View Full Version : [SOLVED] What is Complete Integrability?
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 <40e7fa9d\\$1@news.sentex.net>, <tessel@tum.bot> wrote:\n\n>On Mon, 28 Jun 2004, Gerard Westendorp wrote:\n\n>> tessel@tum.bot wrote:\n\n>> Maybe "completely integrable Hamiltonian systems" are a bit like\n>> "warped" linear systems, whereas the other ones are not.\n\n>Dunno--- what are warped linear systems?\n\nI think he means something like this: if you take a harmonic\noscillator with lots of degrees of freedom and apply any\nsymplectomorphism to its phase space, you get a completely\nintegrable system. This is true. In other words, you can get\nlots of completely integrable systems by taking linear systems\nand "warping" them. But, you don\'t get *all* completely\nintegrable systems this way! However, this is close enough to\nbeing true that studying how it fails has led to interesting insights.\n\n(Btw: a linear Hamiltonian system is one whose Hamiltonian is\nquadratic. If we also assume this quadratic form is positive\ndefinite, it\'s a harmonic oscillator. The other ones are a bit\nmore annoying to study, since their surfaces of constant energy\nneedn\'t be compact - e.g. hyperboloids instead of ellipsoids.)\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 <40e7fa9d$1@news.sentex.net>, <tessel@tum.bot> wrote:
>On Mon, 28 Jun 2004, Gerard Westendorp wrote:
>> tessel@tum.bot wrote:
>> Maybe "completely integrable Hamiltonian systems" are a bit like
>> "warped" linear systems, whereas the other ones are not.
>Dunno--- what are warped linear systems?
I think he means something like this: if you take a harmonic
oscillator with lots of degrees of freedom and apply any
symplectomorphism to its phase space, you get a completely
integrable system. This is true. In other words, you can get
lots of completely integrable systems by taking linear systems
and "warping" them. But, you don't get *all* completely
integrable systems this way! However, this is close enough to
being true that studying how it fails has led to interesting insights.
(Btw: a linear Hamiltonian system is one whose Hamiltonian is
quadratic. If we also assume this quadratic form is positive
definite, it's a harmonic oscillator. The other ones are a bit
more annoying to study, since their surfaces of constant energy
needn't be compact - e.g. hyperboloids instead of ellipsoids.)
Gerard Westendorp
Aug23-04, 04:05 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>\n\nJohn Baez wrote:\n[..]\n\n\n>>>Maybe "completely integrable Hamiltonian systems" are a bit like\n>>>"warped" linear systems, whereas the other ones are not.\n>>>\n>\n>>Dunno--- what are warped linear systems?\n>>\n>\n> I think he means something like this: if you take a harmonic\n> oscillator with lots of degrees of freedom and apply any\n> symplectomorphism to its phase space, you get a completely\n> integrable system. This is true. In other words, you can get\n> lots of completely integrable systems by taking linear systems\n> and "warping" them. But, you don\'t get *all* completely\n> integrable systems this way! However, this is close enough to\n> being true that studying how it fails has led to interesting insights.\n\n\nYes, this is what I meant!\nBeing not a mathematician, I probably mess up terminology.\n\nImagine the phase space of a chain of masses and springs.\nThis will have 2N degrees of freedom, which are basically N\neigenfrequency modes that you can superpose. The phase space\nwill be a 2N dimensional space, and each eigenfrequancy\nwill have a bunch of concentric circles (each circle depicting\nthe time evolution of a solution) and each bunch of concentric\ncircles is lying in its own 2D plane of 2N-space.\n\nThe set of all lines that trace out solutions will look\nlike a kind of bunch of\nconcentric hyper-tori. Some properties:\n\n-no lines will intersect\n-they all orbit the origin (but not always periodically; a bit\nlike the old toy "spirograph") .\n-You could figure out a special metric such that the distance\nto the origin on all curves stays constant. (The conservation\nof total energy)\n\nNow what would happen if we made the springs *very slightly*\nnon-linear? It would seem reasonable that phase space will\nget only slightly deformed. Notably, the *topology* of\nthe set of lines will not change. A mapping from the old\nto the new phase space will be well-behaved, it should\nbe an isomorphism, with the arrows gently waving like\na field of barley on a summer evening.\n\n[As we turn up the non-linearity, we might expect our\nbarley field of arrows to get knotted, disconnected,\nsplit, ...]\n\nWe know that the superposition\nprinciple no longer holds, because the system is non-linear.\nBut because the topology of the phase space is not changed,\nwe can create a mapping from the non-linear space to the\nlinear space. Using this mapping, there is in fact a kind\nof pseudo-superposition. If we used awkward coordinates,\nthis pseudo-superposition might be hidden from us at\nfirst.\n\nOK, so more questions:\n\nIs the Inverse Scattering Transform some kind of\n"warped superposition principle" , in the sense I\ntry to sketch above?\n\nThe fact that solitons are stable and appear to move\nthrough each other when they "collide" is what you would\nexpect from a "warped linear system".\n\nSo what about the KdV equation, can be map it onto\na linear system?\n\nGerard\n\n\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>John Baez wrote:
[..]
>>>Maybe "completely integrable Hamiltonian systems" are a bit like
>>>"warped" linear systems, whereas the other ones are not.
>>>
>
>>Dunno--- what are warped linear systems?
>>
>
> I think he means something like this: if you take a harmonic
> oscillator with lots of degrees of freedom and apply any
> symplectomorphism to its phase space, you get a completely
> integrable system. This is true. In other words, you can get
> lots of completely integrable systems by taking linear systems
> and "warping" them. But, you don't get *all* completely
> integrable systems this way! However, this is close enough to
> being true that studying how it fails has led to interesting insights.
Yes, this is what I meant!
Being not a mathematician, I probably mess up terminology.
Imagine the phase space of a chain of masses and springs.
This will have 2N degrees of freedom, which are basically N
eigenfrequency modes that you can superpose. The phase space
will be a 2N dimensional space, and each eigenfrequancy
will have a bunch of concentric circles (each circle depicting
the time evolution of a solution) and each bunch of concentric
circles is lying in its own 2D plane of 2N-space.
The set of all lines that trace out solutions will look
like a kind of bunch of
concentric hyper-tori. Some properties:
-no lines will intersect
-they all orbit the origin (but not always periodically; a bit
like the old toy "spirograph") .
-You could figure out a special metric such that the distance
to the origin on all curves stays constant. (The conservation
of total energy)
Now what would happen if we made the springs *very slightly*
non-linear? It would seem reasonable that phase space will
get only slightly deformed. Notably, the *topology* of
the set of lines will not change. A mapping from the old
to the new phase space will be well-behaved, it should
be an isomorphism, with the arrows gently waving like
a field of barley on a summer evening.
[As we turn up the non-linearity, we might expect our
barley field of arrows to get knotted, disconnected,
split, ...]
We know that the superposition
principle no longer holds, because the system is non-linear.
But because the topology of the phase space is not changed,
we can create a mapping from the non-linear space to the
linear space. Using this mapping, there is in fact a kind
of pseudo-superposition. If we used awkward coordinates,
this pseudo-superposition might be hidden from us at
first.
OK, so more questions:
Is the Inverse Scattering Transform some kind of
"warped superposition principle" , in the sense I
try to sketch above?
The fact that solitons are stable and appear to move
through each other when they "collide" is what you would
expect from a "warped linear system".
So what about the KdV equation, can be map it onto
a linear system?
Gerard
<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>\nGerard Westendorp <westy31@xs4all.nl> wrote in message news:<412872BF.8090405@xs4all.nl>...\n\n> So what about the KdV equation, can be map it onto\n> a linear system?\n\n\n\nYour model with anharmonic oscillators becomes the Fermi-Pasta_Ulam\nmodel, which turns out to be described by a discretized KdV equation,\nwhich is integrable in terms of solitons.\n\n\n[Moderator\'s note: Quoted text trimmed by moderator. Please quote reasonably.\nSee http://www-stud.uni-essen.de/~sb0264/HowToPost.html -usc]\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>Gerard Westendorp <westy31@xs4all.nl> wrote in message news:<412872BF.8090405@xs4all.nl>...
> So what about the KdV equation, can be map it onto
> a linear system?
Your model with anharmonic oscillators becomes the Fermi-Pasta_Ulam
model, which turns out to be described by a discretized KdV equation,
which is integrable in terms of solitons.
[Moderator's note: Quoted text trimmed by moderator. Please quote reasonably.
See http://www-stud.uni-essen.de/~sb0264/HowToPost.html -usc]
Ralph E. Frost
Aug24-04, 04:58 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>"Gerard Westendorp" <westy31@xs4all.nl> wrote in message\nnews:412872BF.8090405@xs4all.nl...\n>\n.. ..\n> Now what would happen if we made the springs *very slightly*\n> non-linear? It would seem reasonable that phase space will\n> get only slightly deformed. Notably, the *topology* of\n> the set of lines will not change. A mapping from the old\n> to the new phase space will be well-behaved, it should\n> be an isomorphism, with the arrows gently waving like\n> a field of barley on a summer evening.\n\nIn the more unified models the anharmonic wafting is far more widespread\nthan just in fields of barley, isn\'t it?\n\nThe very breeze itself, the tides, rains, climatic swings, solar maxima,\nannual light/dark cycles, laminar and turbulent flow... the list in nature\nappears quite endless.\n\nIt almost makes me want to ask why the anharmonic system is not taught in\nschools first and before the linear harmonic instances --Are they mostly\nman-made?-- are covered.\n\n\n>\n> [As we turn up the non-linearity, we might expect our\n> barley field of arrows to get knotted, disconnected,\n> split, ...]\n\nTurbulent? Changing state?\n\n>\n> We know that the superposition\n> principle no longer holds, because the system is non-linear.\n> But because the topology of the phase space is not changed,\n> we can create a mapping from the non-linear space to the\n> linear space.\n\nBut, why would one want to do this or think this to be important?\n\n--\nRalph Frost\nImagine a single internal analog language made of ordered water ...and\nits variants.\n\n"...Love one another..." John 15:12\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>"Gerard Westendorp" <westy31@xs4all.nl> wrote in message
news:412872BF.8090405@xs4all.nl...
>
....
> Now what would happen if we made the springs *very slightly*
> non-linear? It would seem reasonable that phase space will
> get only slightly deformed. Notably, the *topology* of
> the set of lines will not change. A mapping from the old
> to the new phase space will be well-behaved, it should
> be an isomorphism, with the arrows gently waving like
> a field of barley on a summer evening.
In the more unified models the anharmonic wafting is far more widespread
than just in fields of barley, isn't it?
The very breeze itself, the tides, rains, climatic swings, solar maxima,
annual light/dark cycles, laminar and turbulent flow... the list in nature
appears quite endless.
It almost makes me want to ask why the anharmonic system is not taught in
schools first and before the linear harmonic instances --Are they mostly
man-made?-- are covered.
>
> [As we turn up the non-linearity, we might expect our
> barley field of arrows to get knotted, disconnected,
> split, ...]
Turbulent? Changing state?
>
> We know that the superposition
> principle no longer holds, because the system is non-linear.
> But because the topology of the phase space is not changed,
> we can create a mapping from the non-linear space to the
> linear space.
But, why would one want to do this or think this to be important?
--
Ralph Frost
Imagine a single internal analog language made of ordered water ...and
its variants.
"...Love one another..." John 15:12
John Baez
Aug24-04, 07:28 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 <412872BF.8090405@xs4all.nl>,\nGerard Westendorp <westy31@xs4all.nl> wrote:\n\n>John Baez wrote:\n\n>> Chris Hillman wrote:\n\n>>> Gerard Westendorp wrote:\n\n> >>>Maybe "completely integrable Hamiltonian systems" are a bit like\n> >>>"warped" linear systems, whereas the other ones are not.\n\n> >>Dunno--- what are warped linear systems?\n> >\n> > I think he means something like this: if you take a harmonic\n> > oscillator with lots of degrees of freedom and apply any\n> > symplectomorphism to its phase space, you get a completely\n> > integrable system. This is true. In other words, you can get\n> > lots of completely integrable systems by taking linear systems\n> > and "warping" them.\n\n>Yes, this is what I meant!\n\nActually from what you say below, my guess is *not* what\nyou meant. I thought you were "warping" the oscillator\nby applying a coordinate transformation to the phase space -\nthe space of p\'s and q\'s.\n\n(To be any good, this coordinate transformation should\npreserve the Poisson brackets. So, it should be what people\ncall a "canonical transformation" or "symplectomorphism".)\n\nBut you seem to be "warping" it in another way - namely\nby adding a small arbitrary term to the Hamiltonian.\n\n>Imagine the phase space of a chain of masses and springs.\n>This will have 2N degrees of freedom, which are basically N\n>eigenfrequency modes that you can superpose.\n\nIn short, you start with an oscillator...\n\n>Now what would happen if we made the springs *very slightly*\n>non-linear?\n\n.... and then add a small extra term to the Hamiltonian -\na small perturbation.\n\nSo: is your way of "warping" the system always equivalent\nto my way?\n\nYou claim it is, at least for small perturbations:\n\n>It would seem reasonable that phase space will\n>get only slightly deformed. Notably, the *topology* of\n>the set of lines will not change. A mapping from the old\n>to the new phase space will be well-behaved, it should\n>be an isomorphism, with the arrows gently waving like\n>a field of barley on a summer evening.\n>\n>[As we turn up the non-linearity, we might expect our\n>barley field of arrows to get knotted, disconnected,\n>split, ...]\n\nThat\'s a good guess! You are reinventing the idea behind\nthe KAM theorem - the Kolmogorov-Arnold-Moser theorem, that is.\nThere\'s a subtlety called the "small denominators problem"\nwhich you haven\'t noticed, but apart from that I believe your\nclaim is correct if one interprets your phrase\n\n*very slightly* nonlinear\n\nin a charitable way.\n\nThe sublety is this: if your original harmonic oscillator has\neigenfrequencies omega_1, ..., omega_n that are in resonance:\n\na_1 omega_1 + ... + a_n omega_n = 0 for some integers a_1, ..., a_n\n\nthen even a slight perturbation of the Hamiltonian can drastically\nchange the behavior of the system.\n\nYou can get some more information here:\n\nhttp://mathworld.wolfram.com/Kolmogorov-Arnold-MoserTheorem.html\n\nand here:\n\nhttp://www-nonlinear.physik.uni-bremen.de/nlp/publications/ChaosHTML/r14richter/node7.html\n\nThe golden ratio plays an important role in this game, since it\'s the\n"most irrational number" - so oscillators with eigenmodes having this\nfrequency ratio are especially robust to perturbations.\n\n\n\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 <412872BF.8090405@xs4all.nl>,
Gerard Westendorp <westy31@xs4all.nl> wrote:
>John Baez wrote:
>> Chris Hillman wrote:
>>> Gerard Westendorp wrote:
> >>>Maybe "completely integrable Hamiltonian systems" are a bit like
> >>>"warped" linear systems, whereas the other ones are not.
> >>Dunno--- what are warped linear systems?
> >
> > I think he means something like this: if you take a harmonic
> > oscillator with lots of degrees of freedom and apply any
> > symplectomorphism to its phase space, you get a completely
> > integrable system. This is true. In other words, you can get
> > lots of completely integrable systems by taking linear systems
> > and "warping" them.
>Yes, this is what I meant!
Actually from what you say below, my guess is *not* what
you meant. I thought you were "warping" the oscillator
by applying a coordinate transformation to the phase space -
the space of p's and q's.
(To be any good, this coordinate transformation should
preserve the Poisson brackets. So, it should be what people
call a "canonical transformation" or "symplectomorphism".)
But you seem to be "warping" it in another way - namely
by adding a small arbitrary term to the Hamiltonian.
>Imagine the phase space of a chain of masses and springs.
>This will have 2N degrees of freedom, which are basically N
>eigenfrequency modes that you can superpose.
In short, you start with an oscillator...
>Now what would happen if we made the springs *very slightly*
>non-linear?
.... and then add a small extra term to the Hamiltonian -
a small perturbation.
So: is your way of "warping" the system always equivalent
to my way?
You claim it is, at least for small perturbations:
>It would seem reasonable that phase space will
>get only slightly deformed. Notably, the *topology* of
>the set of lines will not change. A mapping from the old
>to the new phase space will be well-behaved, it should
>be an isomorphism, with the arrows gently waving like
>a field of barley on a summer evening.
>
>[As we turn up the non-linearity, we might expect our
>barley field of arrows to get knotted, disconnected,
>split, ...]
That's a good guess! You are reinventing the idea behind
the KAM theorem - the Kolmogorov-Arnold-Moser theorem, that is.
There's a subtlety called the "small denominators problem"
which you haven't noticed, but apart from that I believe your
claim is correct if one interprets your phrase
*very[/itex] slightly* nonlinear
in a charitable way.
The sublety is this: if your original harmonic oscillator has
eigenfrequencies \omega_1, ..., \omega_n that are in resonance:
a_1 \omega_1 + ... + a_n \omega_n = for some integers a_1, ..[itex]., a_n
then even a slight perturbation of the Hamiltonian can drastically
change the behavior of the system.
You can get some more information here:
http://mathworld.wolfram.com/Kolmogorov-Arnold-MoserTheorem.html
and here:
http://www-nonlinear.physik.uni-bremen.de/nlp/publications/ChaosHTML/r14richter/node7.html
The golden ratio plays an important role in this game, since it's the
"most irrational number" - so oscillators with eigenmodes having this
frequency ratio are especially robust to perturbations.
<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>\nDickT <rthompson10@new.rr.com> writes\n>Your model with anharmonic oscillators becomes the Fermi-Pasta_Ulam\n>model, which turns out to be described by a discretized KdV equation,\n>which is integrable in terms of solitons.\n\nThis takes me another step. One can perform a fourier transform and\nconvert any waveform to a sum of sine waves. One can also do the\nequivalent operation and obtain it as a sum of square waves, and indeed\nany repeating function can be so used.\n\nOne might hope that one could attempt to use solitons as the summing\nfunction, but I doubt that would typically work. OTOH one might expect a\nsubset of functions that would work. This is presumably what you mean by\n\'a discretised KdV equation which is integrable in terms of solitons\'.\n\nI find it quite extraordinary and hugely encouraging that solitons turn\nup naturally in the basic equations of QM and am enthused that work in\nthis area continues apace. If (as I currently believe) elementary\nparticles are waves, no more, then the particulate nature (in particular\ntheir identicality) would seem to be ideally expressed by a soliton-like\nbehaviour.\n\nPity the maths is so challenging....\n\n--\nOz\nThis post is worth absolutely nothing and is probably fallacious.\n\nBTOPENWORLD address about to cease. DEMON address no longer in use.\n>>Use oz@farmeroz.port995.com<<\nozacoohdb@despammed.com still functions.\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>DickT <rthompson10@new.rr.com> writes
>Your model with anharmonic oscillators becomes the Fermi-Pasta_Ulam
>model, which turns out to be described by a discretized KdV equation,
>which is integrable in terms of solitons.
This takes me another step. One can perform a fourier transform and
convert any waveform to a sum of sine waves. One can also do the
equivalent operation and obtain it as a sum of square waves, and indeed
any repeating function can be so used.
One might hope that one could attempt to use solitons as the summing
function, but I doubt that would typically work. OTOH one might expect a
subset of functions that would work. This is presumably what you mean by
'a discretised KdV equation which is integrable in terms of solitons'.
I find it quite extraordinary and hugely encouraging that solitons turn
up naturally in the basic equations of QM and am enthused that work in
this area continues apace. If (as I currently believe) elementary
particles are waves, no more, then the particulate nature (in particular
their identicality) would seem to be ideally expressed by a soliton-like
behaviour.
Pity the maths is so challenging....
--
Oz
This post is worth absolutely nothing and is probably fallacious.
BTOPENWORLD address about to cease. DEMON address no longer in use.
>>Use oz@farmeroz.port995.com<<
ozacoohdb@despammed.com still functions.
Gerard Westendorp
Aug26-04, 04:31 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>Ralph E. Frost wrote:\n\n[..]\n\n>>We know that the superposition\n>>principle no longer holds, because the system is non-linear.\n>>But because the topology of the phase space is not changed,\n>>we can create a mapping from the non-linear space to the\n>>linear space.\n>>\n>\n> But, why would one want to do this or think this to be important?\n\nLinear systems are pretty well understood, whereas\nnon-linear ones are not. So if we can find a relation\nbetween linear systems and certain types of non-linear\nsystems, we may understand the non-linear ones better.\n\nTry this experiment (or maybe you know it already):\nMake 2 pendula of equal length, using weights and string.\nThen interconnect them with some string near the top:\n\n\\ /\n---\n| |\n| |\n| |\n| |\n0 0\n\nThen, swing one of the pendula, and see what happens\nover time.\n\nThis system of coupled osilators can be neatly described\nby linear theory for small amplitudes. For large amplitudes,\nit gets trickier.\n\nA longer chain of these things will give a nice Sine-Gordon\nwave. Hmmm... I might even make a 2 dimensional version\non my ceiling (a grid of coupled pendula). *if* I have time.\n\n\nGerard\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>Ralph E. Frost wrote:
[..]
>>We know that the superposition
>>principle no longer holds, because the system is non-linear.
>>But because the topology of the phase space is not changed,
>>we can create a mapping from the non-linear space to the
>>linear space.
>>
>
> But, why would one want to do this or think this to be important?
Linear systems are pretty well understood, whereas
non-linear ones are not. So if we can find a relation
between linear systems and certain types of non-linear
systems, we may understand the non-linear ones better.
Try this experiment (or maybe you know it already):
Make 2 pendula of equal length, using weights and string.
Then interconnect them with some string near the top:
\ /
---
| |
| |
| |
| |
Then, swing one of the pendula, and see what happens
over time.
This system of coupled osilators can be neatly described
by linear theory for small amplitudes. For large amplitudes,
it gets trickier.
A longer chain of these things will give a nice Sine-Gordon
wave. Hmmm... I might even make a 2 dimensional version
on my ceiling (a grid of coupled pendula). *if* I have time.
Gerard
Gerard Westendorp
Aug26-04, 04:31 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>John Baez wrote:\n[..]\n\n> Actually from what you say below, my guess is *not* what\n> you meant. I thought you were "warping" the oscillator\n> by applying a coordinate transformation to the phase space -\n> the space of p\'s and q\'s.\n>\n> (To be any good, this coordinate transformation should\n> preserve the Poisson brackets. So, it should be what people\n> call a "canonical transformation" or "symplectomorphism".)\n>\n> But you seem to be "warping" it in another way - namely\n> by adding a small arbitrary term to the Hamiltonian.\n\nWell, I wasn\'t really envisioning a specific form for\nthe Hamiltonian. Any perturbation that can be smoothly\nreduced to zero. For example,\n\nH = 1/2 p^2 + 1/(2+a) x^(2+a)\n\nFor a = 0, we get a harmonic oscillator. If a = small,\nwe should be able to define a map from the non-linear\ncase to the linear one.\n\nI believe you can create 2 new variables\n\n(X, P) <--> (x, p)\n\nSuch that (X\'P - P\'X) is a constant of motion.\n(I lost the scrap of paper on which I derived this,\nI could redo it it if anyone cares)\nThis is I *think* this means there is a\n"symplectomorphism" lurking somewhere, but I may\nneed to do more homework on that.\n\nI must say I still don\'t quite understand why symplectic\nforms are so important. I started the thread\n"symplectic structures for dummies" a while ago. I learned\na bit since then, like the formal definitions of a symplectic\nform and a Poison bracket, and some standard stuff about\nLagrangians, Hamiltonians, but I still feel I am missing the\npoint.\n\nThe way I see it, you have dynamic equations in real variables.\nSo if you want these variable not to grow exponentially in\ntime or decay exponentially in time, all eigenvalues of the\ntime evolution operator must be complex conjugate pairs of\nnorm 1.\n\nBecause of this pair formation, all variables have a\n"canonical conjugate". These pairs seem to pop up\nall the time in physics.\nBut for non-linear systems,\neigenvalues and diagonalisation of the time evolution\noperator only works for a short time interval.\nNow I would like to say something about how the\nsymplectic structure puts all this into perspective,\nbut I am still in the mist...\n\n[..]\n\n> The sublety is this: if your original harmonic oscillator has\n> eigenfrequencies omega_1, ..., omega_n that are in resonance:\n>\n> a_1 omega_1 + ... + a_n omega_n = 0 for some integers a_1, ..., a_n\n>\n> then even a slight perturbation of the Hamiltonian can drastically\n> change the behavior of the system.\n\nAha, that makes sense intuitively. I will think about it.\n\n[..]\n\n> The golden ratio plays an important role in this game, since it\'s the\n> "most irrational number" - so oscillators with eigenmodes having this\n> frequency ratio are especially robust to perturbations.\n\nWow!\n\nGerard\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>John Baez wrote:
[..]
> Actually from what you say below, my guess is *not* what
> you meant. I thought you were "warping" the oscillator
> by applying a coordinate transformation to the phase space -
> the space of p's and q's.
>
> (To be any good, this coordinate transformation should
> preserve the Poisson brackets. So, it should be what people
> call a "canonical transformation" or "symplectomorphism".)
>
> But you seem to be "warping" it in another way - namely
> by adding a small arbitrary term to the Hamiltonian.
Well, I wasn't really envisioning a specific form for
the Hamiltonian. Any perturbation that can be smoothly
reduced to zero. For example,
H = 1/2 p^2 + 1/(2+a) x^(2+a)
For a = 0, we get a harmonic oscillator. If a = small,
we should be able to define a map from the non-linear
case to the linear one.
I believe you can create 2 new variables
(X, P) <--> (x, p)
Such that (X'P - P'X) is a constant of motion.
(I lost the scrap of paper on which I derived this,
I could redo it it if anyone cares)
This is I *think* this means there is a
"symplectomorphism" lurking somewhere, but I may
need to do more homework on that.
I must say I still don't quite understand why symplectic
forms are so important. I started the thread
"symplectic structures for dummies" a while ago. I learned
a bit since then, like the formal definitions of a symplectic
form and a Poison bracket, and some standard stuff about
Lagrangians, Hamiltonians, but I still feel I am missing the
point.
The way I see it, you have dynamic equations in real variables.
So if you want these variable not to grow exponentially in
time or decay exponentially in time, all eigenvalues of the
time evolution operator must be complex conjugate pairs of
norm 1.
Because of this pair formation, all variables have a
"canonical conjugate". These pairs seem to pop up
all the time in physics.
But for non-linear systems,
eigenvalues and diagonalisation of the time evolution
operator only works for a short time interval.
Now I would like to say something about how the
symplectic structure puts all this into perspective,
but I am still in the mist...
[..]
> The sublety is this: if your original harmonic oscillator has
> eigenfrequencies \omega_1, ..., \omega_n that are in resonance:
>
> a_1 \omega_1 + ... + a_n \omega_n = for some integers a_1, ..., a_n
>
> then even a slight perturbation of the Hamiltonian can drastically
> change the behavior of the system.
Aha, that makes sense intuitively. I will think about it.
[..]
> The golden ratio plays an important role in this game, since it's the
> "most irrational number" - so oscillators with eigenmodes having this
> frequency ratio are especially robust to perturbations.
Wow!
Gerard
Thomas Sauvaget
Aug27-04, 06:10 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>\nGerard Westendorp <westy31@xs4all.nl> wrote in message news:<412D1FE9.7020809@xs4all.nl>...\n\n\n> > The sublety is this: if your original harmonic oscillator has\n> > eigenfrequencies omega_1, ..., omega_n that are in resonance:\n> >\n> > a_1 omega_1 + ... + a_n omega_n = 0 for some integers a_1, ..., a_n\n> >\n> > then even a slight perturbation of the Hamiltonian can drastically\n> > change the behavior of the system.\n>\n> Aha, that makes sense intuitively. I will think about it.\n>\n> [..]\n>\n> > The golden ratio plays an important role in this game, since it\'s the\n> > "most irrational number" - so oscillators with eigenmodes having this\n> > frequency ratio are especially robust to perturbations.\n>\n> Wow!\n\nAt this point you really want to read the lovely book:\nV.I.Arnold, Mathematical Methods of Classical Mechanics -- 2nd\nedition.\n\nApplications to celestial mechanics are reviewed here:\n<http://www.mat.uniroma3.it/users/chierchia/PREPRINTS/kam.pdf>\n\nA nice color picture of a near-integrable system with 2 DOFs is here:\n<http://www.ericjhellergallery.com/index.pl?page=image;iid=1>.\nEach circle is a cut of a 2-torus (the so-called KAM tori), the rest\nbeing the \'chaotic sea\'.\n--\nthomas.\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>Gerard Westendorp <westy31@xs4all.nl> wrote in message news:<412D1FE9.7020809@xs4all.nl>...
> > The sublety is this: if your original harmonic oscillator has
> > eigenfrequencies \omega_1, ..., \omega_n that are in resonance:
> >
> > a_1 \omega_1 + ... + a_n \omega_n = for some integers a_1, ..., a_n
> >
> > then even a slight perturbation of the Hamiltonian can drastically
> > change the behavior of the system.
>
> Aha, that makes sense intuitively. I will think about it.
>
> [..]
>
> > The golden ratio plays an important role in this game, since it's the
> > "most irrational number" - so oscillators with eigenmodes having this
> > frequency ratio are especially robust to perturbations.
>
> Wow!
At this point you really want to read the lovely book:
V.I.Arnold, Mathematical Methods of Classical Mechanics -- 2nd
edition.
Applications to celestial mechanics are reviewed here:
<http://www.mat.uniroma3.it/users/chierchia/PREPRINTS/kam.pdf>
A nice color picture of a near-integrable system with 2 DOFs is here:
<http://www.ericjhellergallery.com/index.pl?page=image;iid=1>.
Each circle is a cut of a 2-torus (the so-called KAM tori), the rest
being the 'chaotic sea'.
--
thomas.
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