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Re: [Phys-l] Phys-l Digest, Vol 60, Issue 36 Landau on Lagrangian



The Principle of least action works for all of known physics. A Lagrangian
can be generated for all known physics - classical mechanics, special
relativity, general relativity, and quantum mechanics.

Calculus of variations can use first, second, and higher order derivatives
-although these won't always come into play based on a specific Lagrangian.

The Lagrangian is deduced from the particular physics of interest.
KE=(1/2)mv^2 for a single particle. Leibniz defined KE as the product of the
mass of an object and its velocity squared. This is an observation and
assumption. Minimizing the Action of the Lagrangian L = (1/2)mv^2 - U
generates Newton's equations. The Lagrangian itself is not attached to a
given area of physics - it is formulated by the physical assumptions of a
given problem. The symmetry of the problem generates the conservation laws.

Al Bernstein

-----Original Message-----
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Sent: Wednesday, January 27, 2010 9:00 AM
To: phys-l@carnot.physics.buffalo.edu
Subject: Phys-l Digest, Vol 60, Issue 36

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Today's Topics:

1. Re: PV question (Carl Mungan)
2. Re: Landau on Lagrangian (David Bowman)
3. Re: MIT world video, "Teaching the Second Law" (John Clement)
4. Re: Landau on Lagrangian (Stefan Jeglinski)
5. Re: MIT world video, "Teaching the Second Law" (Brian Whatcott)
6. Re: A geek's observations on "Avatar" (Marc "Zeke" Kossover)
7. Re: Landau on Lagrangian (Bob Sciamanda)
8. frequency: a modest proposal (James McLean)
9. Re: frequency: a modest proposal (John Denker)
10. Re: frequency: a modest proposal (curtis osterhoudt)
11. Re: frequency: a modest proposal (Jack Uretsky)


----------------------------------------------------------------------

Message: 1
Date: Tue, 26 Jan 2010 13:51:06 -0500
From: Carl Mungan <mungan@usna.edu>
Subject: Re: [Phys-l] PV question
To: phys-l@carnot.physics.buffalo.edu
Message-ID: <a06240815c784e4a339f5@131.122.75.125>
Content-Type: text/plain; charset="us-ascii" ; format="flowed"

Certainly the "usual" system would not behave in this manner, being
in a state of mechanical but not thermodynamic equilibrium. The
specific system I had in mind is the usual system with a gimmick.
Place a second heat-incapacious thermally-insulating piston in the
middle, and put different gases, say one monatomic and the other
diatomic, on the two sides of that piston. One can now carry out
reversible processes on this system that leave the two sides at
different temperatures. Consider an adiabatic compression of the
system from an initial state of thermodynamic equilibrium, for
example. Which gas gets warmer?

Leigh

The monatomic gas. Defining t to be the ratio of the final and
initial temperature of a gas, then t_monatomic = t_diatomic^1.4. Okay
I accept that example of yours as a valid demonstration of a system
with a single well-defined V and P but not T (but only because T has
two piecewise uniform values).

But I don't see how this demonstrates your previous statement: "If
the process is reversible, the system must be near thermodynamic
equilibrium...."

Let's accept your two-piston system as not remaining in thermodynamic
equilibrium for the sake of continuing the discussion. (At least not
the whole system, although piecemeal the parts of the system
certainly are. So John D might quibble here. Anyhow, let's just go
on.) But it still certainly looks to me like your compression IS
reversible. We did a slow, dissipation-free adiabatic compression. I
can undo it with a slow, dissipation-free adiabatic expansion in such
a fashion that I also undo all changes in the environment. That has
to fit any reasonable definition of reversible, right? If so, your
statement doesn't stand up.... -Carl
--
Carl E Mungan, Assoc Prof of Physics 410-293-6680 (O) -3729 (F)
Naval Academy Stop 9c, 572C Holloway Rd, Annapolis MD 21402-1363
mailto:mungan@usna.edu http://usna.edu/Users/physics/mungan/


------------------------------

Message: 2
Date: Tue, 26 Jan 2010 15:56:17 -0500
From: David Bowman <David_Bowman@georgetowncollege.edu>
Subject: Re: [Phys-l] Landau on Lagrangian
To: 'Forum for Physics Educators' <phys-l@carnot.physics.buffalo.edu>
Message-ID:

<91E065A961D85D469AF93084ABC41FF801937A1E@EXCHANGE.georgetowncollege.edu>

Content-Type: text/plain; charset="us-ascii"

Regarding Bob S's response:

I wrote: . . .
But even If I grant you these requirements and accept that they
imply that the mechanical L = L(v^2), you still have to show that
this implies: KE(translation) is proportional to v^2.

To which David Bowman responded:
That was already done (more than once). That comes from the
invariance of the EOM under Galilean boosts once it is granted
that the resulting equations be no higher than 2nd order and
the symmetries of space & time homogeneity and spatial isotropy
are imposed. Please reread the part of the argument about how
only the expression v^2 obeys v^2 = v'^2 + dF/dt when
v = v' + v_0 and F = F(r',v',t). Other nonlinear functions of
v^2 do not have this property.

Those arguments only attempt to show that L = L(v^2).

No, they do not. Please pay attention.

The homogeneity in space and time constrain L to not depend explicitly on t
or r. The 2nd order of the EOM requirement constrains L to now depend only
on v and not on any higher derivatives of r WRT t. The isotropy of space
then forces L to depend only on the square magnitude v^2. Now get this:
*The invariance under Galilean boosts constrains this L(v^2) to now be
PROPORTIONAL TO THE FIRST POWER of v^2* (up to an irrelevant additive
constant). This is because any other function of v^2 that is nonlinear
(i.e. not proportional to it) will not have the needed property that L(v^2)
= L(v'^2) + dF/dt where F is some function F(r',v',t) when v = v' + v_0.
This property is needed to make the EOM invariant under Galilean boosts. We
then define the mass as twice this proportionality constant. If Hamilton's
principle is taken to be a *minimum* principle rather than merely a
principle of stationarity then the constant mass must be positive as well.

They do not address the proposition: KE(translation) is
proportional to v^2.

That is *precisely* what the point of the invariance-under-boosts part of
the argument does address.

How does the concept of KE even enter the development?

It is the resulting Lagrangian of a free particle after the assumption of
2nd order EOM and all the symmetries are imposed (after subtracting off any
possible arbitrary irrelevant additive constant that one may have included,
i.e. the 'rest Lagrangian').

Bob Sciamanda

David Bowman


------------------------------

Message: 3
Date: Tue, 26 Jan 2010 16:30:49 -0600
From: "John Clement" <clement@hal-pc.org>
Subject: Re: [Phys-l] MIT world video, "Teaching the Second Law"
To: "'Forum for Physics Educators'"
<phys-l@carnot.physics.buffalo.edu>
Message-ID: <4B12F632C5D0442593EE87FB2D720D2D@Clement>
Content-Type: text/plain; charset="us-ascii"

The thing that is really infuriating is that nobody mentioned any research
on teaching thermo. There is research on this, and there is even a TCI
Thermo Concept Inventory. I have found articles about the TCI.

I doubt that a traditional physicist would actually do a better job. Some
of the panelists did express the opinion that they achieved better
understanding, but there was no data to back it up, so the opinion could be
an illusion.

I have not quite finished the video, but after I do finish it, I might have
to revise my opinion, but I really doubt it. There are even engineering
educators who are working in the TCI, but that sort of thing does not seem
to have penetrated most engineering professors.

John M. Clement
Houston, TX

On 01/25/2010 08:39 PM, curtis osterhoudt wrote:

http://mitworld.mit.edu/video/540/

Horrifying. Macabre.

Some professors weigh in on teaching the 2nd law of Thermo.

Apparently no physics professors. Also nobody from
the cryptography, communications, machine learning,
or statistics communities.

Interesting takes on where students get hung up.

Interesting? I found it infuriating. My reaction
was, "Duh, if you run through a cactus plantation
at night, you're going to get hung up." That is
to say, the nine panelists described at least eight
different ways of making the subject seem infinitely
more complicated and less useful than it really is.

No wonder the students get hung up.

I had no idea the situation was so bad. Maybe I'm
naive, but I would have preferred to stay naive.

Now I need to go watch The Exorcist or The Shining,
something to get the image of that MIT video out of
my head. Otherwise I'm gonna have nightmares.

_______________________________________________
Forum for Physics Educators
Phys-l@carnot.physics.buffalo.edu
https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l



------------------------------

Message: 4
Date: Tue, 26 Jan 2010 17:44:58 -0500
From: Stefan Jeglinski <jeglin@4pi.com>
Subject: Re: [Phys-l] Landau on Lagrangian
To: Forum for Physics Educators <phys-l@carnot.physics.buffalo.edu>
Message-ID: <p06240807c785109472a4@[192.168.9.139]>
Content-Type: text/plain; charset="us-ascii" ; format="flowed"

>Those arguments only attempt to show that L = L(v^2).

No, they do not. Please pay attention.

One of my implicit intents in starting this thread was to talk about
the Lagrangian alone. As such, it is the function L that appears in
the kernel of the integral assigned to S (the action). The question
was a fleshing out of L&L's (and other's) treatment of the form L
would take for a free particle, based on considerations of symmetry
and the key point about Galilean boosts.

By "form" was meant its dependence on r, t, r-dot, r-doubledot (or
not), not the question of whether it was ever to be written as KE
(Bob's sticking point I think) or (I hope I'm right here) JD's
sticking point thinking I meant that L&L was using these arguments to
derive explicit versions of L.

To me, the question has been answered really well by DB. Once
asserting (yes, an assertion, but a really interesting one) a
stationary action, the math alone suffices to yield the
Euler-Lagrange equation, which itself suggests that constants of
motion exist, given certain explicit versions of L.

This is technically the end of the question. To suggest that L is
written in the form of T - U, and/or that T=KE and U=PE, is a further
step. But the mere fact that L=L(v^2) has been "deduced" is highly
suggestive of what at least one of them (T or U) must look like for a
particle. To go further and find other explicit additions to L, such
as jdotA, is a separate question, not addressed. However, I'm willing
to bet that symmetry arguments can also be used to deduce analogous
generalities for L wrt fields?

The question of how must one write L so that the Euler-Lagrange
equation yields N2, or anything else desired, is unresolved. But to
me, the seemingly trivial step of "deducing" that L=L(v^2), without
invoking N2, is at least incrementally better than asserting, without
any justification other than it gives N2, that L = 1/2mv^2 (free
particle).

If I've still missed some subtleties, I'm happy to be told so!


Stefan Jeglinski



------------------------------

Message: 5
Date: Tue, 26 Jan 2010 18:17:15 -0600
From: Brian Whatcott <betwys1@sbcglobal.net>
Subject: Re: [Phys-l] MIT world video, "Teaching the Second Law"
To: Forum for Physics Educators <phys-l@carnot.physics.buffalo.edu>
Message-ID: <4B5F860B.3070901@sbcglobal.net>
Content-Type: text/plain; charset=ISO-8859-1; format=flowed

curtis osterhoudt wrote:
Some professors weigh in on teaching the 2nd law of Thermo. Interesting
takes on where students get hung up.

http://mitworld.mit.edu/video/540/



/************************************
Down with categorical imperative!
flutzpah@yahoo.com
************************************/

Before I fell asleep at the wheel, I heard one speaker offer that
entropy is a sticking point with students on account of its time
directionality. But, that one day, it might turn into a conservation law
like energy-mass.

Which reminds me: there is at least one metaphysicist who notes that the
universal property of entropy, which is to increase with time, leads
immediately to the postdiction that in former ages, universal entropy
was progressively lower with -T ending (beginning?) with a very cold
very structured very small thing.... This conception is so unlikely that
there might rather be some
uber-gestalt of constant entropy, from which, some universe of high
entropy might appear, while another universe might issue, better
corresponding to the low entropy predecession that we envisage.

Brian W


------------------------------

Message: 6
Date: Tue, 26 Jan 2010 16:32:46 -0800 (PST)
From: "Marc \"Zeke\" Kossover" <zeke_kossover@yahoo.com>
Subject: Re: [Phys-l] A geek's observations on "Avatar"
To: Forum for Physics Educators <phys-l@carnot.physics.buffalo.edu>
Message-ID: <55510.51242.qm@web53405.mail.re2.yahoo.com>
Content-Type: text/plain; charset=us-ascii



Earlier we talked about recycling 3D glasses. Slate has an article about
that in today's Explainer column.

http://www.slate.com/id/2242548/

Some are washed in house others (RealD) are washed at the factory. Some are
tossed.

Zeke Kossover





------------------------------

Message: 7
Date: Tue, 26 Jan 2010 23:09:52 -0500
From: "Bob Sciamanda" <treborsci@verizon.net>
Subject: Re: [Phys-l] Landau on Lagrangian
To: "Forum for Physics Educators" <phys-l@carnot.physics.buffalo.edu>
Message-ID: <B93EC2BB5C364A2DB6BC1F4B7D7AD432@Bob>
Content-Type: text/plain; format=flowed; charset=iso-8859-1;
reply-type=original


Good night, David

Bob Sciamanda
Physics, Edinboro Univ of PA (Em)
treborsci@verizon.net
http://mysite.verizon.net/res12merh/

--------------------------------------------------
From: "David Bowman" <David_Bowman@georgetowncollege.edu>
Sent: Tuesday, January 26, 2010 3:56 PM
To: "'Forum for Physics Educators'" <phys-l@carnot.physics.buffalo.edu>
Subject: Re: [Phys-l] Landau on Lagrangian

Regarding Bob S's response:

I wrote: . . .
But even If I grant you these requirements and accept that they
imply that the mechanical L = L(v^2), you still have to show that
this implies: KE(translation) is proportional to v^2.

To which David Bowman responded:
That was already done (more than once). That comes from the
invariance of the EOM under Galilean boosts once it is granted
that the resulting equations be no higher than 2nd order and
the symmetries of space & time homogeneity and spatial isotropy
are imposed. Please reread the part of the argument about how
only the expression v^2 obeys v^2 = v'^2 + dF/dt when
v = v' + v_0 and F = F(r',v',t). Other nonlinear functions of
v^2 do not have this property.

Those arguments only attempt to show that L = L(v^2).

No, they do not. Please pay attention.

The homogeneity in space and time constrain L to not depend explicitly on
t or r. The 2nd order of the EOM requirement constrains L to now depend
only on v and not on any higher derivatives of r WRT t. The isotropy of
space then forces L to depend only on the square magnitude v^2. Now get
this: *The invariance under Galilean boosts constrains this L(v^2) to now

be PROPORTIONAL TO THE FIRST POWER of v^2* (up to an irrelevant additive
constant). This is because any other function of v^2 that is nonlinear
(i.e. not proportional to it) will not have the needed property that
L(v^2) = L(v'^2) + dF/dt where F is some function F(r',v',t) when v = v'
+ v_0. This property is needed to make the EOM invariant under Galilean
boosts. We then define the mass as twice this proportionality constant.
If Hamilton's principle is taken to be a *minimum* principle rather than
merely a principle of stationarity then the constant mass must be positive

as well.

They do not address the proposition: KE(translation) is
proportional to v^2.

That is *precisely* what the point of the invariance-under-boosts part of
the argument does address.

How does the concept of KE even enter the development?

It is the resulting Lagrangian of a free particle after the assumption of
2nd order EOM and all the symmetries are imposed (after subtracting off
any possible arbitrary irrelevant additive constant that one may have
included, i.e. the 'rest Lagrangian').

Bob Sciamanda

David Bowman
_______________________________________________
Forum for Physics Educators
Phys-l@carnot.physics.buffalo.edu
https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l



------------------------------

Message: 8
Date: Wed, 27 Jan 2010 03:33:53 -0500
From: James McLean <mclean@geneseo.edu>
Subject: [Phys-l] frequency: a modest proposal
To: Physics List <phys-l@carnot.physics.buffalo.edu>
Message-ID: <4B5FFA71.6020902@geneseo.edu>
Content-Type: text/plain; charset=ISO-8859-1; format=flowed

Greetings all,

I've always been mildly disturbed by the definition of the unit hertz as
equivalent to 1/second. (Such as can be found, for example, at
<http://physics.nist.gov/cuu/Units/units.html> and
<http://physics.nist.gov/cuu/Units/SIdiagram.html>.

What is missing from these web pages is any consideration of the concept
of "phase." Here are some propositions with which I think everyone will
agree...

(1) The unit 'radian' is just a special name given to the number 1, when
used for measuring plane angles. (This is straight off the NIST sites.)
(2) Phase is properly measured in units such as degrees or radians.
That is, phase is measured with the same units as plane angles.
(3) Phase therefore has the same physical dimension as plane angles.
(4) The rate of change of phase for a periodic signal can be properly
expressed in units of radian/s. This is usually called "angular frequency".
(5) Combining (1) and (4), angular frequency can properly be expressed
in units of 1/s.
(6) A periodic signal with and angular frequency of 1 s^{-1} does not
have a frequency of 1 Hz. I'd like to have a more specific name for
that second quantity; it seems like either "cyclic frequency" or
"counting frequency" would do nicely. "Counting frequency" extends
nicely to the concept of frequency of discrete objects.

Given these facts, it seems that one is faced with two choices:
--------------------------
(A) Declare that angular frequency and counting frequency are two
separate concepts, which just happen to be measured in the same units.
In this view, it is fine to have a single signal that has an angular
frequency of 2pi/s and a counting frequency of 1/s.

It seems that this is essentially the choice currently in vogue. And I
guess it is not too outlandish; there is already the example of activity
(SI unit: becquerel) as yet another separate concept which shares the
same unit.
---------------------------
(B) BUT, the two frequencies seem very closely linked. Option (A) is
almost like declaring length and width to be separate concepts. (Of
course, length and width aren't separate because of the rotation
operation. So that analogy isn't perfect..)

So why not define the unit 'cycle' = 2pi radians, and then make
1 Hz = 1 cycle/s = 2pi radians/s = 2pi/s ?

Then, periodic signals would simply have one physical characteristic,
'frequency,' which could be expressed in either unit. The equation
omega=2pi*f becomes a simple unit conversion, with units of
radians/cycle on the 2pi. This very naturally extends the concept of
phase to periodic non-sinusoidal signals, which I'm not sure is the case
currently.

For the most part, I don't think that this would necessitate changes in
how business is currently conducted. Counting frequency calculations
would remain the same (using Hz), and angular frequency calculations
would remain the same. It is just that a very natural link is added
between them.

The one change I can think of is that the proper base unit for
wavelength (as we now know it) would become meter/cycle. It would be
strictly proper to write lambda = 10 m/cyc = 1.59 m, and also the wave
number could be defined as k = 1/lambda. Those might look like big
departures from current practice, but actually people would just never
refer to wavelength in pure meters, and the wavenumber definition would
be written k=(2pi rad/cycle)/lambda.

Can anyone see any problems with this second option, other than the near
impossibility of changing tradition?

Cheers,
-- James
--
Dr. James McLean phone: (585) 245-5897
Dept. of Physics and Astronomy FAX: (585) 245-5116
SUNY Geneseo email: mclean@geneseo.edu
1 College Circle web: http://www.geneseo.edu/~mclean
Geneseo, NY 14454-1401


------------------------------

Message: 9
Date: Wed, 27 Jan 2010 05:48:44 -0700
From: John Denker <jsd@av8n.com>
Subject: Re: [Phys-l] frequency: a modest proposal
To: Forum for Physics Educators <phys-l@carnot.physics.buffalo.edu>
Message-ID: <4B60362C.7060101@av8n.com>
Content-Type: text/plain; charset=UTF-8

On 01/27/2010 01:33 AM, James McLean wrote in part:

... why not define the unit 'cycle' = 2pi radians, and then make
1 Hz = 1 cycle/s = 2pi radians/s = 2pi/s ?

That makes sense ... and is indeed the only definition
I have ever used, or ever heard of, until now.

I've always been mildly disturbed by the definition of the unit hertz as
equivalent to 1/second. (Such as can be found, for example, at
<http://physics.nist.gov/cuu/Units/units.html> and
<http://physics.nist.gov/cuu/Units/SIdiagram.html>.

I'm surprised. I've never noticed that before. It
looks like a bug to me.

Can anyone see any problems with this [radian] option, other than the near

impossibility of changing tradition?

It's not a problem. It's not even a change. Radians
and radians per second (not cycles per second) are
already traditional throughout mathematics, throughout
electrical engineering, and in every physics book I
can think of.

I cannot imagine any argument in favor of cycles/sec
as equivalent to 1/sec. I have to assume that those
two NIST pages are just mistakes.

Suggestion: Send a short note to the NIST guys and
suggest they repair the web page to show Hz as
2? radians per second? Or call 'em on the phone.

I would hope a very short note would suffice. That
is, I hope it is not necessary to argue the point.
It should suffice to say that 2? radians per second
is consistent with the other units on the page, and
s^-1 is not consistent. I cannot imagine there would
be any sort of counterargument.

You could start with Thomas O'Brian, chief of the
Time and Frequency division. Whether or not he's
the exact right guy to handle this, it's his job to
know who the right guy is.

http://physics.nist.gov/cgi-bin/StaffOrg/stafftable.pl?direct_group=847&gues
tbox=1&edit=0



------------------------------

Message: 10
Date: Wed, 27 Jan 2010 07:21:31 -0800 (PST)
From: curtis osterhoudt <flutzpah@yahoo.com>
Subject: Re: [Phys-l] frequency: a modest proposal
To: Forum for Physics Educators <phys-l@carnot.physics.buffalo.edu>
Message-ID: <127281.61224.qm@web65615.mail.ac4.yahoo.com>
Content-Type: text/plain; charset=us-ascii

I must admit that I don't really see the problem with the current use of
"frequency", "angular frequency", and "phase". This might be because I work
in acoustics, and everyone uses these terms (and "phase angle") without much
trouble, and understands that there is just a difference of multiplier
between them. Which term one uses is based on whom one is talking to
(non-technical visitors are much more comfortable with "f"). I will say that
students first encountering the terms must be made to be careful, especially
when ambiguity between "counting frequency" and "angular frequency" can
arise.

We *always* use Hz to be equivalent to s^-1, to the extent that things
like sound speeds are often written as "the speed of sound in water is
approximately 1.485 mm MHz"; this makes the concepts of wavenumbers and
wavelengths immediately apparent.

It seems to me that the analogy might be a little more apt if one uses
"Avogadro's number", which is, after all, just a prefix for "lots and lots
of things".

/************************************
Down with categorical imperative!
flutzpah@yahoo.com
************************************/




________________________________
From: James McLean <mclean@geneseo.edu>
To: Physics List <phys-l@carnot.physics.buffalo.edu>
Sent: Wed, January 27, 2010 1:33:53 AM
Subject: [Phys-l] frequency: a modest proposal

Greetings all,

I've always been mildly disturbed by the definition of the unit hertz as
equivalent to 1/second. (Such as can be found, for example, at
<http://physics.nist.gov/cuu/Units/units.html> and
<http://physics.nist.gov/cuu/Units/SIdiagram.html>.

What is missing from these web pages is any consideration of the concept
of "phase." Here are some propositions with which I think everyone will
agree...

(1) The unit 'radian' is just a special name given to the number 1, when
used for measuring plane angles. (This is straight off the NIST sites.)
(2) Phase is properly measured in units such as degrees or radians.
That is, phase is measured with the same units as plane angles.
(3) Phase therefore has the same physical dimension as plane angles.
(4) The rate of change of phase for a periodic signal can be properly
expressed in units of radian/s. This is usually called "angular frequency".
(5) Combining (1) and (4), angular frequency can properly be expressed
in units of 1/s.
(6) A periodic signal with and angular frequency of 1 s^{-1} does not
have a frequency of 1 Hz. I'd like to have a more specific name for
that second quantity; it seems like either "cyclic frequency" or
"counting frequency" would do nicely. "Counting frequency" extends
nicely to the concept of frequency of discrete objects.

Given these facts, it seems that one is faced with two choices:
--------------------------
(A) Declare that angular frequency and counting frequency are two
separate concepts, which just happen to be measured in the same units.
In this view, it is fine to have a single signal that has an angular
frequency of 2pi/s and a counting frequency of 1/s.

It seems that this is essentially the choice currently in vogue. And I
guess it is not too outlandish; there is already the example of activity
(SI unit: becquerel) as yet another separate concept which shares the
same unit.
---------------------------
(B) BUT, the two frequencies seem very closely linked. Option (A) is
almost like declaring length and width to be separate concepts. (Of
course, length and width aren't separate because of the rotation
operation. So that analogy isn't perfect..)

So why not define the unit 'cycle' = 2pi radians, and then make
1 Hz = 1 cycle/s = 2pi radians/s = 2pi/s ?

Then, periodic signals would simply have one physical characteristic,
'frequency,' which could be expressed in either unit. The equation
omega=2pi*f becomes a simple unit conversion, with units of
radians/cycle on the 2pi. This very naturally extends the concept of
phase to periodic non-sinusoidal signals, which I'm not sure is the case
currently.

For the most part, I don't think that this would necessitate changes in
how business is currently conducted. Counting frequency calculations
would remain the same (using Hz), and angular frequency calculations
would remain the same. It is just that a very natural link is added
between them.

The one change I can think of is that the proper base unit for
wavelength (as we now know it) would become meter/cycle. It would be
strictly proper to write lambda = 10 m/cyc = 1.59 m, and also the wave
number could be defined as k = 1/lambda. Those might look like big
departures from current practice, but actually people would just never
refer to wavelength in pure meters, and the wavenumber definition would
be written k=(2pi rad/cycle)/lambda.

Can anyone see any problems with this second option, other than the near
impossibility of changing tradition?

Cheers,
-- James
--
Dr. James McLean phone: (585) 245-5897
Dept. of Physics and Astronomy FAX: (585) 245-5116
SUNY Geneseo email: mclean@geneseo.edu
1 College Circle web: http://www.geneseo.edu/~mclean
Geneseo, NY 14454-1401
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------------------------------

Message: 11
Date: Wed, 27 Jan 2010 09:54:24 -0600 (CST)
From: Jack Uretsky <jlu@hep.anl.gov>
Subject: Re: [Phys-l] frequency: a modest proposal
To: Forum for Physics Educators <phys-l@carnot.physics.buffalo.edu>
Message-ID: <alpine.LRH.2.00.1001270951470.8646@theory.hep.anl.gov>
Content-Type: TEXT/PLAIN; format=flowed; charset=US-ASCII

There's more to it than that, as I've pointed out in a math arXiv:
the derivative of the sine is the cosine - only if the argument is in
radians (doesn't work in degrees).
Regards,
Jack

"Trust me. I have a lot of experience at this."
General Custer's unremembered message to his men,
just before leading them into the Little Big Horn Valley






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