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I think there is some confusion here regarding symmetry and Noether's
theorem.
For our purposes Noether's theorem follows from the Lagrangian equation of
motion for the generalized coordinate q and its time derivative dq/dt = q':
d/dt (DL{q,q'...}/Dq') = DL{q,q'...}/Dq, where L(q,q'...) is the Lagrangian
and D/D refers to a partial derivative.
The momentum conjugate to the coordinate q is defined as p = DL{q,q'...}/Dq'
.
The above equation can thus be written as:
dp/dt = DL{q,q'...}/Dq . There is one such equation for each generalized
coordinate and its conjugate momentum.
This says that if the Lagrangian is independent of a coordinate q (ie. the
RHS = 0), then its conjugate momentum p is a constant in time. This, for our
purposes, is Noether's theorem.
Thus if L is independent of a linear coordinate, then the conjugate linear
momentum is constant in time; and if L is independent of an angular
coordinate then the conjugate angular momentum is constant in time. This
applies to any system describable by a Lagrangian - closed or not closed.
Thus, the angular momentum of a simple pendulum is not constant, because its
Lagrangian explicitly depends upon the angle THETA. Yet a closed system
which includes a pendulum will conserve momenta.
The homogeneity and isotropy of space and time are (usually unspoken)
axioms, built into Newton's vectorial laws of motion.
They thus (quietly) assume that the physics of a closed system will be
independent of the choice of origin of the physicist or of his choice of
angular orientation for his coordinate system.
Bob Sciamanda
Physics, Edinboro Univ of PA (Em)
trebor@winbeam.com
http://www.winbeam.com/~trebor/
----- Original Message ----- From: "John Denker" <jsd@av8n.com>
To: "Forum for Physics Educators" <phys-l@carnot.physics.buffalo.edu>
Sent: Tuesday, August 19, 2008 9:10 PM
Subject: Re: [Phys-l] rotational dynamics +- straight-line dynamics
On 08/19/2008 03:40 PM, Steve Highland wrote:
I guess what's bothered me over the years is why it is legal to take F=ma
and basically do "r x 'F=ma'" to create the rotational analog for any
choice
of r. Is this purely mathematical, or is there some physics in it, too?
What reasoning assigns each force vector (and the ma vector as well) a
particular location in space? What is it that tells us where they act?
Wow. That is quite a deep question.
This is one of those quintessential "phys-l" questions that needs
to be looked at from multiple viewpoints. There are
-- issues at the formal, fundamental physics level, and also
-- issues at the introductory, pedagogical level.
There is some conflict between the levels.
===========
Here is my take on the fundamentals:
As for the assertion that you can derive the rotational laws from the
straight-line laws: I'm not convinced. Asserting that the derivation
is "in all the textbooks" does not impress me; the derivation always
includes some sketchy assumptions.
Here is the counterargument: Conservation of straight-line momentum is
connected (via Noether's theorem) to a particular symmetry in the laws
of physics, namely invariance with respect to an overall translation.
Meanwhile, conservation of angular momentum is connected to a different
symmetry, namely invariance with respect to rotation.
Is anyone bold enough to tell me that invariance with respect to
translation
implies invariance with respect to rotation? How much would you like to
bet? It is super-easy to come up with systems that exhibit one symmetry
but
not the other. Example: polaroid material. Example: any low-symmetry
crystal.
The thing that actually works is the *converse*: If you start with
conservation of angular momentum, it implies conservation of straight-line
momentum. Hint: choose a pivot point infinitely far away.
To repeat: I am not convinced that F=ma is a complete description of
the basic physics, or that the rotational laws are a corollary of the
straight-line laws.
==============
As applied to extended objects (not restricted to point particles), you
can have one or the other of the following notions, not both:
a) Force is a vector
b) F=ma is a complete description of the dynamics.
My preference is to stick with the conventional notion (a). That means
that force, per se, has a direction and magnitude, period.
If you want a usable description of the dynamics, you need something more.
You need a notion that doesn't yet have a name AFAICT. We can call it
an "ultraforce" for now. It has two elements: a plain old force vector,
and a place of attachment.
ultraforce := {force vector, attachment}
Keeping track of the ultraforces is tantamount to keeping track of the
angular momentum, since torque = dL/dt always.
-- It implies keeping track of the forces (and *not* conversely).
-- It implies keeping track of the linear momentum (and *not* conversely).
There's more I could say about this is anybody is interested.
_______________________________________________
Forum for Physics Educators
Phys-l@carnot.physics.buffalo.edu
https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l
_______________________________________________
Forum for Physics Educators
Phys-l@carnot.physics.buffalo.edu
https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l