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Re: Kepler clarification

On Wed, 1 May 1996, Rauber, Joel Phys wrote:


I need some more clarification of the problem, before I spend time and
effort working. I want to make sure that the conditions aren't self-defing
the situation such that I can't do it. Correct any of these
statements/clarifications as need be.

1) I'm to derive the following Kepler law

period^2 = (4 pi (earth-sun distance)^3)/G(M+m)

in a frame of reference where the earth and sun are stationary.

2) Will you allow me to do it for circular orbits, ...

What orbits? There are no orbits, circular or otherwise, in the reference
frame specified for the problem. The conditions of the problem imply
constant distance between Earth and Sun -- neither Earth or Sun are
moving relative to the frame.

...the only case where the
conversion to the inertial frame from the non-inertial frame is relatively
simple(or vice versa); we ultimately have to convert from one frame to other
to compare answers; so we must have knowledge of the two frame in order to
do that.
...

I'm not asking you to compare anything. I'm asking for the derivation of a
SINGLE formula, for either M or M+m, derived by applying Newton's laws in
the specified frame, and expressed in terms of quantities measured
relative to that frame. Asking for specifications relative to an inertial
frame just backs up my statement, made from the beginning of this thread,
that Newton's laws, correctly stated, involve the proviso, "relative to an
inertial frame," or they're not valid and cannot be used to produce accurate
results. Your claim, as I have understood it, is that you do not need that
proviso, you can apply Newton's laws in any frame you want and get
correct results if only you are allowed to introduce pseudoforces. Do it.

What I'm hoping to elicit is an example of applying Newton's laws in a
noninertial frame and getting a valid formula for M or M+m, whichever you
think you can validly compute. I have never seen it done. My own claim
specifically results in non-Newtonian laws applying relative to noninertial
frames, and to find what they are I would have to start with an inertial
frame, apply Newton's laws, then transform any kinematical
quantities (and ONLY kinematical quantities, i. e., position, velocity,
acceleration) to the frame I want to work in, and then I can see what
laws work in that frame -- they will not be Newton's laws, but they
will be the correct formulas for the frame I have chosen to measure the
kinematical data in.

(By the way, someone brought up Lagrangian and Hamiltonian methods; they
work exactly the same way -- you must start with valid expressions for
L = T - V or H = T + V, or all is lost, i. e., your formulas must
reduce to the Newtonian expressions for T and V , or Einstein's
relativistically corrected versions, relative to some inertial frame or
you will be in trouble.)


3) My physicists have figured out the inverse square law of gravity; and
have figured out the centrifugal force law, namely there exists an outward
pointing force on a mass in this frame of reference equal to
mass*const*(distance from coordinate origin) and that the const has been
verified by experiment to be (4 pi^2)/(period). Where period is the time
it takes the stars to orbit the coordinate system.

4) The dynamical law I'll use is the 2nd law in this non-inertial frame:

Namely

applied to any mass in the system of analysis

m*a = sum of forces,

where in this situation described above sum of forces is the sum of our
experimentally determined law of gravity and law of centrifugal force.


Are the above a well posed statement of the problem, if not how should I
change them?
...

OK. See what you can come up with. It should be interesting.

A. R. Marlow E-MAIL: marlow@beta.loyno.edu
Department of Physics PHONE: (504) 865 3647 (Office)
Loyola University 865 2245 (Home)
New Orleans, LA 70118 FAX: (504) 865 2453