the inverse problem
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A short self-review by Phil, dated 12.10.16, in the support documents for his "new frames" mechanics write-up. He goes section by section to check whether each treats frames S and S' neutrally, notes where S is at rest or inertial, and decides the G Rule and velocity/acceleration relations hold even if both frames are non-inertial. He separates the Forward Problem into Group 1 kinematic equations and Group 2 force and torque equations, and defines the Inverse Problem in Section 13.
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The Inverse Problem PhL 12.10.16
I am unhappy with my presentation of this subject for several reasons.
First, what is my statement of "the Forward Problem" ?
In Fig 1 Frame S is at rest, that is true. I fail to say in my Fig 1 discussion that Frame S is fixed to the paper. I should add something there maybe, and say it is the forward problem.
The first use of the word "forward" is at the start of Section 12. I imply that Sections 6.7.8 and 11 were all about this forward problem which I never really defined. Let's scan a bit
1.1 either frame could be at rest
1.2 same
1.3 same
1.4 same
1.5 same
1.6 concerns (da/dt) = ω x a and uses ω ≡ dφ/dt , but either frame could be rotating I guess
1.7 notion of ∂S versus ∂S', either frame could be at rest.
1.8 notations for time derivatives.
What about the ang mom section here? Seems frame motion neutral.
1.9 neutral
1.10 commutation? neutral
2. G Rule. Is written as ∂Sa = ∂S'a + ω x a so you obtain a Frame S thing on the left, but I think the discussion here is neutral.
3. observer discussion is neutral, though I do say only if inertial do you have F = ma.
4.1 Fig 4.1.1 has Frame S at rest relative to paper. Frame S' is rotating.
4.2 Just a new picture of same thing. Frame S at rest rel to paper, Frame S' is rotating.
4.3 about b, think OK
4.4 Special Case 1: I think Frame S could move, as long as I don't do F = ma things are neutral.
4.5 Special Case 2
4.7 turntable. Here Frame S really is at rest.
4.8 Earth. here Frame S is really at rest.
4.8 camera platform seems neutral.
5. Here I state the Goal which I should call the Forward Problem. I think the equations that solve the Forward Problem allow that either or both frames be non-inertial!
6. Determ velocities. Trying to see why these equations allow either or both frames to be non-inertial. Ingredients are the G Rule and r = r' + b . This entire chapter is valid even if both frames are non-inertial. In other words, the paper could be a non-inertial frame in Fig (4.2.1). The G Rule talks only about the relation between two frames, it does not require either frame to be inertial.
I have added a Comment in the G Rule section emphasizing that that Rule is valid even if Frames S and S' are both non-inertial. I think that is a helpful thing to put on the table.
I have added another Comments claiming that all the v and a relations derived in Section 6 are valid evern if both frames are non-inertial.
8.1 Here I state that Frame S is inertial for the first time so that F = ma.
8.7 I will later examine how inertial frame fits into this sun-earth discussion.
11.2 is where I first deal with ang mom L vectors and I use Fig 11.2.1.
If I were to not use Ffict and write things out, I would have
L(c) = L'(c') + m (r'-c') x [ (ω x r') + S] (11.2.14)
(c) = '(c') + m (r'-c') x (S + ω x (ω x r') + 2 ω x v' + x r')
– m (' + ω x c' + S) x ( v' + ω x r' + S) + m ' x v' . (11.2.15)
where I have added the mass back in These equations are like v = v' + other terms for the first, and like
a = a' + other terms for the second. I claim then that, since only the G rule and veloc and accel relations go into these results, they are both valid even of both Frame S and Frame S' are non-inertial frames!
Ang mom -- always show the mass:
1.8.11 is first appearance of ang mom. Added m here
Sec 11 return:
11.1 OK as is
11.2 OK, I think things are all fixed up here.
Section 12. Here then is where I first use the phrase Forward Problem. I make the point that both Frames could be non-inertial.
So I divide the Forward Problem into two sets of equations:
Group 1. Equations involving v, a and L(c) which are valid if both frames are non-inertial
Group 2. Equations involving force and torque which are valid only if Frame S is inertial.
13. Here I now define the Inverse Problem. I only mention Group 1 equations here. For the inverse problem, Frame S is still inertial
Well, for this group Frame S is Inertial and Frame S' is not.
Added clarifying comment near start of Section 13 as to why Ffict and torque is NOT included in the lists which define the two problems.
I am now OK up to the start of Section 13.3.