principle of least action [Archive]

alistair

Jun4-04, 02:49 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>EDWIN F.TAYLOR (http://www.eftaylor.com/pub/FmaAJPguest5.pdf)thinks\nstudents should be taught Newtonian mechanics from the point of view\nof the principle of least action because it is generally more useful\nand interesting than the standard way that Newtonian mechanics is\ntaught.Is he right?\nHere are some quotes from Richard Feynman and Freeman Dyson:\n\n\nWhen I was in high school, my physics teacher—whose name was Mr.\nBader—called me down one day after physics class and said, "You look\nbored; I want to tell you something interesting." Then he told me\nsomething which I found absolutely fascinating, and have, since then,\nalways found fascinating. . . the\nprinciple of least action.\n\n—Richard Feynman\n\nThirty-one years ago [1949!], Dick Feynman told me about his "sum over\nhistories" version of quantum mechanics. "The electron does anything\nit likes," he said. "It just goes in any direction at any speed,\nforward or backward in time, however it likes, and then you add up the\namplitudes and it gives you the wave-function." I said to him, "You\'re\ncrazy." But he wasn\'t.\n—Freeman J. Dyson, 198013\n\nThe text above and the text below is from EDWIN F TAYLOR\'S pdf file\nlisted at the start of this post:\n\nFeynman\'s approach is well documented.In brief, Feynman tells us how\nto evaluate the contribution of each alternative worldline to the\nprobability that a particle emitted at event A will be detected at a\nlater event B. The key point is that as the mass of a particle\nincreases, the set of worldlines between A and B that contribute\nsignificantly to the probability of detection at B shrinks to a narrow\npencil around the worldline of least action.15 In the limit of large\nmass this pencil contracts seamlessly to the single worldline\npredicted by the Principle of Least Action. Newtonian mechanics\nbecomes an obvious special case of quantum mechanics (Figure 1). Using\nthe Feynman approach, students can wield the power of quantum\nmechanics earlier in their careers than is possible when they start\nwiththe Schrödinger equation.\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>EDWIN F.TAYLOR (http://www.eftaylor.com/pub/FmaAJPguest5.pdf)thinks
students should be taught Newtonian mechanics from the point of view
of the principle of least action because it is generally more useful
and interesting than the standard way that Newtonian mechanics is
taught.Is he right?
Here are some quotes from Richard Feynman and Freeman Dyson:

When I was in high school, my physics teacher—whose name was Mr.
Bader—called me down one day after physics class and said, "You look
bored; I want to tell you something interesting." Then he told me
something which I found absolutely fascinating, and have, since then,
always found fascinating. . . the
principle of least action.

—Richard Feynman

Thirty-one years ago [1949!], Dick Feynman told me about his "sum over
histories" version of quantum mechanics. "The electron does anything
it likes," he said. "It just goes in any direction at any speed,
forward or backward in time, however it likes, and then you add up the
amplitudes and it gives you the wave-function." I said to him, "You're
crazy." But he wasn't.
—Freeman J. Dyson, 198013

The text above and the text below is from EDWIN F TAYLOR'S pdf file
listed at the start of this post:

Feynman's approach is well documented.In brief, Feynman tells us how
to evaluate the contribution of each alternative worldline to the
probability that a particle emitted at event A will be detected at a
later event B. The key point is that as the mass of a particle
increases, the set of worldlines between A and B that contribute
significantly to the probability of detection at B shrinks to a narrow
pencil around the worldline of least action.15 In the limit of large
mass this pencil contracts seamlessly to the single worldline
predicted by the Principle of Least Action. Newtonian mechanics
becomes an obvious special case of quantum mechanics (Figure 1). Using
the Feynman approach, students can wield the power of quantum
mechanics earlier in their careers than is possible when they start
withthe Schrödinger equation.

Doug Sweetser

Jun6-04, 04:26 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>alistair wrote:\n\n> EDWIN F.TAYLOR (http://www.eftaylor.com/pub/FmaAJPguest5.pdf)thinks\n> students should be taught Newtonian mechanics from the point of view\n> of the principle of least action because it is generally more useful\n> and interesting than the standard way that Newtonian mechanics is\n> taught.Is he right?\n\nYou would need to alter math education in parallel because the students\nwould need to understand the calculus of variations. It is a challenge\nto teach calculus! To make things appear more concrete, the goal in\nintegral calculus is to find a number. With the calculus of\nvariations, it is to find a minimum or maximum function. I had a year\nof calculus in high school, and a year and a half at MIT. Although I\nheard of the topic, I do not believe we spent any time on it.\n\nTaylor is right, physics and math education should evolve toward\nteaching this topic.\n\n\ndoug\nquaternions.com\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>alistair wrote:

> EDWIN F.TAYLOR (http://www.eftaylor.com/pub/FmaAJPguest5.pdf)thinks
> students should be taught Newtonian mechanics from the point of view
> of the principle of least action because it is generally more useful
> and interesting than the standard way that Newtonian mechanics is
> taught.Is he right?

You would need to alter math education in parallel because the students
would need to understand the calculus of variations. It is a challenge
to teach calculus! To make things appear more concrete, the goal in
integral calculus is to find a number. With the calculus of
variations, it is to find a minimum or maximum function. I had a year
of calculus in high school, and a year and a half at MIT. Although I
heard of the topic, I do not believe we spent any time on it.

Taylor is right, physics and math education should evolve toward
teaching this topic.

doug
quaternions.com

Patrick Powers

Jun7-04, 04:54 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>\nalistair@goforit64.fsnet.co.uk (alistair) wrote in message news:<861c1b21.0406020852.2b283160@posting.google. com>...\n> EDWIN F.TAYLOR (http://www.eftaylor.com/pub/FmaAJPguest5.pdf)thinks\n> students should be taught Newtonian mechanics from the point of view\n> of the principle of least action because it is generally more useful\n> and interesting than the standard way that Newtonian mechanics is\n> taught.Is he right?\n\n\nI found this to be poor pedagogy. I understand their motivation, but\nthey are replacing an relatively intuitive method with an abstract one\nthat takes a fair amount of skill to use. At the the stage of\nintroductory physics this is quite wrong: the student has neither\nintuition or skill. Newtonian physics is more than enough for many of\nthem. I found particulary irritating the use of the name Principle of\nLeast Action, which is more than unintuitive, it is just plain wrong\nand certain to actually inhibit understanding.\n\nI looked up the reference for the derivation of the Lagrange equations\nin an introductory physics text. The author doesn\'t even try to\nprovide any explanation of the meaning of the symbols and ideas in\nuse. "Here is the formula: it works: use it."\n\nI think a skillful educator might be able to replace good old Isaac.\nBut this is not the way.\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>alistair@goforit64.fsnet.co.uk (alistair) wrote in message news:<861c1b21.0406020852.2b283160@posting.google.com>...
> EDWIN F.TAYLOR (http://www.eftaylor.com/pub/FmaAJPguest5.pdf)thinks
> students should be taught Newtonian mechanics from the point of view
> of the principle of least action because it is generally more useful
> and interesting than the standard way that Newtonian mechanics is
> taught.Is he right?

I found this to be poor pedagogy. I understand their motivation, but
they are replacing an relatively intuitive method with an abstract one
that takes a fair amount of skill to use. At the the stage of
introductory physics this is quite wrong: the student has neither
intuition or skill. Newtonian physics is more than enough for many of
them. I found particulary irritating the use of the name Principle of
Least Action, which is more than unintuitive, it is just plain wrong
and certain to actually inhibit understanding.

I looked up the reference for the derivation of the Lagrange equations
in an introductory physics text. The author doesn't even try to
provide any explanation of the meaning of the symbols and ideas in
use. "Here is the formula: it works: use it."

I think a skillful educator might be able to replace good old Isaac.
But this is not the way.

Frank Hellmann

Jun8-04, 04:03 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>\nfrisbieinstein@yahoo.com (Patrick Powers) wrote in message news:<9511688f.0406060505.39e8730a@posting.google. com>...\n> alistair@goforit64.fsnet.co.uk (alistair) wrote in message news:<861c1b21.0406020852.2b283160@posting.google. com>...\n> > EDWIN F.TAYLOR (http://www.eftaylor.com/pub/FmaAJPguest5.pdf)thinks\n> > students should be taught Newtonian mechanics from the point of view\n> > of the principle of least action because it is generally more useful\n> > and interesting than the standard way that Newtonian mechanics is\n> > taught.Is he right?\n>\n>\n> I found this to be poor pedagogy. I understand their motivation, but\n....\n\nObviously you would want to teach them the calculus of variations\nfirst.\n\nAbstraction is usefull because it means simplification. abstractly\nthings are easy and simple, yet no education system actually teaches\nabstraction as a way of thinking or an approach to problems. It\'s\nalways stuck to intuition which is clunky and complicated and not\nactually as intuitive as the abstract thing once it\'s seen clearly iun\nit\'s abstraction.\n\nLook at the way Landau Lifschitz introduces Classical Mechanics it is\nbeautifull but without the proper background in some of the ideas of\nthe calculus of variation (and generall mathematic capacity) it\'s\nquite tough to follow.\nAssuming the background though it\'s pedagogically and physically\nhighly elegant, efficient and beautifull.\n\n(No wonder so many people dislike maths and physics, they never break\nthrough into abstraction and always are stuck with the unintuitive\n"intuitive pedagogical" approach. Which means they never develop an\nactuall mathematical intuition. In no other subject are the kids left\nalone so much, and this carries on to undergrad students)\n\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>frisbieinstein@yahoo.com (Patrick Powers) wrote in message news:<9511688f.0406060505.39e8730a@posting.google.com>...
> alistair@goforit64.fsnet.co.uk (alistair) wrote in message news:<861c1b21.0406020852.2b283160@posting.google.com>...
> > EDWIN F.TAYLOR (http://www.eftaylor.com/pub/FmaAJPguest5.pdf)thinks
> > students should be taught Newtonian mechanics from the point of view
> > of the principle of least action because it is generally more useful
> > and interesting than the standard way that Newtonian mechanics is
> > taught.Is he right?
>
>
> I found this to be poor pedagogy. I understand their motivation, but
....

Obviously you would want to teach them the calculus of variations
first.

Abstraction is usefull because it means simplification. abstractly
things are easy and simple, yet no education system actually teaches
abstraction as a way of thinking or an approach to problems. It's
always stuck to intuition which is clunky and complicated and not
actually as intuitive as the abstract thing once it's seen clearly iun
it's abstraction.

Look at the way Landau Lifschitz introduces Classical Mechanics it is
beautifull but without the proper background in some of the ideas of
the calculus of variation (and generall mathematic capacity) it's
quite tough to follow.
Assuming the background though it's pedagogically and physically
highly elegant, efficient and beautifull.

(No wonder so many people dislike maths and physics, they never break
through into abstraction and always are stuck with the unintuitive
"intuitive pedagogical" approach. Which means they never develop an
actuall mathematical intuition. In no other subject are the kids left
alone so much, and this carries on to undergrad students)

---
frank

Oz

Jun8-04, 02:23 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>\nFrank Hellmann <C.i.m@gmx.net> writes\n>\n>Abstraction is usefull because it means simplification. abstractly\n>things are easy and simple, yet no education system actually teaches\n>abstraction as a way of thinking or an approach to problems. It\'s always\n>stuck to intuition which is clunky and complicated and not actually as\n>intuitive as the abstract thing once it\'s seen clearly iun it\'s\n>abstraction.\n\nIsn\'t this the wrong way round?\nSurely one can only reach intuition via abstraction?\n\nAfter all intuition isn\'t (or certainly shouldn\'t be) \'guessing\'.\nIt should be operating a simplified model (ie an abstraction) in an\n\'analogue\' way to obtain a rough prediction.\n\n>Look at the way Landau Lifschitz introduces Classical Mechanics it is\n>beautifull but without the proper background in some of the ideas of the\n>calculus of variation (and generall mathematic capacity) it\'s quite\n>tough to follow. Assuming the background though it\'s pedagogically and\n>physically highly elegant, efficient and beautifull.\n\nUsing, or more, appreciating, mathematical tools requires one should\ngrok them well enough for the symbols and processes to *mean* something.\nRegrettably this usually takes rather a lot of hard graft (hence my\ntotal ignorance).\n\n--\nOz\nThis post is worth absolutely nothing and is probably fallacious.\n\nBTOPENWORLD address about to cease. DEMON address no longer in use.\n>>Use oz@farmeroz.port995.com (whitelist check on first posting)<<\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 <C.i.m@gmx.net> writes
>
>Abstraction is usefull because it means simplification. abstractly
>things are easy and simple, yet no education system actually teaches
>abstraction as a way of thinking or an approach to problems. It's always
>stuck to intuition which is clunky and complicated and not actually as
>intuitive as the abstract thing once it's seen clearly iun it's
>abstraction.

Isn't this the wrong way round?
Surely one can only reach intuition via abstraction?

After all intuition isn't (or certainly shouldn't be) 'guessing'.
It should be operating a simplified model (ie an abstraction) in an
'analogue' way to obtain a rough prediction.

>Look at the way Landau Lifschitz introduces Classical Mechanics it is
>beautifull but without the proper background in some of the ideas of the
>calculus of variation (and generall mathematic capacity) it's quite
>tough to follow. Assuming the background though it's pedagogically and
>physically highly elegant, efficient and beautifull.

Using, or more, appreciating, mathematical tools requires one should
grok them well enough for the symbols and processes to *mean* something.
Regrettably this usually takes rather a lot of hard graft (hence my
total ignorance).

--
Oz
This post is worth absolutely nothing and is probably fallacious.

BTOPENWORLD address about to cease. DEMON address no longer in use.
>>Use oz@farmeroz.port995.com (whitelist check on first posting)<<

Patrick Powers

Jun9-04, 04:26 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>\nalistair@goforit64.fsnet.co.uk (alistair) wrote in message news:<861c1b21.0406020852.2b283160@posting.google. com>...\n> EDWIN F.TAYLOR (http://www.eftaylor.com/pub/FmaAJPguest5.pdf)thinks\n> students should be taught Newtonian mechanics from the point of view\n> of the principle of least action because it is generally more useful\n> and interesting than the standard way that Newtonian mechanics is\n> taught.Is he right?\n\nIn place of criticism of this document allow me to make a proposal of\nmy own for an introductory college physics course.\n\nPrerequisite of statistics. Start with Feynman\'s "QED: The Strange\nTheory of Light and Matter" because this is easier to understand if\nyou know no physics and have no preconceptions to get in the way. So\nthere is a very good reason to do this first. The book contains no\nmathematics at all. I\'d introduce experimental data right away to\nshow how common concepts are like energy actually statistical and not\ndirectly observable. By this I mean we can observe the effect of\nenergy on matter but can\'t see it. If we continuously observe its\neffects it behaves differently. I\'d also show qualitatively how very\nbasic concepts like inertia come from the theory, answering questions\nlike "why do things move in a straight line"?\n\nThe book also does an excellent job of introducing the intuition\nbehind the Principle of Stationary Action (there must be a better\nname. The Path Of Noncancellation or something. ). After this you\ncould move on to an electron moving in a field and show that the path\nis curved. Once this has been done it becomes quite plausible that\nthe Principle can be applied more widely. Maybe you could derive a\nLagrangian by integrating the Newtonian solution instead of just\npulling it out of a hat.\n\nRecall that Feynman himself taught the now-famous introductory physics\nclass. He also declared the class to be a failure at its ostensible\ngoal, that of teaching freshmen and sophomores. So this is not an\neasy task.\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>alistair@goforit64.fsnet.co.uk (alistair) wrote in message news:<861c1b21.0406020852.2b283160@posting.google.com>...
> EDWIN F.TAYLOR (http://www.eftaylor.com/pub/FmaAJPguest5.pdf)thinks
> students should be taught Newtonian mechanics from the point of view
> of the principle of least action because it is generally more useful
> and interesting than the standard way that Newtonian mechanics is
> taught.Is he right?

In place of criticism of this document allow me to make a proposal of
my own for an introductory college physics course.

Prerequisite of statistics. Start with Feynman's "QED: The Strange
Theory of Light and Matter" because this is easier to understand if
you know no physics and have no preconceptions to get in the way. So
there is a very good reason to do this first. The book contains no
mathematics at all. I'd introduce experimental data right away to
show how common concepts are like energy actually statistical and not
directly observable. By this I mean we can observe the effect of
energy on matter but can't see it. If we continuously observe its
effects it behaves differently. I'd also show qualitatively how very
basic concepts like inertia come from the theory, answering questions
like "why do things move in a straight line"?

The book also does an excellent job of introducing the intuition
behind the Principle of Stationary Action (there must be a better
name. The Path Of Noncancellation or something. ). After this you
could move on to an electron moving in a field and show that the path
is curved. Once this has been done it becomes quite plausible that
the Principle can be applied more widely. Maybe you could derive a
Lagrangian by integrating the Newtonian solution instead of just
pulling it out of a hat.

Recall that Feynman himself taught the now-famous introductory physics
class. He also declared the class to be a failure at its ostensible
goal, that of teaching freshmen and sophomores. So this is not an
easy task.

alistair

Jun12-04, 07:20 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>If I was able to choose what was taught in school physics,\nI would concentrate on getting students to learn the broad principles\n-\nnot unlike the way many popular physics books do it.I think the most\nimportant thing to do is to get students interested in physics and if\nyou can do that , they\'ll make the effort to learn the maths and find\nout about the subtleties of the subject.\nMy favourite maths subject at school was mechanics because it seemed\nto apply to the real world.Most students ask questions like "what use\nis this?" and mechanics is a link to what students think is the real\nworld.\nThe maths curriculum and science curriculum in schools need to be\ntaught in\na co-ordinated way so students will think there is a good reason to\nlearn the maths.\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>If I was able to choose what was taught in school physics,
I would concentrate on getting students to learn the broad principles
-
not unlike the way many popular physics books do it.I think the most
important thing to do is to get students interested in physics and if
you can do that , they'll make the effort to learn the maths and find
out about the subtleties of the subject.
My favourite maths subject at school was mechanics because it seemed
to apply to the real world.Most students ask questions like "what use
is this?" and mechanics is a link to what students think is the real
world.
The maths curriculum and science curriculum in schools need to be
taught in
a co-ordinated way so students will think there is a good reason to
learn the maths.

Kirk Gregory Czuhai

Jun12-04, 07:20 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>frisbieinstein@yahoo.com (Patrick Powers) wrote in message news:<9511688f.0406082305.7f722599@posting.google. com>...\n> alistair@goforit64.fsnet.co.uk (alistair) wrote in message news:<861c1b21.0406020852.2b283160@posting.google. com>...\n> > EDWIN F.TAYLOR (http://www.eftaylor.com/pub/FmaAJPguest5.pdf)thinks\n> > students should be taught Newtonian mechanics from the point of view\n> > of the principle of least action because it is generally more useful\n> > and interesting than the standard way that Newtonian mechanics is\n> > taught.Is he right?\n>\n> In place of criticism of this document allow me to make a proposal of\n> my own for an introductory college physics course.\n>\n> Prerequisite of statistics. Start with Feynman\'s "QED: The Strange\n> Theory of Light and Matter" because this is easier to understand if\n> you know no physics and have no preconceptions to get in the way. So\n> there is a very good reason to do this first. The book contains no\n> mathematics at all. I\'d introduce experimental data right away to\n> show how common concepts are like energy actually statistical and not\n> directly observable. By this I mean we can observe the effect of\n> energy on matter but can\'t see it. If we continuously observe its\n> effects it behaves differently. I\'d also show qualitatively how very\n> basic concepts like inertia come from the theory, answering questions\n> like "why do things move in a straight line"?\n>\n> The book also does an excellent job of introducing the intuition\n> behind the Principle of Stationary Action (there must be a better\n> name. The Path Of Noncancellation or something. ). After this you\n> could move on to an electron moving in a field and show that the path\n> is curved. Once this has been done it becomes quite plausible that\n> the Principle can be applied more widely. Maybe you could derive a\n> Lagrangian by integrating the Newtonian solution instead of just\n> pulling it out of a hat.\n>\n> Recall that Feynman himself taught the now-famous introductory physics\n> class. He also declared the class to be a failure at its ostensible\n> goal, that of teaching freshmen and sophomores. So this is not an\n> easy task.\n\nWhoever thinks MOST Freshmen/Sophomores are ready to tackle Lagrange\nMultipliers and variational calulus when many of them are being\nintroduced to blocks sliding down inclined planes and frictionless\npulleys for the first time you can be my guest to see your student\nevaluations at the end of the term of teaching!\n\nAnyway is not the principle of least action just another statement of\nthe fact that a body will travel a geodesic in its traveling between\ntwo points for the energy momentum tensor of the correct geometry of\nthe space-time containing the points? And for many problems a\nHamiltonian formalation of the problem may be more productive leading\nto an easier solution of the problem.\n\nAn arguement could be put forth to bring to the attention right away\nto phyics students such elementary concepts as groups and symmetries,\nlinear algebras, and superposition, isomorphisms, complex vaiables,\netc., etc., etc.!!!\n\nBUT Indeed for a block sliding down an incline plane, F=ma !! W=mg\nsin(theta) etc.\nwe do not want to cause any friction but we have to ensure that some\nof the medical students can pass with a decent enough gpa there basic\nphysics requirements.\npeace and love,\n(kirk) kirk gregory czuhai\nhttp://www.altelco.net/~lovekgc/kirksresume.htm\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>frisbieinstein@yahoo.com (Patrick Powers) wrote in message news:<9511688f.0406082305.7f722599@posting.google.com>...
> alistair@goforit64.fsnet.co.uk (alistair) wrote in message news:<861c1b21.0406020852.2b283160@posting.google.com>...
> > EDWIN F.TAYLOR (http://www.eftaylor.com/pub/FmaAJPguest5.pdf)thinks
> > students should be taught Newtonian mechanics from the point of view
> > of the principle of least action because it is generally more useful
> > and interesting than the standard way that Newtonian mechanics is
> > taught.Is he right?
>
> In place of criticism of this document allow me to make a proposal of
> my own for an introductory college physics course.
>
> Prerequisite of statistics. Start with Feynman's "QED: The Strange
> Theory of Light and Matter" because this is easier to understand if
> you know no physics and have no preconceptions to get in the way. So
> there is a very good reason to do this first. The book contains no
> mathematics at all. I'd introduce experimental data right away to
> show how common concepts are like energy actually statistical and not
> directly observable. By this I mean we can observe the effect of
> energy on matter but can't see it. If we continuously observe its
> effects it behaves differently. I'd also show qualitatively how very
> basic concepts like inertia come from the theory, answering questions
> like "why do things move in a straight line"?
>
> The book also does an excellent job of introducing the intuition
> behind the Principle of Stationary Action (there must be a better
> name. The Path Of Noncancellation or something. ). After this you
> could move on to an electron moving in a field and show that the path
> is curved. Once this has been done it becomes quite plausible that
> the Principle can be applied more widely. Maybe you could derive a
> Lagrangian by integrating the Newtonian solution instead of just
> pulling it out of a hat.
>
> Recall that Feynman himself taught the now-famous introductory physics
> class. He also declared the class to be a failure at its ostensible
> goal, that of teaching freshmen and sophomores. So this is not an
> easy task.

Whoever thinks MOST Freshmen/Sophomores are ready to tackle Lagrange
Multipliers and variational calulus when many of them are being
introduced to blocks sliding down inclined planes and frictionless
pulleys for the first time you can be my guest to see your student
evaluations at the end of the term of teaching!

Anyway is not the principle of least action just another statement of
the fact that a body will travel a geodesic in its traveling between
two points for the energy momentum tensor of the correct geometry of
the space-time containing the points? And for many problems a
Hamiltonian formalation of the problem may be more productive leading
to an easier solution of the problem.

An arguement could be put forth to bring to the attention right away
to phyics students such elementary concepts as groups and symmetries,
linear algebras, and superposition, isomorphisms, complex vaiables,
etc., etc., etc.!!!

BUT Indeed for a block sliding down an incline plane, F=ma !! W=mgsin(\theta) etc.
we do not want to cause any friction but we have to ensure that some
of the medical students can pass with a decent enough gpa there basic
physics requirements.
peace and love,
(kirk) kirk gregory czuhai
http://www.altelco.net/~lovekgc/kirksresume.htm

Alfred Einstead

Jun13-04, 09:39 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>\nalistair@goforit64.fsnet.co.uk (alistair) wrote:\n> EDWIN F.TAYLOR (http://www.eftaylor.com/pub/FmaAJPguest5.pdf)thinks\n> students should be taught Newtonian mechanics from the point of view\n> of the principle of least action because it is generally more useful\n> and interesting than the standard way that Newtonian mechanics is\n> taught.Is he right?\n\nNo. Not all 2nd order systems are deriveable from Lagrangians, and\nnot all Hamiltonian systems from Lagrangian systems. There are\nclear-cut non-trivial, conditions required for a 2nd order system\nto be either of the latter types.\n\n2nd order systems > Lagrangian systems > Hamiltonian systems\n\nis in STRICTLY decreasing order of generality.\n\nThe most general formulation -- and the only one that directly\nconnects to observation (i.e., configuration coordinates) --\nis of the general form, involving:\n(1) Q: the configuration space, a differentiable manifold\n(2) a: Q x T(Q) --> T(T(Q))\na smooth (partial) map, such that\n* domain(a) = union({q} x T_q(Q): q in Q)\n* for each q in Q, v in T_q(Q),\na(q,v) is in T_{q,v}(T_q(Q))\n(3) solutions defined as solution curves to the system:\nq\'\'(t) = a(q(t),q\'(t)).\n\nWhat one may try to do to place this in a more fundamental setting\nis try to link it to a "boundary principle", which would state to\nthe effect:\nFor each t in R, there is an interval I subset of R,\nincluding t, such that for each t-, t+ in I with\nt- < t < t+, and for each (q-, q+) in Q x Q, there\nis a unique q: I -> Q such that q(t-) = q-, q(t+) = q+.\n\nThat is: the principle states that for each space-time boundary\nstate of the system (here: only a time boundary state, since\nevolution is only along the 1-dimensional continuum R), there\nexists a unique unfolding of the system that matches up at the\nboundary.\n\n>From this, maybe with some extra conditions; e.g., that the\nmap (q-,q+) |-> (t in I |-> q(t)) be smooth with respect to\nsome suitably defined manifold over some suitable subset Q^I,\none should be able to prove the EXISTENCE of a 2nd order\ndifferential equation that yields the desired dependence;\ni.e., the existence of a function a: Q x T(Q) -> T(T(Q)) with\nthe above-mentioned properties, that yields the map\n(q-,q+) |-> (t |-> q(t)).\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>alistair@goforit64.fsnet.co.uk (alistair) wrote:
> EDWIN F.TAYLOR (http://www.eftaylor.com/pub/FmaAJPguest5.pdf)thinks
> students should be taught Newtonian mechanics from the point of view
> of the principle of least action because it is generally more useful
> and interesting than the standard way that Newtonian mechanics is
> taught.Is he right?

No. Not all 2nd order systems are deriveable from Lagrangians, and
not all Hamiltonian systems from Lagrangian systems. There are
clear-cut non-trivial, conditions required for a 2nd order system
to be either of the latter types.

2nd order systems > Lagrangian systems > Hamiltonian systems

is in STRICTLY decreasing order of generality.

The most general formulation -- and the only one that directly
connects to observation (i.e., configuration coordinates) --
is of the general form, involving:
(1) Q: the configuration space, a differentiable manifold
(2) a: Q x T(Q) --> T(T(Q))
a smooth (partial) map, such that
* domain(a) = union({q} x T_q(Q): q in Q)
* for each q in Q, v in T_q(Q),a(q,v) is in T_{q,v}(T_q(Q))
(3) solutions defined as solution curves to the system:
q''(t) = a(q(t),q'(t)).

What one may try to do to place this in a more fundamental setting
is try to link it to a "boundary principle", which would state to
the effect:
For each t in R, there is an interval I subset of R,
including t, such that for each t-, t+ in I with
t- < t < t+, and for each (q-, q+) in Q x Q, there
is a unique q: I -> Q such that q(t-) = q-, q(t+) = q+.

That is: the principle states that for each space-time boundary
state of the system (here: only a time boundary state, since
evolution is only along the 1-dimensional continuum R), there
exists a unique unfolding of the system that matches up at the
boundary.

>From this, maybe with some extra conditions; e.g., that the
map (q-,q+) |-> (t in I |-> q(t)) be smooth with respect to
some suitably defined manifold over some suitable subset Q^I,
one should be able to prove the EXISTENCE of a 2nd order
differential equation that yields the desired dependence;
i.e., the existence of a function a: Q x T(Q) -> T(T(Q)) with
the above-mentioned properties, that yields the map
(q-,q+) |-> (t |-> q(t)).

Pmb

Jun14-04, 03:08 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>"Frank Hellmann" <C.i.m@gmx.net> wrote in message\nnews:e7f834be.0406071230.3973a372@posting .google.com...\n>\n> frisbieinstein@yahoo.com (Patrick Powers) wrote in message\nnews:<9511688f.0406060505.39e8730a@postin g.google.com>...\n> > alistair@goforit64.fsnet.co.uk (alistair) wrote in message\nnews:<861c1b21.0406020852.2b283160@postin g.google.com>...\n> > > EDWIN F.TAYLOR (http://www.eftaylor.com/pub/FmaAJPguest5.pdf)thinks\n> > > students should be taught Newtonian mechanics from the point of view\n> > > of the principle of least action because it is generally more useful\n> > > and interesting than the standard way that Newtonian mechanics is\n> > > taught.Is he right?\n> >\n> >\n> > I found this to be poor pedagogy. I understand their motivation, but\n> ...\n>\n> Obviously you would want to teach them the calculus of variations\n> first.\n\nThe author (Edwin F. Taylor) holds that his method does not use the regular\ncalculus of variations at\nall -- it is much simpler -- elementary calculus and some algebra.\n\nPmb\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" <C.i.m@gmx.net> wrote in message
news:e7f834be.0406071230.3973a372@posting.google.c om...
>
> frisbieinstein@yahoo.com (Patrick Powers) wrote in message
news:<9511688f.0406060505.39e8730a@posting.google.com>...
> > alistair@goforit64.fsnet.co.uk (alistair) wrote in message
news:<861c1b21.0406020852.2b283160@posting.google.com>...
> > > EDWIN F.TAYLOR (http://www.eftaylor.com/pub/FmaAJPguest5.pdf)thinks
> > > students should be taught Newtonian mechanics from the point of view
> > > of the principle of least action because it is generally more useful
> > > and interesting than the standard way that Newtonian mechanics is
> > > taught.Is he right?
> >
> >
> > I found this to be poor pedagogy. I understand their motivation, but
> ...
>
> Obviously you would want to teach them the calculus of variations
> first.

The author (Edwin F. Taylor) holds that his method does not use the regular
calculus of variations at
all -- it is much simpler -- elementary calculus and some algebra.

Pmb

Arnold Neumaier

Jun16-04, 03:51 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>Alfred Einstead wrote:\n>\n> No. Not all 2nd order systems are deriveable from Lagrangians, and\n> not all Hamiltonian systems from Lagrangian systems.\n\nAs your display 4 lines later shows, you must have meant\n\'not all Lagrangian systems from Hamiltonian systems\'.\nEvery Hamiltonian system is Lagrangian, with L(x) = p qdot - H(p,q),\nx=(p,q).\n\n> There are\n> clear-cut non-trivial, conditions required for a 2nd order system\n> to be either of the latter types.\n>\n> 2nd order systems > Lagrangian systems > Hamiltonian systems\n>\n> is in STRICTLY decreasing order of generality.\n\nArnold Neumaier\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>Alfred Einstead wrote:
>
> No. Not all 2nd order systems are deriveable from Lagrangians, and
> not all Hamiltonian systems from Lagrangian systems.

As your display 4 lines later shows, you must have meant
'not all Lagrangian systems from Hamiltonian systems'.
Every Hamiltonian system is Lagrangian, with L(x) = p qdot - H(p,q),x=(p,q).

> There are
> clear-cut non-trivial, conditions required for a 2nd order system
> to be either of the latter types.
>
> 2nd order systems > Lagrangian systems > Hamiltonian systems
>
> is in STRICTLY decreasing order of generality.

Arnold Neumaier

alistair

Jul1-04, 04:46 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>Do cosmic ray protons, which shouldn\'t be able to reach the Earth\n(GZK protons), minimise the integral:\n\n/\n|\n| ( kinetic energy + potential energy )dt\n/\n\nwhere the kinetic energy and potential energy of the protons are functions of time.\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>Do cosmic ray protons, which shouldn't be able to reach the Earth
(GZK protons), minimise the integral:

/
|
| ( kinetic energy + potential energy )dt
/

where the kinetic energy and potential energy of the protons are functions of time.

Czuhai Kirk Gregory

May1-05, 05:07 AM

mr. alistair:
perhaps the principle of least action seems quantitatively simple to you and
the likes of people even more so like richard feyman,
but even qualitatively a lot of people would be somewhat lost in a first
course in physics trying to digest the topic i believe. can you belive that?
some people just do not have the mathematical apptitude you may have and
are just struggling then to get through maybe their first exposure to elementary
calculus.
peace and love,
and,
love and peace,
(kirk) kirk gregory czuhai LOVES !
http://KirkGregoryCzuhai.WS