Q: Hamiltonian dynamics [Archive]

Andrew Resnick

May6-04, 12:20 PM

<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>\nI am working through the paper "The rate of entropy change in non-\nhamiltonian systems", L. Andrey, in Phys. Lett. A 111, 45 (1985), and I\nam having trouble understanding some basic concepts.\n\nAccording to the paper, if we define a dynamic system\n\ndx/dt = X(x), where x and X are n-dimensional vectors, we can define\nanother vector field P (\'the flow\'?):\n\nP = X@/@x, @ = partial derivative\n\nThis system is non-hamiltonian if the divergence of P is nonzero:\n\ndiv(P) = @X/@x\n\nI don\'t understand where P came from, and how div(P) is expressed in\nterms of X and x. At least I think these are simple questions, thanks\nin advance....\n--\nAndrew Resnick, Ph. D.\nNational Center for Microgravity Research\nNASA Glenn Research Center\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 am working through the paper "The rate of entropy change in non-
hamiltonian systems", L. Andrey, in Phys. Lett. A 111, 45 (1985), and I
am having trouble understanding some basic concepts.

According to the paper, if we define a dynamic system

dx/dt = X(x),[/itex] where x and X are n-dimensional vectors, we can define
another vector field P ('the flow'?):

P = X@/@x, @ = partial derivative

This system is non-hamiltonian if the divergence of P is nonzero:

[itex]div(P) = @X/@x

I don't understand where P came from, and how div(P) is expressed in
terms of X and x. At least I think these are simple questions, thanks
in advance....
--
Andrew Resnick, Ph. D.
National Center for Microgravity Research
NASA Glenn Research Center

Igor Khavkine

May10-04, 05:02 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\nAndrew Resnick <andy.resnick@NOSPAM.grc.nasaDOTgov> wrote in message news:<20040505123645427-0400@newsread.grc.nasa.gov>...\n> I am working through the paper "The rate of entropy change in non-\n> hamiltonian systems", L. Andrey, in Phys. Lett. A 111, 45 (1985), and I\n> am having trouble understanding some basic concepts.\n>\n> According to the paper, if we define a dynamic system\n>\n> dx/dt = X(x), where x and X are n-dimensional vectors, we can define\n> another vector field P (\'the flow\'?):\n>\n> P = X@/@x, @ = partial derivative\n\nThere is no new vector field P, the paper uses two notations:\nbold X (bX) that denotes the vector field itself, and sum_i X_i @/@x_i\nwhich gives the components X_i of bX in the basis @/@x_i\n(in the paper the sum and the indicies are not show, but they are there).\nThis notation has leaked from differential geometry. Just think of\nthem as unit vectors corresponding to coordinates (although they\nare not always _unit_ vectors). In polar coordinates the\nvectors \\hat{r} and \\hat{\\theta} can also be denoted by @/@r and @/@theta.\n\n> This system is non-hamiltonian if the divergence of P is nonzero:\n>\n> div(P) = @X/@x\n\nWhat you are actually looking at div(bX) = sum_i @X_i/x_i, this is\nthe expression for the divergence in component form.\n\nHope this helps.\n\nIgor\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>Andrew Resnick <andy.resnick@NOSPAM.grc.nasaDOTgov> wrote in message news:<20040505123645427-0400@newsread.grc.nasa.gov>...
> I am working through the paper "The rate of entropy change in non-
> hamiltonian systems", L. Andrey, in Phys. Lett. A 111, 45 (1985), and I
> am having trouble understanding some basic concepts.
>
> According to the paper, if we define a dynamic system
>
> dx/dt = X(x), where x and X are n-dimensional vectors, we can define
> another vector field P ('the flow'?):
>
> P = X@/@x, @ = partial derivative

There is no new vector field P, the paper uses two notations:
bold X (bX) that denotes the vector field itself, and sum_i X_i @/@x_i
which gives the components X_i of bX in the basis @/@x_i
(in the paper the sum and the indicies are not show, but they are there).
This notation has leaked from differential geometry. Just think of
them as unit vectors corresponding to coordinates (although they
are not always _unit_ vectors). In polar coordinates the
vectors \hat{r} and \hat{\theta} can also be denoted by @/@r and @/@\theta.

> This system is non-hamiltonian if the divergence of P is nonzero:
>
> div(P) = @X/@x

What you are actually looking at div(bX) = sum_i @X_i/x_i, this is
the expression for the divergence in component form.

Hope this helps.

Igor

Jeroen

May12-04, 01:30 PM

<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>Andrew Resnick wrote:\n\n>\n> I am working through the paper "The rate of entropy change in non-\n> hamiltonian systems", L. Andrey, in Phys. Lett. A 111, 45 (1985), and I\n> am having trouble understanding some basic concepts.\n>\n> According to the paper, if we define a dynamic system\n>\n> dx/dt = X(x), where x and X are n-dimensional vectors, we can define\n> another vector field P (\'the flow\'?):\n>\n> P = X@/@x, @ = partial derivative\n>\n> This system is non-hamiltonian if the divergence of P is nonzero:\n>\n> div(P) = @X/@x\n>\n> I don\'t understand where P came from, and how div(P) is expressed in\n> terms of X and x. At least I think these are simple questions, thanks\n> in advance....\n\nI\'m not sure what you mean by a non-Hamiltonian system (could you give a\ndefinition?), however the second question, about the divergence, does ring\na bell.\n\nP is a vector, the basic of the vector space is given by the partial\nderivatives @/@x (there are n of them). So writing\nP = X @/@x\nis the same as\nP = X^i e_i\nwith e_i the basis ( i = 1,...,n ) of the vector space.\n\nThe divergence is then simply:\ndiv(P) = \\sum_{i=1,..,n} @/@x^i P_i = \\sum_{i=1,..,n} @X^i/@x^i\n\nI have a feeling your question is related to Hamiltonian vector fields and\nsymplectic geometry. A vector field v is called Hamiltonian if its inner\nproduct with the symplectic form \\omega is an exact one-form:\ni_v \\omega = d h\nh is the Hamiltonian corresponding to the flow v. Since d^2 = 0 we have\nd ( i_v \\omega ) = 0,\nit somehow appears to me that i_v \\omega is related to P.\n\nbest,\nJeroen\n\nbest,\nJeroen\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>Andrew Resnick wrote:

>
> I am working through the paper "The rate of entropy change in non-
> hamiltonian systems", L. Andrey, in Phys. Lett. A 111, 45 (1985), and I
> am having trouble understanding some basic concepts.
>
> According to the paper, if we define a dynamic system
>
> dx/dt = X(x), where x and X are n-dimensional vectors, we can define
> another vector field P ('the flow'?):
>
> P = X@/@x, @ = partial derivative
>
> This system is non-hamiltonian if the divergence of P is nonzero:
>
> div(P) = @X/@x
>
> I don't understand where P came from, and how div(P) is expressed in
> terms of X and x. At least I think these are simple questions, thanks
> in advance....

I'm not sure what you mean by a non-Hamiltonian system (could you give a
definition?), however the second question, about the divergence, does ring
a bell.

P is a vector, the basic of the vector space is given by the partial
derivatives @/@x (there are n of them). So writing
P = X @/@x
is the same as
P = X^i e_i
with e_i the basis ( i = 1,...,n ) of the vector space.

The divergence is then simply:
div(P) = \sum_{i=1,..,n} @/@x^i P_i = \sum_{i=1,..,n} @X^i/@x^i

I have a feeling your question is related to Hamiltonian vector fields and
symplectic geometry. A vector field v is called Hamiltonian if its inner
product with the symplectic form \omega is an exact one-form:
i_v \omega = d h
h is the Hamiltonian corresponding to the flow v. Since d^2 = we have
d ( i_v \omega ) = 0,
it somehow appears to me that i_v \omega is related to P.

best,
Jeroen

best,
Jeroen

Patrick Van Esch

May12-04, 01:40 PM

<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>Andrew Resnick <andy.resnick@NOSPAM.grc.nasaDOTgov> wrote in message news:<20040505123645427-0400@newsread.grc.nasa.gov>...\n> I am working through the paper "The rate of entropy change in non-\n> hamiltonian systems", L. Andrey, in Phys. Lett. A 111, 45 (1985), and I\n> am having trouble understanding some basic concepts.\n>\n> According to the paper, if we define a dynamic system\n>\n> dx/dt = X(x), where x and X are n-dimensional vectors, we can define\n> another vector field P (\'the flow\'?):\n\nAre you sure that P is not simply X, which is already the "flow" ?\n\n>\n> P = X@/@x, @ = partial derivative\n>\n> This system is non-hamiltonian if the divergence of P is nonzero:\n>\n> div(P) = @X/@x\n\nI\'m probably being too naive here, but I guess that the vector field P\nis to be identified with the derivatives of the Hamiltonian if the\nsystem were hamiltonian:\n\ndp/dt = -@H/@q = X_p(p,q)\ndq/dt = @H/@p = X_q(p,q)\n\nP_p = @X_p/@p = -@^2 H / @p @q\nP_q = @ X_q/@q = @^2 H / @q @p\n\nP_p + P_q = @X_p / @p + @X_q / @q = 0\n\nSo the flow (X) is divergenceless in phase space (that\'s Liouville\'s\ntheorem!).\nIf you find a dynamics for which the flow is not divergenceless, it\ncannot be hamiltonian (I don\'t know if the reverse is true).\n\nBut I\'m confused with your definition of P, and the divergence of it.\n\ncheers,\nPatrick.\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>Andrew Resnick <andy.resnick@NOSPAM.grc.nasaDOTgov> wrote in message news:<20040505123645427-0400@newsread.grc.nasa.gov>...
> I am working through the paper "The rate of entropy change in non-
> hamiltonian systems", L. Andrey, in Phys. Lett. A 111, 45 (1985), and I
> am having trouble understanding some basic concepts.
>
> According to the paper, if we define a dynamic system
>
> dx/dt = X(x), where x and X are n-dimensional vectors, we can define
> another vector field P ('the flow'?):

Are you sure that P is not simply X, which is already the "flow" ?

>
> P = X@/@x, @ = partial derivative
>
> This system is non-hamiltonian if the divergence of P is nonzero:
>
> div(P) = @X/@x

I'm probably being too naive here, but I guess that the vector field P
is to be identified with the derivatives of the Hamiltonian if the
system were hamiltonian:

dp/dt = -@H/@q = X_p(p,q)dq/dt = @H/@p = X_q(p,q)P_p = @X_p/@p = -@^2 H / @p @qP_q = @ X_q/@q = @^2 H / @q @pP_p + P_q = @X_p / @p + @X_q / @q =

So the flow (X) is divergenceless in phase space (that's Liouville's
theorem!).
If you find a dynamics for which the flow is not divergenceless, it
cannot be hamiltonian (I don't know if the reverse is true).

But I'm confused with your definition of P, and the divergence of it.

cheers,
Patrick.