_1 = 2\pi q/\omega_2$.
They all wind around $p$ times in $\theta_1$ and $q$ times in $\theta_2$.\\
If this is not a {\it porism}, what is? Most frequency pairs $(\omega_1, \omega_2) $ are not rationally related, but if a single trajectory is closed, all others in the same torus will also be closed, with the {\it same} period.\\
As it is well known, the elliptical billiard is an integrable Hamiltonian system. In the physical plane the trajectories are bouncing straight rays, but they are in fact projections of winding curves in Lagrangian tori.
The nice coordinates for this problem are the confocal conic coordinates. One obtains two possible regimes, separated by trajectories that pass through the foci. Caustics are confocal ellipses or hyperbolae depending on the regime. \\
% As pointed out in several places (see eg. \cite{Dragovic}, \cite{Tabashnikov...})
{\it Poncelet's porism} can be derived as a consequence of the fact that any pair of conics, say two nested ellipses,
can be obtained by a projective transformation from two confocal ones. The oldest reference we found is Darboux \cite{Darboux}, 1870. He atributes to Chasles the proof that the all closed trajectories of the same type have the same length. But this is also an immediate consequence of integrability. Fix the energy, corresponding to unit velocity. Then $L=T$, a common value o all. As we discuss in the historical note, it would be probably evident to Jacobi. %, had him cared about billiards.
\newpage
$$
$$
Include or not? The famous figures in the elliptical biliard, resonant vs. nonresonant
$$
$$
\newpage
\section{Our experiments: summary} % triangles on the elliptical billiard}
We report here some of the dynamic experiments with the elliptical billiard we have been doing. We take the simplest case of frequency ratio 1:3, % $1:3$,
the triangular trajectories. The reader is invited to exert his/hers imagination to imagine this family of triangles (video1) as winding curves in the Lagrange torus in four space. For short we call any triangle in this family a {\it triangle orbit}. A triangle orbit with vertices $A,B,C$ is determined by the position of one of its vertices, say $A$. As one runs $A$ once around the ellipse, the triangle orbit appears three times.\\
One may not know this: for any triangle one can associate more than $ 40,000^+ $ points, called {\it triangle centers}! They must be well defined under Euclidian motions and similitudes. Of course, three of them are the barycenter, the incenter and the circumcenter, that one
learns about in high school. Some of the triangle centers, such as the barycenter, are projectively well defined, but most aren't. An enciclopaedia of the {\it triangle centers} was produced by Kimberling \cite{Kimberling}. See also \cite{eric}.\\
Our project is the following: fix an elliptical billiard by the ratio $a:b$ of the ellipse axis, say with $b=1, a>1$. As one varies the triangle orbits, how does each and every one of their triangle centers move? Which of the resulting curves are in some sense special? Are there interesting bifurcations are $a$ varies?
\newpage
\section*{Historical notes}
\subsection*{The birth of integrable systems, and a missed opportunity}
\noindent It was probably a cold but a starry Friday night in K\"onisberg. A hundred years before Euler was jaywalking around its seven bridges\footnote{Now Kalingrad,
with less bridges, his problem has a solution.}. It was 28 December 1838, just a few days after Christmas. But Carl Gustav Jacob Jacobi felt warm. He was writing
% in finding a coordinate system in the triaxial ellipsoid in which the geodesic equations
%that the geodesics of the triaxial ellipsoid could be explicitly found by quadratures. Two days latter
a note to his twenty year older colleague, Friedrich Wilhelm Bessel, the leading astronomer of %and geodesist of
Prussia:
%, boasting:
\begin{quote}
``The day before yesterday, I reduced to quadrature the problem of geodesic lines on an ellipsoid with three unequal axes. They are the simplest formulas in the world, Abelian integrals, which become the well known elliptic integrals if 2 axes are set equal ".
\end{quote}
%(December 28 1838; quoted from
%wikipedia\footnote{\url{https://en.wikipedia.org/wiki/Geodesics_on_an_ellipsoid#Geodesics_on_a_triaxial_ellipsoid}}).
The short paper he published in Crelle's journal \cite{Jacobi1, Jacobi2} is said to mark the birth of the theory of {\it integrable Hamiltonian systems}. He further developed the ideas, now called the Hamilton-Jacobi method in his 1842-1843 winter lectures (Vorlesungen \cite{Jacobivorlesungen}). Not much later, Neumann (\cite{Neumann}, 1859) and others found some special yet interesting mechanical problems solvable by separation of variables.\\
%We will come back to {\it porisms} in a moment, but let's keep in the track.
Had Jacobi made the smaller axis of the ellipsoid go to zero, he would have gotten an explicit integration of the elliptic billiard, in planar confocal coordinates, exactly the same Euler used for his solution of the fixed two centers problem \cite{Euler}.
But Jacobi did not care about doing it. Most likely what he wanted was to impress his mentor, which was also a geodesist, and of course Gauss, who did only find geodesics for an ellipsoid of revolution. \\
[ At his time mathematicians could not justify an addiction to {\it billiards}. This changed after Birkhoff's Acta Mathematica 1927 paper \cite{Birkhoff1927} and his book \cite{Birkhoff}. Playing pools is now a respectable endeavour \cite{Tabashnikovbook}.
Birkhoff, an analyst by heart, somewhat contradictorily further developed the {\it qualitative methods} introduced by Poincar\'e in his studies about the three body problem. ] \\
Ten years before the starry night where he found the geodesic curves of the triaxial ellipsoid, Jacobi was developing the foundations of elliptic functions. He then published a paper \cite{Jacobi2}, where
he used the the addition properties of his (then) new functions on two geometric problems - the porism by Steiner and specially that of Poncelet. Modern lgebraic geometry owes al lot to Poncelet and Jacobi. \\
It is curious that ten years latter Jacobi did not think of the billiard to revisit Poncelet's porism.
\subsection*{Digression. The Liouville-Arnold theorem and KAM theory}
For quite a long time, in the eighteen and nineteen centuries, astronomers were essentially producing by brute force a lot of integrable systems in order to approximate complicated celestial mechanics problems. \\
Some perceived the difference of resonant vs nonresonant invariant tori when their integrable systems were perturbed. But it
is perhaps fair to say that few people before Poincar\'e\footnote{Perhaps Lagrange himself? History is hard!} thought geometrically of the phase space of Hamiltonian systems as one does now. \\
Poincar\'e agonized when a ``resonant mistake'' was pointed out in his original King Oscar prize paper. In fixing the mistake he unveiled the causes of chaotic behavior in Hamiltonian systems: what happens when perturbations destroy the resonant tori. He certainly guessed KAM theory, which shows the robustness of sufficiently nonresonant tori under small perturbations (\cite{Chenciner 1}, section 2.6; \cite{Chenciner2}).\\
But what is an integrable system and KAM theory after all? In a two page note Liouville (\cite{Liouville}, 1855) explained that for a n-degrees of freedom Hamiltonian system, integrability results from the existence of n independent functions all whose Poisson brackets vanish. Liouville's result was revisited by Mineur in 1935 \cite{Mineur} for use in Bohr-Sommerfeld quantization. But it was in Arnold and Avez book \cite{ArnoldAvez} that the fundamental nature of the result was clearly stated. We feel sorry for Mineur.\\
In the early 1960's Arnold developed perturbation methods outlined by his youth mentor Kolmogorov. Roughly, the idea was to use the
frequencies as the independent quantities instead of the actions. This become known as KAM theory. \\
M is for Moser, who independently studied specially the two degrees of freedom situation. Moser recounted a bit of it in his ``cult article'' ({\it Is the solar system stable?}, \cite{Moser}) that appeared in the first issue of Math. Intelligencer. \\
Looking up the story would lead as astray, but we can refer to \cite{story}.
\newpage
\begin{figure}[b]
\centering
\includegraphics[scale=0.4]{letterjacobi-bessel.png}
%\caption[Matrioscas]{https://www.fromrussia.com/russian-dolls/nesting-dolls-matryoshkas}
\label{fig:letter}
\end{figure}
\bibliography{referenciasjair}
\bibliographystyle{ieeetr}
% \end{quote}
% http://thebestbilliards.com/index.php
\end{document}
tp://olivernash.org/2018/07/08/poring-over-poncelet/
https://en.wikipedia.org/wiki/Jean-Victor_Poncelet 1788 Ð 22 December 1867
https://en.wikipedia.org/wiki/Feuerbach_point
https://en.wikipedia.org/wiki/Karl_Wilhelm_Feuerbach
https://en.wikipedia.org/wiki/Nagel_point
https://en.wikipedia.org/wiki/Christian_Heinrich_von_Nagel
https://en.wikipedia.org/wiki/Triangle_center
--
https://en.wikipedia.org/wiki/University_of_Knigsberg
https://en.wikipedia.org/wiki/Seven_Bridges_of_Knigsberg
December 28 1838
The day before yesterday, I reduced to quadrature the problem of geodesic lines on an ellipsoid with three unequal axes. They are the simplest formulas in the world, Abelian integrals, which become the well known elliptic integrals if 2 axes are set equal.
Knigsberg, 28th Dec. '38. It was a Friday.
https://en.wikipedia.org/wiki/Knigsberg
(Few traces of the former Knigsberg remain today. After 1945 Kaliningrad.
https://en.wikipedia.org/wiki/University_of_Knigsberg
https://en.wikipedia.org/wiki/Carl_Gustav_Jacob_Jacobi 1804-1851
https://en.wikipedia.org/wiki/Friedrich_Bessel
https://en.wikipedia.org/wiki/Geodesics_on_an_ellipsoid#Geodesics_on_a_triaxial_ellipsoid
page 385 Gesammelte werke, Volume 7
Carl Gustav Jakob Jacobi
G. Reimer, 1891
----
OK REFERENCIADOS
Jacobi, C. G. J. (1839). "Note von der geodtischen Linie auf einem Ellipsoid und den verschiedenen Anwendungen einer merkwrdigen analytischen Substitution" [The geodesic on an ellipsoid and various applications of a remarkable analytical substitution]. Journal fr die Reine und Angewandte Mathematik (Crelles Journal) (in German). 19 (19): 309Ð313. doi:10.1515/crll.1839.19.309.
Letter to Bessel, Dec. 28, 1838. French translation (1841).
JACOBI
De la ligne godsique sur un ellipsode, et des diffrents usages dÕune transformation analytique remarquable.
Journal de mathmatiques pures et appliques 1re srie, tome 6 (1841), p. 267-272.
available at http://sites.mathdoc.fr/JMPA/PDF/JMPA_1841_1_6_A20_0.pdf
Journal fr die reine und angewandte Mathematik, Volume 19, 1839
editado por Carl Wilhelm Borchardt, Leopold Kronecker, Lazarus Fuchs, Kurt Wilhelm Sebastian Hensel, Helmut Hasse
------
Jacobi and Poncelet
http://www.numdam.org/item/AFST_2013_6_22_2_353_0/
We give an exposition of unpublished fragments of Gauss where he discovered (using a work of Jacobi) a remarkable connection between Napier pentagons on the sphere and Poncelet pentagons on the plane. As a corollary we find a parametrization in elliptic functions of the classical dilogarithm five-term relation.
--
https://link.springer.com/article/10.1007/BF02924855
Rendiconti del Seminario Matematico e Fisico di Milano
December 1985, Volume 54, Issue 1, pp 145Ð158 | Cite as
The closure theorem of Poncelet
Authors
Authors and affiliations
H. J. M. Bos
A report on a joint study together with C. Kers, F. Oort and D. W. Raven on historical and mathematical aspects of Poncelet's closure theorem. Proofs of the theorem by Griffiths (1976), Jacobi (1828) and Poncelet himself (1822) are discussed and a new result is reported concerning a certain one-parameter family of curves. This family of curves arises naturally from arguments in Poncelet's original proof and it offers an interesting case of strong non-commutativity of dualizing and specializing.
--
https://link.springer.com/chapter/10.1007%2F978-3-0348-5438-2_53
http://dx.doi.org/10.1007/978-3-0348-5438-2_53
Schoenberg, I. J. (1983). On Jacobi-BertrandÕs proof of a Theorem of Poncelet. Studies in Pure Mathematics, 623Ð627. doi:10.1007/978-3-0348-5438-2_53
O. Nash says G-Harris proof is the same as Jacobi's!
Jacobi's elliptic functions and criptography!
--
https://arxiv.org/abs/1202.0002
A vector bundle proof of Poncelet theorem
Jean Valls (LMA-PAU)
(Submitted on 31 Jan 2012)
In the town of Saratov where he was prisonner, Poncelet, continuing the work of Euler and Steiner on polygons simultaneously inscribed in a circle and circumscribed around an other circle, proved the following generalization : "Let C and D be two smooth conics in the projective complex plane. If D passes through the n(n-1)/2 vertices of a complete polygon with n sides tangent to C then D passes through the vertices of infinitely many such polygons." According to Marcel Berger this theorem is the nicest result about the geometry of conics. Even if it is, there are few proofs of it. To my knowledge there are only three. The first proof, published in 1822 and based on infinitesimal deformations, is due to Poncelet. Later, Jacobi proposed a new proof based on finite order points on elliptic curves; his proof, certainly the most famous, is explained in a modern way and in detail by Griffiths and Harris. In 1870 Weyr proved a Poncelet theorem in space (more precisely for two quadrics) that implies the one above when one quadric is a cone; this proof is explained by Barth and Bauer. Our aim in this short note is to involve vector bundles techniques to propose a new proof of this celebrated result. Poncelet did not appreciate Jacobi's for the reason that it was too far from the geometric intuition. I guess that he would not appreciate our proof either for the same reason.
In 2011 we uploaded a video to youtube about an obscure phenomenon:
the incenter of 3-periodic (triangular) orbits in an elliptic billiard
has an elliptic locus. During the next few years the video was watched
by a few billiard experts, and proofs were published about it. In
early 2019 we created a webpage to summarize their findings which led
to a 2-week frenzy of visual experiments where we stumbled upon 5 new
amazing properties of 3-periodic orbits, including unknown invariants
and a stationary point. In this talk we will explore them, as well as
advocate visual experimentation as an important tool for mathematical
discovery.