John Baez
Aug18-04, 04:08 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nIn article <40e7fa99\\$1@news.sentex.net>, <tessel@tum.bot> wrote:\n\n>You raised another question to which I have yet to either figure out the\n>answer on my own, or else to find answered in some book.\n>\n>Namely: what is "completely integrable" all about, and why oh why are\n>there -two- distinct concepts known by the same name in one field?!\n\nAh, if there were only just *two*... wouldn\'t that make things\nlovely and simple?\n\nDream on!\n\nSounds like you need to look at this book:\n\nV. E. Zakharov, ed., What is Integrability?, Springer Verlag, Berlin, 1991.\n\nBut, don\'t expect *one* answer to the question! Expect *lots*.\n\nThe point is this: once upon a time a physics problem was said\nto be "completely integrable" if you could reduce the problem of\nsolving it down to the problem of computing a bunch of integrals.\n\nOther words for this notion include "reducible to quadratures"\nand - even vaguer - "exactly solvable".\n\nPeople invented lots of tricks for showing that problems were "completely\nintegrable", and a physical system susceptible to any of these tricks\ngot lumped in the class of completely integrable systems. You mention\ntwo tricks. There are a bunch more.\n\nThey also invented criteria for *guessing* that problems were completely\nintegrable without actually proving it - most notably the "Painleve test".\n\nSorting out the logical relations among all these tricks and criteria\nis a bit like cleaning up the Augean stables, the only difference being\nthat there is gold to be found here, not just horse ****.\n\nBut, it\'s a big mess.\n\nI\'ve seen a big diagram of different concepts in this subject\nand arrows relating them (unfortunately not morphisms in any\nwell-defined category), which takes up more than one page, and\nmainly serves to prove nobody knows how everything fits together!\n\nI wish I could find it on the web, just to intimidate you.\n\n>So far, I have seen two definitions which certainly don\'t look equivalent,\n>at least not to me, at least not yet, at least not without thought.\n\nI\'ve never seen anyone claim they\'re *equivalent*.\n\n>The first definition goes by the long name "completely integrable in the\n>sense of Liouville", and applies to a n-dimensional Hamiltonian system.\n>If we can find 2n independent conserved integrals which are pairwise\n>involutory (vanishing Poisson bracket), this system is completely\n>integrable. (I think need some more conditions on these integrals, like\n>analyticity.) This notion is at least recognizably a special case of\n>"integrable system".\n\nRight, and there\'s a wonderful general theory of these, related\nto toric varieties, KAM theory, and other good things... so this\nis *my* favorite precise definition of complete integrability.\n\n(Analyticity is only necessary for certain things, like hooking\nup to algebraic geometry.)\n\n>The second usage (I hesitate to say "definition") seems silly. Let me\n>give it a long name too: a system is "completely integrable in the sense\n>of Lax" if it is solvable by the IST.\n\nWhere "IST" means "inverse scattering transform".\n\nI think this *implies* complete integrability in the first\nsense, at least under some conditions. I can\'t imagine any\nway to get the converse.\n\nI wish I knew more about the Lax pair/inverse scattering transform/\nLie-Baecklund stuff, especially about how it relates to loop\ngroups and affine Lie algebras. Someone like Graeme Segal or\nHitchin wrote a book about this, which James Dolan was carrying\naround for a long time with little discernible effect. Whoever\nwrote it, it was some mathematician whom you\'d expect to be good\nat getting to the bottom of things!\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <40e7fa99$1@news.sentex.net>, <tessel@tum.bot> wrote:
>You raised another question to which I have yet to either figure out the
>answer on my own, or else to find answered in some book.
>
>Namely: what is "completely integrable" all about, and why oh why are
>there -two- distinct concepts known by the same name in one field?!
Ah, if there were only just *two*... wouldn't that make things
lovely and simple?
Dream on!
Sounds like you need to look at this book:
V. E. Zakharov, ed., What is Integrability?, Springer Verlag, Berlin, 1991.
But, don't expect *one* answer to the question! Expect *lots*.
The point is this: once upon a time a physics problem was said
to be "completely integrable" if you could reduce the problem of
solving it down to the problem of computing a bunch of integrals.
Other words for this notion include "reducible to quadratures"
and - even vaguer - "exactly solvable".
People invented lots of tricks for showing that problems were "completely
integrable", and a physical system susceptible to any of these tricks
got lumped in the class of completely integrable systems. You mention
two tricks. There are a bunch more.
They also invented criteria for *guessing* that problems were completely
integrable without actually proving it - most notably the "Painleve test".
Sorting out the logical relations among all these tricks and criteria
is a bit like cleaning up the Augean stables, the only difference being
that there is gold to be found here, not just horse ****.
But, it's a big mess.
I've seen a big diagram of different concepts in this subject
and arrows relating them (unfortunately not morphisms in any
well-defined category), which takes up more than one page, and
mainly serves to prove nobody knows how everything fits together!
I wish I could find it on the web, just to intimidate you.
>So far, I have seen two definitions which certainly don't look equivalent,
>at least not to me, at least not yet, at least not without thought.
I've never seen anyone claim they're *equivalent*.
>The first definition goes by the long name "completely integrable in the
>sense of Liouville", and applies to a n-dimensional Hamiltonian system.
>If we can find 2n independent conserved integrals which are pairwise
>involutory (vanishing Poisson bracket), this system is completely
>integrable. (I think need some more conditions on these integrals, like
>analyticity.) This notion is at least recognizably a special case of
>"integrable system".
Right, and there's a wonderful general theory of these, related
to toric varieties, KAM theory, and other good things... so this
is *my* favorite precise definition of complete integrability.
(Analyticity is only necessary for certain things, like hooking
up to algebraic geometry.)
>The second usage (I hesitate to say "definition") seems silly. Let me
>give it a long name too: a system is "completely integrable in the sense
>of Lax" if it is solvable by the IST.
Where "IST" means "inverse scattering transform".
I think this *implies* complete integrability in the first
sense, at least under some conditions. I can't imagine any
way to get the converse.
I wish I knew more about the Lax pair/inverse scattering transform/
Lie-Baecklund stuff, especially about how it relates to loop
groups and affine Lie algebras. Someone like Graeme Segal or
Hitchin wrote a book about this, which James Dolan was carrying
around for a long time with little discernible effect. Whoever
wrote it, it was some mathematician whom you'd expect to be good
at getting to the bottom of things!
>You raised another question to which I have yet to either figure out the
>answer on my own, or else to find answered in some book.
>
>Namely: what is "completely integrable" all about, and why oh why are
>there -two- distinct concepts known by the same name in one field?!
Ah, if there were only just *two*... wouldn't that make things
lovely and simple?
Dream on!
Sounds like you need to look at this book:
V. E. Zakharov, ed., What is Integrability?, Springer Verlag, Berlin, 1991.
But, don't expect *one* answer to the question! Expect *lots*.
The point is this: once upon a time a physics problem was said
to be "completely integrable" if you could reduce the problem of
solving it down to the problem of computing a bunch of integrals.
Other words for this notion include "reducible to quadratures"
and - even vaguer - "exactly solvable".
People invented lots of tricks for showing that problems were "completely
integrable", and a physical system susceptible to any of these tricks
got lumped in the class of completely integrable systems. You mention
two tricks. There are a bunch more.
They also invented criteria for *guessing* that problems were completely
integrable without actually proving it - most notably the "Painleve test".
Sorting out the logical relations among all these tricks and criteria
is a bit like cleaning up the Augean stables, the only difference being
that there is gold to be found here, not just horse ****.
But, it's a big mess.
I've seen a big diagram of different concepts in this subject
and arrows relating them (unfortunately not morphisms in any
well-defined category), which takes up more than one page, and
mainly serves to prove nobody knows how everything fits together!
I wish I could find it on the web, just to intimidate you.
>So far, I have seen two definitions which certainly don't look equivalent,
>at least not to me, at least not yet, at least not without thought.
I've never seen anyone claim they're *equivalent*.
>The first definition goes by the long name "completely integrable in the
>sense of Liouville", and applies to a n-dimensional Hamiltonian system.
>If we can find 2n independent conserved integrals which are pairwise
>involutory (vanishing Poisson bracket), this system is completely
>integrable. (I think need some more conditions on these integrals, like
>analyticity.) This notion is at least recognizably a special case of
>"integrable system".
Right, and there's a wonderful general theory of these, related
to toric varieties, KAM theory, and other good things... so this
is *my* favorite precise definition of complete integrability.
(Analyticity is only necessary for certain things, like hooking
up to algebraic geometry.)
>The second usage (I hesitate to say "definition") seems silly. Let me
>give it a long name too: a system is "completely integrable in the sense
>of Lax" if it is solvable by the IST.
Where "IST" means "inverse scattering transform".
I think this *implies* complete integrability in the first
sense, at least under some conditions. I can't imagine any
way to get the converse.
I wish I knew more about the Lax pair/inverse scattering transform/
Lie-Baecklund stuff, especially about how it relates to loop
groups and affine Lie algebras. Someone like Graeme Segal or
Hitchin wrote a book about this, which James Dolan was carrying
around for a long time with little discernible effect. Whoever
wrote it, it was some mathematician whom you'd expect to be good
at getting to the bottom of things!