Deterministic models Text - Physics Forums Library

Phil Gardner

Jul19-04, 04:09 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\n\nIn the simplest deterministic models of particle interactions of\nclassical physics all the variables that appear in the equations of\nmotion are functions of a single independent variable, t (the time\nregistered by a stationary clock), and a set of constants of the\nmotion. As soon as we move beyond the Newtonian two body problem the\nequations of motion become pretty intractable.\n\nI assume that for any interaction model that gives explicit equations\nof motion the variables within them are still functions of a single\nindependent variable. Is this correct?\n\nPhil Gardner\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 the simplest deterministic models of particle interactions of
classical physics all the variables that appear in the equations of
motion are functions of a single independent variable, t (the time
registered by a stationary clock), and a set of constants of the
motion. As soon as we move beyond the Newtonian two body problem the
equations of motion become pretty intractable.

I assume that for any interaction model that gives explicit equations
of motion the variables within them are still functions of a single
independent variable. Is this correct?

Phil Gardner

Spud

Jul20-04, 04:25 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\npej_dg@dodo.com.au (Phil Gardner) wrote in message news:<ea961a86.0407180323.65981f46@posting.google.com>...\n> I assume that for any interaction model that gives explicit equations\n> of motion the variables within them are still functions of a single\n> independent variable. Is this correct?\n>\n> Phil Gardner\n\nI would say yes unless anyone else knows of any observations predicted\nby theory to contradict your statement.\n\nSpud\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>pej_dg@dodo.com.au (Phil Gardner) wrote in message news:<ea961a86.0407180323.65981f46@posting.google.com>...
> I assume that for any interaction model that gives explicit equations
> of motion the variables within them are still functions of a single
> independent variable. Is this correct?
>
> Phil Gardner

I would say yes unless anyone else knows of any observations predicted
by theory to contradict your statement.

Spud

tessel@tum.bot

Jul22-04, 05:16 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>On Mon, 19 Jul 2004, Phil Gardner wrote:\n\n> In the simplest deterministic models of particle\n\nPoint particles? Classical or quantum? What about more general\nparticle-like phenomena such as solitons, phonons, etc.? Or instantons?\n\n> interactions\n\nGravitational? Inverse square only or the more general "force laws"\ndiscussed for example in Landau & Lifschitz, Mechanics?\n\nWhat about EM interactions? Any other "forces" or "fields"?\n\nWhat about important models derived from fundamental physical equations\nusing approximations or idealizations? For example, do you wish to\ninclude things like the interactions of "kinks" and "antikinks" (certain\nsolutions to the sine-Gordon equation, a second order nonlinear wave\nequation) being discussed in the soliton thread? Or the FPU and Toda\nlattice?\n\nAnd what about "toy models"? These can be well worth discussing too!\n\nMaybe you have in mind "Lagrangian mechanics"? Or "Hamiltonian\nmechanics"? Or Hamiltonian dynamical systems? Finite dimensional? Or\npossibly infinite dimensional? (Note that infinite dimensional systems\nmay very well describe phenomena happening on the "stage" of ordinary\nthree dimensional euclidean space; e.g. a separable Hilbert space as in\nquantum mechanics might be used to discuss scattering of "waves" on the\nreal line by a potential, as described by the time independent\nSchroedinger equation; see the soliton thread.)\n\n> of classical\n\nMeaning "nonquantum"? Or "nonrelativistic"? Or something else?\n\n> physics all the variables that appear in the equations of motion are\n> functions of a single independent variable, t (the time registered by a\n^^^^^^^^\n\n> stationary clock),\n^^^^^^^^^^^^^^^^^\n\n???\n\nI\'m confused because this qualification doesn\'t make sense in -either-\nGalilean kinematics or in str, still less gtr. But it\'s good that you\ntried to clarify something you here which you must have decided needed\nclarification :-/\n\n> and a set of constants of the motion. As soon as we move beyond the\n> Newtonian two body problem the equations of motion become pretty\n> intractable.\n\nWell, the whole point of the soliton thread is that "completely\nintegrable" Hamiltonian systems are tractable. Some Newtonian\ngravitational n-body problems are intractable by this criterion, but\nvarious important nongravitational/nonfundamental models like the Toda\nlattice -are- tractable.\n\n> I assume that for any interaction model that gives explicit equations\n> of motion\n\nBe aware that in modern physics, "equations of motion" is commonly used to\nmean a much more general notion than you may have in mind.\n\n> the variables within them are still functions of a single independent\n> variable. Is this correct?\n\nI hope it is now clear that you probably need to clarify what you mean by\n"interaction model", "equations of motion", etc., before anyone here can\ngive a meaningful answer to your question. But of course, your revised\nquestion will only make sense if your definition of "interaction model" is\nnot "a model giving a system of equations with one independent variable\nwhich we can take to be Newton\'s universal time".\n\n"T. Essel" (hiding somewhere in cyberspace)\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>On Mon, 19 Jul 2004, Phil Gardner wrote:

> In the simplest deterministic models of particle

Point particles? Classical or quantum? What about more general
particle-like phenomena such as solitons, phonons, etc.? Or instantons?

> interactions

Gravitational? Inverse square only or the more general "force laws"
discussed for example in Landau & Lifschitz, Mechanics?

What about EM interactions? Any other "forces" or "fields"?

What about important models derived from fundamental physical equations
using approximations or idealizations? For example, do you wish to
include things like the interactions of "kinks" and "antikinks" (certain
solutions to the sine-Gordon equation, a second order nonlinear wave
equation) being discussed in the soliton thread? Or the FPU and Toda
lattice?

And what about "toy models"? These can be well worth discussing too!

Maybe you have in mind "Lagrangian mechanics"? Or "Hamiltonian
mechanics"? Or Hamiltonian dynamical systems? Finite dimensional? Or
possibly infinite dimensional? (Note that infinite dimensional systems
may very well describe phenomena happening on the "stage" of ordinary
three dimensional euclidean space; e.g. a separable Hilbert space as in
quantum mechanics might be used to discuss scattering of "waves" on the
real line by a potential, as described by the time independent
Schroedinger equation; see the soliton thread.)

> of classical

Meaning "nonquantum"? Or "nonrelativistic"? Or something else?

> physics all the variables that appear in the equations of motion are
> functions of a single independent variable, t (the time registered by a
^^^^^^^^

> stationary clock),
^^^^^^^^^^^^^^^^^

???

I'm confused because this qualification doesn't make sense in -either-
Galilean kinematics or in str, still less gtr. But it's good that you
tried to clarify something you here which you must have decided needed
clarification :-/

> and a set of constants of the motion. As soon as we move beyond the
> Newtonian two body problem the equations of motion become pretty
> intractable.

Well, the whole point of the soliton thread is that "completely
integrable" Hamiltonian systems are tractable. Some Newtonian
gravitational n-body problems are intractable by this criterion, but
various important nongravitational/nonfundamental models like the Toda
lattice -are- tractable.

> I assume that for any interaction model that gives explicit equations
> of motion

Be aware that in modern physics, "equations of motion" is commonly used to
mean a much more general notion than you may have in mind.

> the variables within them are still functions of a single independent
> variable. Is this correct?

I hope it is now clear that you probably need to clarify what you mean by
"interaction model", "equations of motion", etc., before anyone here can
give a meaningful answer to your question. But of course, your revised
question will only make sense if your definition of "interaction model" is
not "a model giving a system of equations with one independent variable
which we can take to be Newton's universal time".

"T. Essel" (hiding somewhere in cyberspace)