Re: Can energy conservation be derived from Newton's motion laws [Archive]

Dan Platt

Dec16-04, 08:08 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>You mean, can this be derived from Newton\'s laws without an assumption\nof a gradient force? Not really. There is something very fundamental\nabout the connection between gradient forces and conservation of energy,\nand that has to be added to Newton\'s laws to get conservation of energy\nto work. Further, as more layers (thermodynamics, electrodynamics,\nrelativity) are added, this question becomes more and more entwined with\nother aspects of physics (conservation of energy and momentum of charged\nparticles in the influence of electromagnetic fields: the field can\ncarry and transfer momentum, energy, and angular momentum) -- there is\nsomething significant and fundamental to and in physics about the extra\nrequirement of a gradient force in Newton\'s laws. But Newton\'s laws are\nnot really complete, either. The third law is sufficient to get\nconservation of momentum, but it does so by injecting instantaneous\nforces. Conservation of momentum is now recognized to be more\nfundamental than action-reaction.\n\nDan\n\n\nPoakfield@msn.com wrote:\n\n> Hi! Could someone please tell me if the law of conservation of energy\n> can be derived only from Newton\'s motion laws? Thanks.\n>\n> Peter\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>You mean, can this be derived from Newton's laws without an assumption
of a gradient force? Not really. There is something very fundamental
about the connection between gradient forces and conservation of energy,
and that has to be added to Newton's laws to get conservation of energy
to work. Further, as more layers (thermodynamics, electrodynamics,
relativity) are added, this question becomes more and more entwined with
other aspects of physics (conservation of energy and momentum of charged
particles in the influence of electromagnetic fields: the field can
carry and transfer momentum, energy, and angular momentum) -- there is
something significant and fundamental to and in physics about the extra
requirement of a gradient force in Newton's laws. But Newton's laws are
not really complete, either. The third law is sufficient to get
conservation of momentum, but it does so by injecting instantaneous
forces. Conservation of momentum is now recognized to be more
fundamental than action-reaction.

Dan

Poakfield@msn.com wrote:

> Hi! Could someone please tell me if the law of conservation of energy
> can be derived only from Newton's motion laws? Thanks.
>
> Peter
>

Strong_Field

Dec17-04, 07:51 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"Dan Platt" <DanP57@ispwest.com> wrote in message\nnews:t08wd.2407\\$ou2.622@fe09.lga...\n> You mean, can this be derived from Newton\'s laws without an assumption\n> of a gradient force?\n\nPardon my ignorance but what is a gradient force?\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>"Dan Platt" <DanP57@ispwest.com> wrote in message
news:t08wd.2407$ou2.622@fe09.lga...
> You mean, can this be derived from Newton's laws without an assumption
> of a gradient force?

Pardon my ignorance but what is a gradient force?

Igor Khavkine

Dec19-04, 06:48 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 Fri, 17 Dec 2004 13:51:13 +0000, Strong_Field wrote:\n> "Dan Platt" <DanP57@ispwest.com> wrote in message\n> news:t08wd.2407\\$ou2.622@fe09.lga...\n\n>> You mean, can this be derived from Newton\'s laws without an assumption\n>> of a gradient force?\n>\n> Pardon my ignorance but what is a gradient force?\n\nIt is the same as a conservative force, which means that we can write\nthe force F = - grad V, where grad is the gradient operator and V is some\nscalar function called the potential. All the derivations of energy\nconservation you\'ve gotten so far make this assumption.\n\nIgor\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>On Fri, 17 Dec 2004 13:51:13 +0000, Strong_Field wrote:
> "Dan Platt" <DanP57@ispwest.com> wrote in message
> news:t08wd.2407$ou2.622@fe09.lga...

>> You mean, can this be derived from Newton's laws without an assumption
>> of a gradient force?
>
> Pardon my ignorance but what is a gradient force?

It is the same as a conservative force, which means that we can write
the force F = - grad V, where grad is the gradient operator and V is some
scalar function called the potential. All the derivations of energy
conservation you've gotten so far make this assumption.

Igor

Dan Platt

Dec19-04, 06:51 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>A force which can be computed from a gradient of a scalar function, as in\n\nF = -grad V.\n\nIt is equivalent to the condition that the work W = integral F.dx\n\nbetween two endpoints along a path of a mass is independent of the path,\nin which case, W = -Delta V (difference between endpoints).\n\nAs far as Newton\'s laws is concerned, it is a special case. As far as\nphysics is concerned, it is almost centrally important.\n\nDan\n\nStrong_Field wrote:\n\n> "Dan Platt" <DanP57@ispwest.com> wrote in message\n> news:t08wd.2407\\$ou2.622@fe09.lga...\n>\n>>You mean, can this be derived from Newton\'s laws without an assumption\n>>of a gradient force?\n>\n>\n> Pardon my ignorance but what is a gradient force?\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>A force which can be computed from a gradient of a scalar function, as in

F = -grad V.

It is equivalent to the condition that the work W = integral F.dx

between two endpoints along a path of a mass is independent of the path,
in which case, W = -\Delta V (difference between endpoints).

As far as Newton's laws is concerned, it is a special case. As far as
physics is concerned, it is almost centrally important.

Dan

Strong_Field wrote:

> "Dan Platt" <DanP57@ispwest.com> wrote in message
> news:t08wd.2407$ou2.622@fe09.lga...
>
>>You mean, can this be derived from Newton's laws without an assumption
>>of a gradient force?
>
>
> Pardon my ignorance but what is a gradient force?
>

Hendrik van Hees

Dec20-04, 10:01 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>Igor Khavkine wrote:\n\n> It is the same as a conservative force, which means that we can write\n> the force F = - grad V, where grad is the gradient operator and V is\n> some scalar function called the potential. All the derivations of\n> energy conservation you\'ve gotten so far make this assumption.\n\nAlso the classical theory of the motion of point charges in given\n(external) static electro-magnetic fields obeys the energy conservation\nlaw, and the magnetic force is not a potential force.\n\nWithin analytical mechanics (Hamilton\'s principle) all Hamiltonians,\nwhich are not explicitly dependent on time, lead for sure to equations\nof motion which fulfill the energy conservation law (and vice versa),\naccording to Noether\'s theorem. Whether this is the most general case,\nI\'m not sure. In other words, I do not know, wheter one can construct\nan eom which is energy conserving, where the conservation does not\nfollow from a Noether symmetry.\n\n--\nHendrik van Hees Cyclotron Institute\nPhone: +1 979/845-1411 Texas A&M University\nFax: +1 979/845-1899 Cyclotron Institute, MS-3366\nhttp://theory.gsi.de/~vanhees/ College Station, TX 77843-3366\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>Igor Khavkine wrote:

> It is the same as a conservative force, which means that we can write
> the force F = - grad V, where grad is the gradient operator and V is
> some scalar function called the potential. All the derivations of
> energy conservation you've gotten so far make this assumption.

Also the classical theory of the motion of point charges in given
(external) static electro-magnetic fields obeys the energy conservation
law, and the magnetic force is not a potential force.

Within analytical mechanics (Hamilton's principle) all Hamiltonians,
which are not explicitly dependent on time, lead for sure to equations
of motion which fulfill the energy conservation law (and vice versa),
according to Noether's theorem. Whether this is the most general case,
I'm not sure. In other words, I do not know, wheter one can construct
an eom which is energy conserving, where the conservation does not
follow from a Noether symmetry.

--
Hendrik van Hees Cyclotron Institute
Phone: +1 979/845-1411 Texas A&M University
Fax: +1 979/845-1899 Cyclotron Institute, MS-3366
http://theory.gsi.de/~vanhees/ College Station, TX 77843-3366

Strong_Field

Dec20-04, 10:02 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>"Dan Platt" <DanP57@ispwest.com> wrote in message\nnews:t08wd.2407\\$ou2.622@fe09.lga...\n.. ..\n> The third law is sufficient to get\n> conservation of momentum, but it does so by injecting instantaneous\n> forces. Conservation of momentum is now recognized to be more\n> fundamental than action-reaction.\n\nWhat is the mathematical statement of this "third law" that I hear about\nall the time? Does it have a mathematical representation? Or do\nphysicists by tradition take the picturesque example Newton gave in his\nbook literally about a horse pulling a stone etc??\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>"Dan Platt" <DanP57@ispwest.com> wrote in message
news:t08wd.2407$ou2.622@fe09.lga...
....
> The third law is sufficient to get
> conservation of momentum, but it does so by injecting instantaneous
> forces. Conservation of momentum is now recognized to be more
> fundamental than action-reaction.

What is the mathematical statement of this "third law" that I hear about
all the time? Does it have a mathematical representation? Or do
physicists by tradition take the picturesque example Newton gave in his
book literally about a horse pulling a stone etc??

Dan Platt

Dec22-04, 06:00 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>Newton\'s example of the horse and the stone,\n(http://history.hanover.edu/courses/excerpts/347newt.html) is a\nsufficient representation of Newton\'s third law.\n\nNewton\'s mechanics were and are not a complete formulation of physics.\nThey contain some issues that came to be recognized to be mistakes in\nformulation (Mach\'s "Science of Mechanics" addresses some of these\nissues, such as absolute space -- absolute reference frames, but also\naren\'t developed as thoroughly as they came to be later). Particularly,\nNewton proposed an absolute "space" (reference frame), but you cannot\nconstruct a way to discover that frame from the structure of Newton\'s\nlaws (ironic given his famous statement: "hypothesis non fingo").\n\nNewton\'s Laws provided a good context to develop testable questions on\nwhich physics could be and was built, and from which more central pieces\n(such as energy conservation) could be recognized and understood.\n\nDan\n\nStrong_Field wrote:\n\n> "Dan Platt" <DanP57@ispwest.com> wrote in message\n> news:t08wd.2407\\$ou2.622@fe09.lga...\n> ...\n>\n>>The third law is sufficient to get\n>>conservation of momentum, but it does so by injecting instantaneous\n>>forces. Conservation of momentum is now recognized to be more\n>>fundamental than action-reaction.\n>\n>\n> What is the mathematical statement of this "third law" that I hear about\n> all the time? Does it have a mathematical representation? Or do\n> physicists by tradition take the picturesque example Newton gave in his\n> book literally about a horse pulling a stone etc??\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>Newton's example of the horse and the stone,
(http://history.hanover.edu/courses/excerpts/347newt.html) is a
sufficient representation of Newton's third law.

Newton's mechanics were and are not a complete formulation of physics.
They contain some issues that came to be recognized to be mistakes in
formulation (Mach's "Science of Mechanics" addresses some of these
issues, such as absolute space -- absolute reference frames, but also
aren't developed as thoroughly as they came to be later). Particularly,
Newton proposed an absolute "space" (reference frame), but you cannot
construct a way to discover that frame from the structure of Newton's
laws (ironic given his famous statement: "hypothesis non fingo").

Newton's Laws provided a good context to develop testable questions on
which physics could be and was built, and from which more central pieces
(such as energy conservation) could be recognized and understood.

Dan

Strong_Field wrote:

> "Dan Platt" <DanP57@ispwest.com> wrote in message
> news:t08wd.2407$ou2.622@fe09.lga...
> ...
>
>>The third law is sufficient to get
>>conservation of momentum, but it does so by injecting instantaneous
>>forces. Conservation of momentum is now recognized to be more
>>fundamental than action-reaction.
>
>
> What is the mathematical statement of this "third law" that I hear about
> all the time? Does it have a mathematical representation? Or do
> physicists by tradition take the picturesque example Newton gave in his
> book literally about a horse pulling a stone etc??
>

PanSynthesis@netscape.net

Dec22-04, 06:01 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>> Within analytical mechanics (Hamilton\'s principle) all Hamiltonians,\n> which are not explicitly dependent on time, lead for sure to\nequations\n> of motion which fulfill the energy conservation law (and vice versa),\n\n>I do not know, wheter one can construct\n> an eom which is energy conserving, where the conservation does not\n> follow from a Noether symmetry.\n\nLook at the book "Classical Field Theory" by Davison Soper. He gives\nthe Lagrangian for a damped harmonic oscillator, and a very complicated\nfunction it is -- but time independent. Requiring this Lagranian to be\nstationary yields the harmonic oscillator eom with linear damping. The\ncorresponding Hamiltonian function is a constant of the motion, but it\nis not the oscillator\'s energy which, of course, decreases.\n\nThe author also claims that Maxwell\'s equations, modified in the\nnatural way to allow for magnetic charge, have never been derived from\na Lagrangian -- presumably because for unknown reasons they cannot be.\nThis despite the fact that this modified em theory conserves energy and\nin all other respects seems to be a viable classical field theory.\n-Arnie King\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>> Within analytical mechanics (Hamilton's principle) all Hamiltonians,
> which are not explicitly dependent on time, lead for sure to
equations
> of motion which fulfill the energy conservation law (and vice versa),

>I do not know, wheter one can construct
> an eom which is energy conserving, where the conservation does not
> follow from a Noether symmetry.

Look at the book "Classical Field Theory" by Davison Soper. He gives
the Lagrangian for a damped harmonic oscillator, and a very complicated
function it is -- but time independent. Requiring this Lagranian to be
stationary yields the harmonic oscillator eom with linear damping. The
corresponding Hamiltonian function is a constant of the motion, but it
is not the oscillator's energy which, of course, decreases.

The author also claims that Maxwell's equations, modified in the
natural way to allow for magnetic charge, have never been derived from
a Lagrangian -- presumably because for unknown reasons they cannot be.
This despite the fact that this modified em theory conserves energy and
in all other respects seems to be a viable classical field theory.
-Arnie King

Frank Hellmann (Certhas -at- gmail -dot- com)

Dec23-04, 05:46 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>PanSynthesis@netscape.net wrote:\n> > Within analytical mechanics (Hamilton\'s principle) all Hamiltonians,\n> > which are not explicitly dependent on time, lead for sure to\n> equations\n> > of motion which fulfill the energy conservation law (and vice versa),\n>\n> >I do not know, wheter one can construct\n> > an eom which is energy conserving, where the conservation does not\n> > follow from a Noether symmetry.\n>\n> Look at the book "Classical Field Theory" by Davison Soper. He gives\n> the Lagrangian for a damped harmonic oscillator, and a very complicated\n> function it is -- but time independent. Requiring this Lagranian to be\n> stationary yields the harmonic oscillator eom with linear damping. The\n> corresponding Hamiltonian function is a constant of the motion, but it\n> is not the oscillator\'s energy which, of course, decreases.\n>\n\nWell in the electromagnetic field the momentum of the theory as p=mv is\nnot conserved, but Noethers theorem is talking about the cannonical\nmomentum which includes the elctromagnetic fields momentum.\nI assume this is a similar case where one needs to ascribe energy to\nsome other aspect of the system.\n\nfrank\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>PanSynthesis@netscape.net wrote:
> > Within analytical mechanics (Hamilton's principle) all Hamiltonians,
> > which are not explicitly dependent on time, lead for sure to
> equations
> > of motion which fulfill the energy conservation law (and vice versa),
>
> >I do not know, wheter one can construct
> > an eom which is energy conserving, where the conservation does not
> > follow from a Noether symmetry.
>
> Look at the book "Classical Field Theory" by Davison Soper. He gives
> the Lagrangian for a damped harmonic oscillator, and a very complicated
> function it is -- but time independent. Requiring this Lagranian to be
> stationary yields the harmonic oscillator eom with linear damping. The
> corresponding Hamiltonian function is a constant of the motion, but it
> is not the oscillator's energy which, of course, decreases.
>

Well in the electromagnetic field the momentum of the theory as p=mv is
not conserved, but Noethers theorem is talking about the cannonical
momentum which includes the elctromagnetic fields momentum.
I assume this is a similar case where one needs to ascribe energy to
some other aspect of the system.

frank

PanSynthesis@netscape.net

Dec24-04, 06:47 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>Frank Hellmann (Certhas -at- gmail -dot- com) wrote:\n\n> PanSynthesis@netscape.net wrote:\n> > Look at the book "Classical Field Theory" by Davison Soper. He gives\n> > the Lagrangian for a damped harmonic oscillator, and a very complicated\n> > function it is -- but time independent. Requiring this Lagranian to be\n> > stationary yields the harmonic oscillator eom with linear damping. The\n> > corresponding Hamiltonian function is a constant of the motion, but it\n> > is not the oscillator\'s energy which, of course, decreases.\n> >\n>\n> Well in the electromagnetic field the momentum of the theory as p=mv is\n> not conserved, but Noethers theorem is talking about the cannonical\n> momentum which includes the elctromagnetic fields momentum.\n> I assume this is a similar case where one needs to ascribe energy to\n> some other aspect of the system.\n\nNo, the point is that, in some instances, the dynamics of systems that\ndon\'t conserve energy can be described by Hamilton\'s principle and a\nLagrangian. In this case, the system is the harmonic oscillator with\nlinear damping. Think of it as being just a certain ODE, a problem in\napplied math. It turns out that this ODE can be derived from Hamilton\'s\nprinciple applied to a certain Lagrangian, giving us a counterexample\nto a statement made by an earlier poster to the effect that energy is\nconserved if the EOM is derived from a Lagrangian. It\'s true that, from\nthe point of view of physics, the linear damping describes the effect\nof many smaller bodies (molecules), and the dynamics of the many-body\nsystem (oscillator + molecules) are conservative. But this goes way\nbeyond my original, mathematical, one-body system. In this narrow\nmathematical view, the "other aspect of the system" you refer to\ndoesn\'t exist.\n\nThe ultimate point of these examples in Soper\'s book is to suggest\nthat, useful as it is, Hamilton\'s principle is not always a reliable\nprocedure for formulating fundamental theories: some systems it admits\nviolate energy conservation (damped oscillator), and some systems it\napparently excludes are physically permissible (magnetic monopoles).\nArnie King\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>Frank Hellmann (Certhas -at- gmail -dot- com) wrote:

> PanSynthesis@netscape.net wrote:
> > Look at the book "Classical Field Theory" by Davison Soper. He gives
> > the Lagrangian for a damped harmonic oscillator, and a very complicated
> > function it is -- but time independent. Requiring this Lagranian to be
> > stationary yields the harmonic oscillator eom with linear damping. The
> > corresponding Hamiltonian function is a constant of the motion, but it
> > is not the oscillator's energy which, of course, decreases.
> >
>
> Well in the electromagnetic field the momentum of the theory as p=mv is
> not conserved, but Noethers theorem is talking about the cannonical
> momentum which includes the elctromagnetic fields momentum.
> I assume this is a similar case where one needs to ascribe energy to
> some other aspect of the system.

No, the point is that, in some instances, the dynamics of systems that
don't conserve energy can be described by Hamilton's principle and a
Lagrangian. In this case, the system is the harmonic oscillator with
linear damping. Think of it as being just a certain ODE, a problem in
applied math. It turns out that this ODE can be derived from Hamilton's
principle applied to a certain Lagrangian, giving us a counterexample
to a statement made by an earlier poster to the effect that energy is
conserved if the EOM is derived from a Lagrangian. It's true that, from
the point of view of physics, the linear damping describes the effect
of many smaller bodies (molecules), and the dynamics of the many-body
system (oscillator + molecules) are conservative. But this goes way
beyond my original, mathematical, one-body system. In this narrow
mathematical view, the "other aspect of the system" you refer to
doesn't exist.

The ultimate point of these examples in Soper's book is to suggest
that, useful as it is, Hamilton's principle is not always a reliable
procedure for formulating fundamental theories: some systems it admits
violate energy conservation (damped oscillator), and some systems it
apparently excludes are physically permissible (magnetic monopoles).
Arnie King

Dan Platt

Dec28-04, 01:52 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>Strong_Field wrote:\n> "Igor Khavkine" <igor.kh@gmail.com> wrote in message\n> news:1103584469.379730.180660@f14g2000cwb.googlegr oups.com...\n>\n>>Strong_Field wrote:\n>>\n>>>"Igor Khavkine" <k_igor_k@lycos.com> wrote in message\n>>>news:pan.2004.12.14.15.56.52.1093@lyco s.com...\n>>>\n>>>>On Tue, 14 Dec 2004 11:35:20 +0000, Poakfield wrote:\n>>>>\n>>>>\n>>>>>Hi! Could someone please tell me if the law of conservation of energy\n>>>\n>>>can\n>>>\n>>>>>be derived only from Newton\'s motion laws? Thanks.\n>>>>\n>>>>Together with the assumption of conservative forces only (F = -grad V).\n>>>>Here is how:\n>>>>\n>>>> m x\'\' = -grad V Newton\'s 2nd law with\n>>>> conservative force.\n>>>> m x\' x\'\' = - x\' grad V Multiplication by x\' on both sides.\n>>>> d/dt (m x\'^2)/2 = -d/dt V Invert product rule (f^2)\' = 2 f\'= f\n>>>> and chain rule g(f)\' = f\' g\'(f).\n>>>> d/dt (m x\'^2/2 + V) = 0 Add dV/dt to both sides.\n>>>> m v^2/2 + V = E, Indefinite integral of both sides\n>>>> with respect to dt (don\'t forget\n>>>> the constant!).\n>>>>\n>>>>where v = x\' and E is some constant.\n>\n>\n> I am trying to understand if this is only a mathematical derivation or\n> if= it has physics content. For instance, I would call, writing (A.B =\n> dot pro= duct) and then deriving (A*B cos x) from it a mathematical\n> derivation. This is only a manipulation of symbols in an identity.\n>\n> If there is an identity, we can always break it apart and label its\n> parts with various names. Then, we can equate the labels we assigned to\n> the par= ts of the identity and recover the original identity. This is\n> the mathematic= al derivation. As far as the physics is concerned there\n> is only one identity. The names mathematicians give to its parts have no\n> physical meaning. (Thi= s is my opinion, it may not be true.)\n\n\nAnother way to look at it is that the mathematical derivations provide\nways to connect physical meanings to each other. Math also provides\nconstraints on the meaning of physical statements (statements can have a\ntighter definition than what is provided for in the math). Sometimes,\nthe math can provide a way to recognize when a physical idea is vacant.\n\nFor example -- Newton postulated an absolute frame/space. However,\nNewton\'s laws are invariant under Gallilean transformations. It is\nimpossible to construct a test from Newtons laws that would detect a\nspecial frame. Since this special frame was supposed to provide a\nfoundation for inertia as posited in his laws of motion, but it was\nimpossible to identify such a frame from the behavior of inertia, it was\npossible to drop the notion of the absolute frame.\n\n>........... [stuff snipped]\n>\n> We can write the first derivative in delta notation as the increment in\n> distance x divided by the increment in time t: delta x / delta t. But\n> the distance moved in the first increment of time equals the\n> acceleration. Therefore, we can write\n>\n> delta x/delta t = [-grad V].\n\nGrad is usually definable this way (x, dx vectors; s, V scalars):\n\nV(x + s dx) = V(x) + s grad V . dx + O(s^2).\n\nSometimes you cannot construct such a linear relationship (for example,\nperturbations of degenerate eigenvalue/eigenvector systems). You can\npick off the components of grad V by selecting dx along the basis vector\ndirections (x, y, z axes). You can take scalar derivatives of V with\nrespect to s, as well, providing an extension of a derivative from\nsimple scalar theorems.\n\n>\n> grad operator is the generalization of the derivative operator into\n> three dimensions. There is no new concept involved. Using the above\n> reasoning w= e can simplify grad V into a delta notation and write it as\n> [increment in p= / increment in q].\n>\n> Can you clarify what p and q is? Is it possible to express grad V as the\n> ratio of increments? What are those increments? Since we are working in\n> simple orthogonal coordinates, this should be possible to do.\n>\n>[snip]\n>\n> I hope all this makes sense. Again I am trying to understand if there is\n> only a mathematical transformation here or if there is a derivation\n> involving physics. Physical notions can be written by various notations.\n> = And it can be useful to look at the problem with different notations.\n> Like in the other thread there is a discussion of how physics would look\n> if it is expressed in projective geometry.\n\nAnother post pointed out that it is possible to construct conserved\nthings that don\'t carry particular physical meaning. On the other hand,\nthe idea of conservation of energy is a fundamental (distinct from\nbasic) physical idea. Perhaps it is more meaningful to ask what kind of\nforce satisfies conservation of energy given Newton\'s laws. If\nconservation of energy is a dominant physical mode of behavior that\npenetrates into thermodynamics, etc, then all of the important forces\nthat will show up in the application of Newton\'s laws will have a\ngradient-like behavior. (Note, in electromagnetic systems, the thing\nthat acts like a conserved momentum and the thing that acts like a\nconserved energy are more complicated than mv and mv^2/2.)\n\n>\n> Well, after writing the above, the other reply by =93Dan Platt=94\n> appeare= d in my newsserver. Combined with what he wrote now it appears\n> to me that F = g= rad V is nothing more than a definition of a function\n> of distance. And this is = the definition of potential. Then already\n> mx=92=92 = -grad V is =93change i= n motion = equals change in\n> height,=94 which is the statement of the conservation of energy.\n\nAs far as Newton\'s laws of motion are concerned, F=-grad V is a special\ncase. As far as physics is concerned, this type of force law is of\ncentral interest.\n\n>\n> So it may be true that there is only a mathematical derivation here.\n> Comments appreciated. Thanks.\n>\n\nHumans are trying to describe the behavior of things that would be doing\nwhat they do regardless of whether humans were there to describe them.\nTo some extent, the notion of physical law is a human construct, which\nis subject to change as our understanding improves. Conservation of\nenergy has survived. Newton\'s laws have survived with modifications\n(both quantum and relativistic) as our understanding has improved. The\nquestion of whether these are "corrected" by these improvements\nimmediately gets to one facet of the point you\'re trying to dig at: the\nconcepts in conversation with these laws of motion and conservation are\nstill intact; the mathematical expression of those ideas have changed\nand have become more tightly connected over the years.\n\nDan\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>Strong_Field wrote:
> "Igor Khavkine" <igor.kh@gmail.com> wrote in message
> news:1103584469.379730.180660@f14g2000cwb.googlegr oups.com...
>
>>Strong_Field wrote:
>>
>>>"Igor Khavkine" <k_{igor_k}@lycos.com> wrote in message
>>>news:pan.2004.12.14.15.56.52.1093@lycos.com...
>>>
>>>>On Tue, 14 Dec 2004 11:35:20 +0000, Poakfield wrote:
>>>>
>>>>
>>>>>Hi! Could someone please tell me if the law of conservation of energy
>>>
>>>can
>>>
>>>>>be derived only from Newton's motion laws? Thanks.
>>>>
>>>>Together with the assumption of conservative forces only (F = -grad V).
>>>>Here is how:
>>>>
>>>> m x'' = -grad V Newton's 2nd law with
>>>> conservative force.
>>>> m x' x'' = - x' grad V Multiplication by x' on both sides.
>>>> d/dt (m x'^2)/2 = -d/dt V Invert product rule (f^2)' = 2 f'= f
>>>> and chain rule g(f)' = f' g'(f).
>>>> d/dt (m x'^2/2 + V) = Add dV/dt to both sides.
>>>> m v^2/2 + V = E, Indefinite integral of both sides
>>>> with respect to dt (don't forget
>>>> the constant!).
>>>>
>>>>where v = x' and E is some constant.
>
>
> I am trying to understand if this is only a mathematical derivation or
> if= it has physics content. For instance, I would call, writing (A.B =
> dot pro= duct) and then deriving (A*B cos x) from it a mathematical
> derivation. This is only a manipulation of symbols in an identity.
>
> If there is an identity, we can always break it apart and label its
> parts with various names. Then, we can equate the labels we assigned to
> the par= ts of the identity and recover the original identity. This is
> the mathematic= al derivation. As far as the physics is concerned there
> is only one identity. The names mathematicians give to its parts have no
> physical meaning. (Thi= s is my opinion, it may not be true.)

Another way to look at it is that the mathematical derivations provide
ways to connect physical meanings to each other. Math also provides
constraints on the meaning of physical statements (statements can have a
tighter definition than what is provided for in the math). Sometimes,
the math can provide a way to recognize when a physical idea is vacant.

For example -- Newton postulated an absolute frame/space. However,
Newton's laws are invariant under Gallilean transformations. It is
impossible to construct a test from Newtons laws that would detect a
special frame. Since this special frame was supposed to provide a
foundation for inertia as posited in his laws of motion, but it was
impossible to identify such a frame from the behavior of inertia, it was
possible to drop the notion of the absolute frame.

>........... [stuff snipped]
>
> We can write the first derivative in \delta notation as the increment in
> distance x divided by the increment in time t: \delta x / \delta t. But
> the distance moved in the first increment of time equals the
> acceleration. Therefore, we can write
>
> \delta x/\delta t = [-grad V].

Grad is usually definable this way (x, dx vectors; s, V scalars):

V(x + s dx) = V(x) + s grad V . dx + O(s^2).

Sometimes you cannot construct such a linear relationship (for example,
perturbations of degenerate eigenvalue/eigenvector systems). You can
pick off the components of grad V by selecting dx along the basis vector
directions (x, y, z axes). You can take scalar derivatives of V with
respect to s, as well, providing an extension of a derivative from
simple scalar theorems.

>
> grad operator is the generalization of the derivative operator into
> three dimensions. There is no new concept involved. Using the above
> reasoning w= e can simplify grad V into a \delta notation and write it as
> [increment in p= / increment in q].
>
> Can you clarify what p and q is? Is it possible to express grad V as the
> ratio of increments? What are those increments? Since we are working in
> simple orthogonal coordinates, this should be possible to do.
>
>[snip]
>
> I hope all this makes sense. Again I am trying to understand if there is
> only a mathematical transformation here or if there is a derivation
> involving physics. Physical notions can be written by various notations.
> = And it can be useful to look at the problem with different notations.
> Like in the other thread there is a discussion of how physics would look
> if it is expressed in projective geometry.

Another post pointed out that it is possible to construct conserved
things that don't carry particular physical meaning. On the other hand,
the idea of conservation of energy is a fundamental (distinct from
basic) physical idea. Perhaps it is more meaningful to ask what kind of
force satisfies conservation of energy given Newton's laws. If
conservation of energy is a dominant physical mode of behavior that
penetrates into thermodynamics, etc, then all of the important forces
that will show up in the application of Newton's laws will have a
gradient-like behavior. (Note, in electromagnetic systems, the thing
that acts like a conserved momentum and the thing that acts like a
conserved energy are more complicated than mv and mv^2/2.)

>
> Well, after writing the above, the other reply by =93Dan Platt=94
> appeare= d in my newsserver. Combined with what he wrote now it appears
> to me that F = g= rad V is nothing more than a definition of a function
> of distance. And this is = the definition of potential. Then already
> mx=92=92 = -grad V is =93change i= n motion = equals change in
> height,=94 which is the statement of the conservation of energy.

As far as Newton's laws of motion are concerned, F=-grad V is a special
case. As far as physics is concerned, this type of force law is of
central interest.

>
> So it may be true that there is only a mathematical derivation here.
> Comments appreciated. Thanks.
>

Humans are trying to describe the behavior of things that would be doing
what they do regardless of whether humans were there to describe them.
To some extent, the notion of physical law is a human construct, which
is subject to change as our understanding improves. Conservation of
energy has survived. Newton's laws have survived with modifications
(both quantum and relativistic) as our understanding has improved. The
question of whether these are "corrected" by these improvements
immediately gets to one facet of the point you're trying to dig at: the
concepts in conversation with these laws of motion and conservation are
still intact; the mathematical expression of those ideas have changed
and have become more tightly connected over the years.

Dan

Strong_Field

Jan4-05, 02:06 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>"Dan Platt" <DanP57@ispwest.com> wrote in message\nnews:cqpqkq0hvj@enews3.newsguy.com...\n\n > Grad is usually definable this way (x, dx vectors; s, V scalars):\n>\n> V(x + s dx) = V(x) + s grad V . dx + O(s^2).\n\nThis seems to corroborate what I wrote initially. Ignoring your functions s\nand O, I don\'t know what they are, your expression simplifies essentially to\ndy/dx = grad V. Does this definition say more than the definition given in\nthe other post as a vector pointing in the greatest increase and having the\nmagnitude of change in velocity?\n\n> ... the idea of conservation of energy is a fundamental (distinct from\n> basic) physical idea. Perhaps it is more meaningful to ask what kind of\n> force satisfies conservation of energy given Newton\'s laws. If\n> conservation of energy is a dominant physical mode of behavior that\n> penetrates into thermodynamics, etc, then all of the important forces\n> that will show up in the application of Newton\'s laws will have a\n> gradient-like behavior.\n\nI think you are trying to say that conservation of energy is the fundamental\naxiom of physics. If so I agree. The equation itself is a statement of the\nconservation of energy. Everything else is derived from the assumption that\nperpetual motion is impossible. Let\'s think about it topologically. In\ntopology equivalent objects can be transformed into each other. It is only\nconvention that the shape A is the master shape. Someone will say no the\nshape B is the master shape. Similarly in physics there is no master\nequation that can be identified as such. In physics any equation can be\nderived from the other. Whatever shape you give to it your topology stays\ninvariant. That’s the energy conservation. The topological invariant. Which\nis the equation itself. Because whatever its “shape” is, the equation is the\ninvariant. Therefore in physics the equation itself is the definition of\nenergy conservation. As such you cannot derive it from any other equation.\nYou can only use energy conservation to prove whatever you want. This also\nshows that deriving energy conservation from Newton amounts to making\nalgebraic transformations.\n\nAnd, conservation of energy is not a “physical mode of behavior.” It is an\naxiom. There is nothing physical about it.\n\n\n> Humans are trying to describe the behavior of things that would be doing\n> what they do regardless of whether humans were there to describe them.\n> To some extent, the notion of physical law is a human construct, which\n> is subject to change as our understanding improves.\n\nYou are talking about a philosophical and epistemological concept of\n“physical law.” I agree that’s a human construct. But that\ndiscussion is not relevant to “Newton’s laws.” Because\nNewton’s Laws are not “physical laws.” They may be laws of\nNewtonian physics, but they are not laws of nature in the sense you are\nthinking. Newton was a serial definer. He had the authority to establish\nhis countless definitions as laws of nature in physics. If you look at\nthe Principia, they are called axioms. In other words, they are\ndefinitions Newton chose to make. By tradition we call them\n“laws” out of respect for Newton. As long as we know that they\nare laws with an asterisk, I guess there is no harm done to science.\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>"Dan Platt" <DanP57@ispwest.com> wrote in message
news:cqpqkq0hvj@enews3.newsguy.com...

> Grad is usually definable this way (x, dx vectors; s, V scalars):
>
> V(x + s dx) = V(x) + s grad V . dx + O(s^2).

This seems to corroborate what I wrote initially. Ignoring your functions s
and O, I don't know what they are, your expression simplifies essentially to
dy/dx = grad V. Does this definition say more than the definition given in
the other post as a vector pointing in the greatest increase and having the
magnitude of change in velocity?

> ... the idea of conservation of energy is a fundamental (distinct from
> basic) physical idea. Perhaps it is more meaningful to ask what kind of
> force satisfies conservation of energy given Newton's laws. If
> conservation of energy is a dominant physical mode of behavior that
> penetrates into thermodynamics, etc, then all of the important forces
> that will show up in the application of Newton's laws will have a
> gradient-like behavior.

I think you are trying to say that conservation of energy is the fundamental
axiom of physics. If so I agree. The equation itself is a statement of the
conservation of energy. Everything else is derived from the assumption that
perpetual motion is impossible. Let's think about it topologically. In
topology equivalent objects can be transformed into each other. It is only
convention that the shape A is the master shape. Someone will say no the
shape B is the master shape. Similarly in physics there is no master
equation that can be identified as such. In physics any equation can be
derived from the other. Whatever shape you give to it your topology stays
invariant. That’s the energy conservation. The topological invariant. Which
is the equation itself. Because whatever its “shape” is, the equation is the
invariant. Therefore in physics the equation itself is the definition of
energy conservation. As such you cannot derive it from any other equation.
You can only use energy conservation to prove whatever you want. This also
shows that deriving energy conservation from Newton amounts to making
algebraic transformations.

And, conservation of energy is not a “physical mode of behavior.” It is an
axiom. There is nothing physical about it.

> Humans are trying to describe the behavior of things that would be doing
> what they do regardless of whether humans were there to describe them.
> To some extent, the notion of physical law is a human construct, which
> is subject to change as our understanding improves.

You are talking about a philosophical and epistemological concept of
“physical law.” I agree that’s a human construct. But that
discussion is not relevant to “Newton’s laws.” Because
Newton’s Laws are not “physical laws.” They may be laws of
Newtonian physics, but they are not laws of nature in the sense you are
thinking. Newton was a serial definer. He had the authority to establish
his countless definitions as laws of nature in physics. If you look at
the Principia, they are called axioms. In other words, they are
definitions Newton chose to make. By tradition we call them
“laws” out of respect for Newton. As long as we know that they
are laws with an asterisk, I guess there is no harm done to science.

Daniel E. Platt

Jan6-05, 04:11 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>Strong_Field wrote:\n> "Dan Platt" <DanP57@ispwest.com> wrote in message\n> news:cqpqkq0hvj@enews3.newsguy.com...\n>\n>\n>>Gra d is usually definable this way (x, dx vectors; s, V scalars):\n>>\n>>V(x + s dx) = V(x) + s grad V . dx + O(s^2).\n>\n>\n> This seems to corroborate what I wrote initially. Ignoring your functio=\nns s\n> and O, I don\'t know what they are, your expression simplifies essential=\nly to\n> dy/dx = grad V. Does this definition say more than the definition giv=\nen in\n> the other post as a vector pointing in the greatest increase and having=\nthe\n> magnitude of change in velocity?\n\nThis was a Taylor (power) series in s, easier than a power series in\nmixed dx components. The O() refers to powers of s^2 and higher (ie,\nsome number times s^2 + another number times s^3 +...). It did look\nlike the kind of thing you said you wanted, along with a method to get\nthere. However, it is not always possible to extract such a power\nseries (the degenerate perturbation series for matrices is an example).\n\n>\n>\n>>... the idea of conservation of energy is a fundamental (distinct from\n>>basic) physical idea. Perhaps it is more meaningful to ask what kind o=\nf\n>>force satisfies conservation of energy given Newton\'s laws. If\n>>conservation of energy is a dominant physical mode of behavior that\n>>penetrates into thermodynamics, etc, then all of the important forces\n>>that will show up in the application of Newton\'s laws will have a\n>>gradient-like behavior.\n>\n>\n> I think you are trying to say that conservation of energy is the fundam=\nental\n> axiom of physics. If so I agree.\n\nI would not even quite say it is an axiom. Physics is descriptive, with\nlots of loose pieces that don\'t quite fit together very well in all\nplaces (though there has been lots of progress in finding ways to make\nthem fit).\n\n> The equation itself is a statement of t=\nhe\n> conservation of energy. Everything else is derived from the assumption =\nthat\n> perpetual motion is impossible.\n\nYou *can* derive an expression of the 2nd law of thermodynamics that way\n(well, you can do it more precisely in terms of heat engines). There is\nno way to construct a probe into the microscopic world without injecting\nsome kind of statistical assumption. It doesn\'t help that the\nstatistical pictureS have multiple forms. For example, the picture of\nan ensemble involves multiple instances of a single experiment. But we\nsee thermodynamic equilibration over one single experiment for a single\nsystem. Then there\'s Birkhoff\'s thm, that states that systems with\ncertain characteristics will always show a time average close to the\nensemble average for all functions... but usually, we don\'t look at all\nfunctions, rather we look at a very special set of macroscopic\nfunctions, and the time we sample is much shorter than the time required\nfor a system to sample most of phase space (which is often rather larger\nthan the age of the universe). So, for some very special but very\ninteresting macroscopic functions, we see thermodynamic behavior; for\nmicroscopic variables, the quantum scattering processes look just like\nindividual events. As a point of interest, you don\'t have to get too\nsmall to run into a circumstance where this type of thing starts to\nbreak down... Mitochondria do a very important biological task in our\ncells, converting ADP to ATP, mediating the reaction with the transfer\nof protons (H+ ions) through transmembrane enzymes called ATPase. The\nquestion is, what is the charge distribution like near the membrane\nwhere this happens? It turns out that the density of the charge\ncarriers is smaller than the Debye screening length you get if you\nassume that chemical potential and smooth particle densities are valid.\nThe mistake is that you only get maybe 6 loose protons in a\nmitochondrion at a time, and the assumption of thermodynamic diffusion\nat that scale doesn\'t quite work.\n\n> Let\'s think about it topologically.=\nIn\n> topology equivalent objects can be transformed into each other. It is o=\nnly\n> convention that the shape A is the master shape. Someone will say no th=\ne\n> shape B is the master shape. Similarly in physics there is no master\n> equation that can be identified as such. In physics any equation can be\n> derived from the other. Whatever shape you give to it your topology sta=\nys\n> invariant. That=92s the energy conservation. The topological invariant.=\nWhich\n> is the equation itself. Because whatever its =93shape=94 is, the equati=\non is the\n> invariant. Therefore in physics the equation itself is the definition o=\nf\n> energy conservation. As such you cannot derive it from any other equati=\non.\n> You can only use energy conservation to prove whatever you want. This a=\nlso\n> shows that deriving energy conservation from Newton amounts to making\n> algebraic transformations.\n\nEnergy conservation is intimately connected to time translation\ninvariance (ie -- the description of the mechanics of a system is\nindependent of when the experiment is performed). It also emerged in a\ndistinct statement in the realization that energy did not disappear in\nnon-conservative mechanical systems -- it was converted to/transfered as\n"heat." These distinct ideas were re-connected through statistical\nmechanical considerations. The notion of energy conservation suggests\nenergy transfer; it isn\'t just mechanical things that can carry energy,\nso can fields (electromagnetism). This connected to issues of\nforce-at-a-distance (instantaneous energy transfer) that would show up\nas a problem, later. One face of protecting the form of the statement\nof electrodynamics as basic physical law was the realization that time\nand space both transformed covariantly for different observers\n(otherwise, you have an absolute space again -- which Newton\'s laws\nprovided no means of detecting by themselves). If so, then energy and\nmomentum also transform in a similar way as space and time. Also,\nformulations of fluid energy density transfer are identical to fluid\nmass transfer... you canot tell the difference.\n\n>\n> And, conservation of energy is not a =93physical mode of behavior.=94 I=\nt is an\n> axiom. There is nothing physical about it.\n>\n\nRather, it is something that gets discovered and re-discovered over and\nover again in new contexts, usually with a face that is not easily\nreconciled with prior pictures. The part that is "physical" is that it\nmust be connected to empirical experience, or else it is simply a\nmathematical diversion.\n\n>\n>\n>>Humans are trying to describe the behavior of things that would be doin=\ng\n>>what they do regardless of whether humans were there to describe them.\n>>To some extent, the notion of physical law is a human construct, which\n>>is subject to change as our understanding improves.\n>\n>\n> You are talking about a philosophical and epistemological concept of\n> =93physical law.=94 I agree that=92s a human construct. But that\n> discussion is not relevant to =93Newton=92s laws.=94 Because\n> Newton=92s Laws are not =93physical laws.=94 They may be laws of\n> Newtonian physics, but they are not laws of nature in the sense you are\n> thinking. Newton was a serial definer. He had the authority to establis=\nh\n> his countless definitions as laws of nature in physics. If you look at\n> the Principia, they are called axioms. In other words, they are\n> definitions Newton chose to make. By tradition we call them\n> =93laws=94 out of respect for Newton. As long as we know that they\n> are laws with an asterisk, I guess there is no harm done to science.\n>\n\nIt doesn\'t take much authority to make an axiom. You have to start\nsomewhere. The notion of physical law is a harder problem: your\ndescription has to conform to observation to the best of your\nexperimental capability. The interplay between math and description\noccasionally reveals striking new insights into the relationships\nbetween ideas (symmetry and conservation, for instance) that otherwise\nwould not be obvious. I think the exciting spots are where things don\'t\nfit together very well... that\'s likely where some understanding is\ngoing to emerge when those issues are resolved.\n\nDan\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>Strong_Field wrote:
> "Dan Platt" <DanP57@ispwest.com> wrote in message
> news:cqpqkq0hvj@enews3.newsguy.com...
>
>
>>Grad is usually definable this way (x, dx vectors; s, V scalars):
>>
>>V(x + s dx) = V(x) + s grad V . dx + O(s^2).
>
>
> This seems to corroborate what I wrote initially. Ignoring your functio=
ns s
> and O, I don't know what they are, your expression simplifies essential=
ly to
> dy/dx = grad V. Does this definition say more than the definition giv=
en in
> the other post as a vector pointing in the greatest increase and having=
the
> magnitude of change in velocity?

This was a Taylor (power) series in s, easier than a power series in
mixed dx components. The O() refers to powers of s^2 and higher (ie,
some number times s^2 + another number times s^3 +...). It did look
like the kind of thing you said you wanted, along with a method to get
there. However, it is not always possible to extract such a power
series (the degenerate perturbation series for matrices is an example).

>
>
>>... the idea of conservation of energy is a fundamental (distinct from
>>basic) physical idea. Perhaps it is more meaningful to ask what kind o=
f
>>force satisfies conservation of energy given Newton's laws. If
>>conservation of energy is a dominant physical mode of behavior that
>>penetrates into thermodynamics, etc, then all of the important forces
>>that will show up in the application of Newton's laws will have a
>>gradient-like behavior.
>
>
> I think you are trying to say that conservation of energy is the fundam=
ental
> axiom of physics. If so I agree.

I would not even quite say it is an axiom. Physics is descriptive, with
lots of loose pieces that don't quite fit together very well in all
places (though there has been lots of progress in finding ways to make
them fit).

> The equation itself is a statement of t=
he
> conservation of energy. Everything else is derived from the assumption =
that
> perpetual motion is impossible.

You *can* derive an expression of the 2nd law of thermodynamics that way
(well, you can do it more precisely in terms of heat engines). There is
no way to construct a probe into the microscopic world without injecting
some kind of statistical assumption. It doesn't help that the
statistical pictureS have multiple forms. For example, the picture of
an ensemble involves multiple instances of a single experiment. But we
see thermodynamic equilibration over one single experiment for a single
system. Then there's Birkhoff's thm, that states that systems with
certain characteristics will always show a time average close to the
ensemble average for all functions... but usually, we don't look at all
functions, rather we look at a very special set of macroscopic
functions, and the time we sample is much shorter than the time required
for a system to sample most of phase space (which is often rather larger
than the age of the universe). So, for some very special but very
interesting macroscopic functions, we see thermodynamic behavior; for
microscopic variables, the quantum scattering processes look just like
individual events. As a point of interest, you don't have to get too
small to run into a circumstance where this type of thing starts to
break down... Mitochondria do a very important biological task in our
cells, converting ADP to ATP, mediating the reaction with the transfer
of protons (H+ ions) through transmembrane enzymes called ATPase. The
question is, what is the charge distribution like near the membrane
where this happens? It turns out that the density of the charge
carriers is smaller than the Debye screening length you get if you
assume that chemical potential and smooth particle densities are valid.
The mistake is that you only get maybe 6 loose protons in a
mitochondrion at a time, and the assumption of thermodynamic diffusion
at that scale doesn't quite work.

> Let's think about it topologically.=
In
> topology equivalent objects can be transformed into each other. It is o=
nly
> convention that the shape A is the master shape. Someone will say no th=
e
> shape B is the master shape. Similarly in physics there is no master
> equation that can be identified as such. In physics any equation can be
> derived from the other. Whatever shape you give to it your topology sta=
ys
> invariant. That=92s the energy conservation. The topological invariant.=
Which
> is the equation itself. Because whatever its =93shape=94 is, the equati=
on is the
> invariant. Therefore in physics the equation itself is the definition o=
f
> energy conservation. As such you cannot derive it from any other equati=
on.
> You can only use energy conservation to prove whatever you want. This a=
lso
> shows that deriving energy conservation from Newton amounts to making
> algebraic transformations.

Energy conservation is intimately connected to time translation
invariance (ie -- the description of the mechanics of a system is
independent of when the experiment is performed). It also emerged in a
distinct statement in the realization that energy did not disappear in
non-conservative mechanical systems -- it was converted to/transfered as
"heat." These distinct ideas were re-connected through statistical
mechanical considerations. The notion of energy conservation suggests
energy transfer; it isn't just mechanical things that can carry energy,
so can fields (electromagnetism). This connected to issues of
force-at-a-distance (instantaneous energy transfer) that would show up
as a problem, later. One face of protecting the form of the statement
of electrodynamics as basic physical law was the realization that time
and space both transformed covariantly for different observers
(otherwise, you have an absolute space again -- which Newton's laws
provided no means of detecting by themselves). If so, then energy and
momentum also transform in a similar way as space and time. Also,
formulations of fluid energy density transfer are identical to fluid
mass transfer... you canot tell the difference.

>
> And, conservation of energy is not a =93physical mode of behavior.=94 I=
t is an
> axiom. There is nothing physical about it.
>

Rather, it is something that gets discovered and re-discovered over and
over again in new contexts, usually with a face that is not easily
reconciled with prior pictures. The part that is "physical" is that it
must be connected to empirical experience, or else it is simply a
mathematical diversion.

>
>
>>Humans are trying to describe the behavior of things that would be doin=
g
>>what they do regardless of whether humans were there to describe them.
>>To some extent, the notion of physical law is a human construct, which
>>is subject to change as our understanding improves.
>
>
> You are talking about a philosophical and epistemological concept of
> =93physical law.=94 I agree that=92s a human construct. But that
> discussion is not relevant to =93Newton=92s laws.=94 Because
> Newton=92s Laws are not =93physical laws.=94 They may be laws of
> Newtonian physics, but they are not laws of nature in the sense you are
> thinking. Newton was a serial definer. He had the authority to establis=
h
> his countless definitions as laws of nature in physics. If you look at
> the Principia, they are called axioms. In other words, they are
> definitions Newton chose to make. By tradition we call them
> =93laws=94 out of respect for Newton. As long as we know that they
> are laws with an asterisk, I guess there is no harm done to science.
>

It doesn't take much authority to make an axiom. You have to start
somewhere. The notion of physical law is a harder problem: your
description has to conform to observation to the best of your
experimental capability. The interplay between math and description
occasionally reveals striking new insights into the relationships
between ideas (symmetry and conservation, for instance) that otherwise
would not be obvious. I think the exciting spots are where things don't
fit together very well... that's likely where some understanding is
going to emerge when those issues are resolved.

Dan

Strong_Field

Jan11-05, 03:49 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>"Daniel E. Platt" <DanP57@optonline.net> wrote in message\nnews:7lXCd.93581\\$pA.48432@fe09.lga...\n >\n> I would not even quite say it is an axiom. Physics is descriptive, with\n> lots of loose pieces that don\'t quite fit together very well in all\n> places (though there has been lots of progress in finding ways to make\n> them fit).\n\nAre there two kinds of conservation of energy in physics? One for mechanical\nsystems and one for orbits?\n\n> ...The part that is "physical" is that it\n> must be connected to empirical experience, or else it is simply a\n> mathematical diversion.\n>\nYes. I think you are right. I was thinking as a concept it was an abstract\nnotion.\n\n> It doesn\'t take much authority to make an axiom.\n\nYes, but it takes authority to make your axiom into a law. Newton\nestablished such an authority.\n\n> ...The notion of physical law is a harder problem: your\n> description has to conform to observation to the best of your\n> experimental capability.\n\nYes. Still, calling a description a law gives it an authority over nature\nwhich is not always justified. Sometimes it may look that Newton was trying\nto legislate nature, more than he was trying to describe it.\n\n> The interplay between math and description\n> occasionally reveals striking new insights into the relationships\n> between ideas (symmetry and conservation, for instance) that otherwise\n> would not be obvious.\n\nI didn\'t exactly understand. Are you thinking that there are two kinds of\nrepresentation, one mathematical and one descriptive, that physicists use?\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>"Daniel E. Platt" <DanP57@optonline.net> wrote in message
news:7lXCd.93581$pA.48432@fe09.lga...
>
> I would not even quite say it is an axiom. Physics is descriptive, with
> lots of loose pieces that don't quite fit together very well in all
> places (though there has been lots of progress in finding ways to make
> them fit).

Are there two kinds of conservation of energy in physics? One for mechanical
systems and one for orbits?

> ...The part that is "physical" is that it
> must be connected to empirical experience, or else it is simply a
> mathematical diversion.
>
Yes. I think you are right. I was thinking as a concept it was an abstract
notion.

> It doesn't take much authority to make an axiom.

Yes, but it takes authority to make your axiom into a law. Newton
established such an authority.

> ...The notion of physical law is a harder problem: your
> description has to conform to observation to the best of your
> experimental capability.

Yes. Still, calling a description a law gives it an authority over nature
which is not always justified. Sometimes it may look that Newton was trying
to legislate nature, more than he was trying to describe it.

> The interplay between math and description
> occasionally reveals striking new insights into the relationships
> between ideas (symmetry and conservation, for instance) that otherwise
> would not be obvious.

I didn't exactly understand. Are you thinking that there are two kinds of
representation, one mathematical and one descriptive, that physicists use?