View Full Version : Degrees of Freedom?
<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>I appologize if this topic has been delt with here before,though I\ncannot find a satisfactory elementary definition in the google\narchives.\nWhat does it mean to say a field has " an infinite degrees of\nfreedom"?\nI will include a small passage from A.O. Barut\'s " Electrodynamics\nand classical theory of fields and particles" to start.\n" The infinite number of degrees of fredom of the field must be\ndescribed by continuous indices. Instead of the coordiates q_1,q_2 ...\n,the dynamicalvariables of the field will be a set of functions\npsi^a(X,t), a=1,.....N where (X,t) are now parameter which,together\nwith a,label the degrees of freedom of the system."\nNow in classical mechanics I understand that the number of\nvariables in the Lagrangian of the system is not synonomous with the\nnumber of degrees of freedom because one can come up with different\nconfiguation space variables. The electromagnetic field in vacuum can\nbe described by either six functions(three of E and 3 of B) or 4\nfunction A^u which are not unique. The free electromagnetic field may\nbe decribed by several different Lagrangians all with a finite number\nof field variables e.g. E and B, or A^u and by a finite number of\nfield equations. So where does the "infinite number of degrees of\nfreedom com from? Cann someone deliniate exactly what the difference\nis in the definition of "degrees of freedom" as it pertains to the\nclassical point particles formulation and classical field theory?\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>I appologize if this topic has been delt with here before,though I
cannot find a satisfactory elementary definition in the google
archives.
What does it mean to say a field has " an infinite degrees of
freedom"?
I will include a small passage from A.O. Barut's " Electrodynamics
and classical theory of fields and particles" to start.
" The infinite number of degrees of fredom of the field must be
described by continuous indices. Instead of the coordiates q_1,q_2 ...
,the dynamicalvariables of the field will be a set of functions
\psi^a(X,t), a=1,.....N where (X,t) are now parameter which,together
with a,label the degrees of freedom of the system."
Now in classical mechanics I understand that the number of
variables in the Lagrangian of the system is not synonomous with the
number of degrees of freedom because one can come up with different
configuation space variables. The electromagnetic field in vacuum can
be described by either six functions(three of E and 3 of B) or 4
function A^u which are not unique. The free electromagnetic field may
be decribed by several different Lagrangians all with a finite number
of field variables e.g. E and B, or A^u and by a finite number of
field equations. So where does the "infinite number of degrees of
freedom com from? Cann someone deliniate exactly what the difference
is in the definition of "degrees of freedom" as it pertains to the
classical point particles formulation and classical field theory?
Frank Hellmann
Aug12-04, 08:29 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\n\njfisher@mn.rr.com (jack) wrote in message news:<f4c1c8af.0408061230.7043d87c@posting.google. com>...\n\n> Cann someone deliniate exactly what the difference\n> is in the definition of "degrees of freedom" as it pertains to the\n> classical point particles formulation and classical field theory?\n\n\nA single particle moving in three dimensions is specified at any time\nby 6 real values (position, momentum), hence 3 degrees of freedom. A\nfield is specified at any time by infinitely many real values (it\'s\nvalues at every point in space), hence infinite degrees of freedom.\nThe space of all real functions constitutes a (rather nasty, I have\nbeen told) vector space. In QM the wavefunction isn\'t actually an\nelement of that space but of a much nicer space in which you don\'t\nneed the uncountable number of reals you need for a full blown\nfunction, but instead we divide out the subspace of all functions with\nintegral zero (for example the function that is zero everywhere except\nat finitely many locations were it takes finite values), and we get a\nHilbert Space with countable infinite dimensionality.\nWhen text books speak of infinitely many degrees of freedom what they\nusually think of is some kind of continuum limit.\nYou start with a linked chain, each member of the chain with a\nvelocity and a position (2 degres of freedom), take the limit of\ninfinitely many memebers and you get a continous string (Field on the\nReal line) with infinit degrees of freedom, the precise mathematical\nsituation is often rather subtle though.\n\nc\n\nfrank.\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>jfisher@mn.rr.com (jack) wrote in message news:<f4c1c8af.0408061230.7043d87c@posting.google.com>...
> Cann someone deliniate exactly what the difference
> is in the definition of "degrees of freedom" as it pertains to the
> classical point particles formulation and classical field theory?
A single particle moving in three dimensions is specified at any time
by 6 real values (position, momentum), hence 3 degrees of freedom. A
field is specified at any time by infinitely many real values (it's
values at every point in space), hence infinite degrees of freedom.
The space of all real functions constitutes a (rather nasty, I have
been told) vector space. In QM the wavefunction isn't actually an
element of that space but of a much nicer space in which you don't
need the uncountable number of reals you need for a full blown
function, but instead we divide out the subspace of all functions with
integral zero (for example the function that is zero everywhere except
at finitely many locations were it takes finite values), and we get a
Hilbert Space with countable infinite dimensionality.
When text books speak of infinitely many degrees of freedom what they
usually think of is some kind of continuum limit.
You start with a linked chain, each member of the chain with a
velocity and a position (2 degres of freedom), take the limit of
infinitely many memebers and you get a continous string (Field on the
Real line) with infinit degrees of freedom, the precise mathematical
situation is often rather subtle though.
c
frank.
Arnold Neumaier
Aug12-04, 08:30 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\njack wrote:\n\n> What does it mean to say a field has " an infinite degrees of\n> freedom"?\n\nit means that the field is an arbitrary element of an infinite-dimensional\nmanifold, and hence needs infinitely many real numbers for its precise\ndescription (by means of continuous operations).\n\n> Now in classical mechanics I understand that the number of\n> variables in the Lagrangian of the system is not synonomous with the\n> number of degrees of freedom because one can come up with different\n> configuation space variables.\n\nNot if you don\'t allow constraints. If you do allow constraints,\nyou must subtract the number of degrees of freedoms fixed by the\nconstraints - then the result is description invariant.\n\nThe underlying mathematics is the implicit function theorem,\nwhich gives nondegeneracy conditions under which a n-dimensional manifold\nconstrained by k conditions results in a (n-k)-dimensional manifold.\n(Think of solving systems of linear equations, or of intersecting\n3D surfaces. Exclude degenerate situations.)\n\n\nArnold Neumaier\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>jack wrote:
> What does it mean to say a field has " an infinite degrees of
> freedom"?
it means that the field is an arbitrary element of an infinite-dimensional
manifold, and hence needs infinitely many real numbers for its precise
description (by means of continuous operations).
> Now in classical mechanics I understand that the number of
> variables in the Lagrangian of the system is not synonomous with the
> number of degrees of freedom because one can come up with different
> configuation space variables.
Not if you don't allow constraints. If you do allow constraints,
you must subtract the number of degrees of freedoms fixed by the
constraints - then the result is description invariant.
The underlying mathematics is the implicit function theorem,
which gives nondegeneracy conditions under which a n-dimensional manifold
constrained by k conditions results in a (n-k)-dimensional manifold.
(Think of solving systems of linear equations, or of intersecting
3D surfaces. Exclude degenerate situations.)
Arnold Neumaier
<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\njack wrote:\n> What does it mean to say a field has " an infinite degrees of\n> freedom"?\n\n....\n\n> function A^u which are not unique. The free electromagnetic field may\n> be decribed by several different Lagrangians all with a finite number\n> of field variables e.g. E and B, or A^u and by a finite number of\n> field equations. So where does the "infinite number of degrees of\n> freedom com from? Cann someone deliniate exactly what the difference\n> is in the definition of "degrees of freedom" as it pertains to the\n> classical point particles formulation and classical field theory?\n\n"Degrees of freedom" is the information content basically. If you\ndescribe a theory about 3 point particles with position and momentum,\nall information is contained in the 3 sets of position/momentum vector\nelements, giving a finite amount of degrees of freedom.\n\nIf you describe a theory of a continous field, you need information\nabout the fields value at every point since it is a function defined\nover, well, a field :)\n\nSo your finite number of continous field variables each contain an\ninfinite amount of information (has infinite degrees of freedom).\n\nIf you make a spatial quantization of the field definition, you reduce\nthe information content to finite levels of course.\n\n/Bjorn\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>jack wrote:
> What does it mean to say a field has " an infinite degrees of
> freedom"?
....
> function A^u which are not unique. The free electromagnetic field may
> be decribed by several different Lagrangians all with a finite number
> of field variables e.g. E and B, or A^u and by a finite number of
> field equations. So where does the "infinite number of degrees of
> freedom com from? Cann someone deliniate exactly what the difference
> is in the definition of "degrees of freedom" as it pertains to the
> classical point particles formulation and classical field theory?
"Degrees of freedom" is the information content basically. If you
describe a theory about 3 point particles with position and momentum,
all information is contained in the 3 sets of position/momentum vector
elements, giving a finite amount of degrees of freedom.
If you describe a theory of a continous field, you need information
about the fields value at every point since it is a function defined
over, well, a field :)
So your finite number of continous field variables each contain an
infinite amount of information (has infinite degrees of freedom).
If you make a spatial quantization of the field definition, you reduce
the information content to finite levels of course.
/Bjorn
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