Section 10 rewrite

10. Notation comparison with Goldstein (1950) and Goldstein, Poole and Safko (2001) In this Section compare our notation for rotating-frame kinematics and non-inertial-frame physics to that of Goldstein on (1950) and Goldstein, Poole and Safko (GFS 2001). Their notation seems to assume the Special Case #1 situation where the rotation axis passes through the inertal frame origin, as for rotating-Earth problems. ATTEMPT USING MY SWAP NOTATION Recall our two equations in swap notation. a' = a + x r + 2 ω x v + ω x (ω x r) + S' S = rotating frame (7.6a)s S' S Euler Coriolis centripetal frame ( r = short vector, r' = long vector) S' = x b + ω x (ω x b) . // Special Case #1 (7.13)s Inserting the second into the first and using r' = b+r gives a' = a + x r' + 2 ω x v + ω x (ω x r') // r' = long vector (*) We may compare this to the Goldstein equation, as = ar + x r + 2 ω x vr + ω x (ω x r) . // Goldstein p 135 (4-105) (**) // GPS p 175 (4.89) In my equation (*) I have r' = long vector. But in (**) I know that r is a short vector because he says this on page 135. This means that (*) and (**) are not compatible. Meanwhile, let's compare these two items vS' = v + ω x r . r = short vector (6.5)s vs = vr + ω x r . r = short vector // Goldstein p 135 (4-104) // GPS p 175 (4.88) I am OK with v = vr in the rotating frame. And r looks right. But in my equation I have vS' = (dr/dt)S' = (dr/dt)space I refer to this as a cross velocity because I am looking at the motion of the short vector r in the rotating frame using a time derivative from the space frame. That is certainly OK to do. With this connection that we have vs = vS', we are 100% OK here, which favors this approach. Maybe he does the same thing with acceleration. He would have as = ∂S'vS'= ∂2S'r = (∂2r/dt2)space exactly what G says! So I want then to connect as = aS'S' = aS' . What do I know about this accel? a'S = a' + x r' + 2 ω x v' + ω x (ω x r') . (7.4) aS' = a + x r + 2 ω x v + ω x (ω x r) . (7.4)s Compare this then to as = ar + x r + 2 ω x vr + ω x (ω x r) . // Goldstein p 135 (4-105) (**) // GPS p 175 (4.89) ATTEMPT USING MY NON-SWAP NOTATION a = a' + x r' + 2 ω x v' + ω x (ω x r') + S S' = rotating frame (7.6a) S S' Euler Coriolis centripetal frame ( r' = short vector, r = long vector) S = x b + ω x (ω x b) . // Special Case #1 (7.13) Inserting the second into the first and using r = b+r' gives a = a' + x r + 2 ω x v' + ω x (ω x r) . (*) (10.1) This may be compared with the Goldstein equation (to which we have added a x r term) as = ar + x r + 2 ω x vr + ω x (ω x r) . // Goldstein p 135 (4-105) (**) // GPS p 175 (4.89) In my equation (*) I have r= long vector. But in (**) I know that r is a short vector because he says this on page 135. This means that (*) and (**) are not compatible. I am unable to find a way to compare the two notations!! Meanwhile, let's compare these two items v'S = v' + ω x r' . (6.5) vs = vr + ω x r . // Goldstein p 135 (4-104) // GPS p 175 (4.88) vs = vr + ω x r . // Goldstein p 135 (4-104) // GPS p 175 (4.88) All my equations above in swap-notation are these ( r = short vector, r' = long vector) a' = a + x r + 2 ω x v + ω x (ω x r) + S' S = rotating frame (7.6a)s S' S Euler Coriolis centripetal frame S' = x b + ω x (ω x b) . // Special Case #1 (7.13)s Inserting the second into the first and using r' = b+r gives a' = a + x r' + 2 ω x v + ω x (ω x r') Now compare this to as = ar + x r + 2 ω x vr + ω x (ω x r) . // Goldstein p 135 (4-105) // GPS p 175 (4.89) So this is IT!! our swap Goldstein notation authors S r name of the rotating frame S' s name of the fixed frame (s = space) ∂S' (d/dt)s time derivative in the fixed frame ∂S (d/dt)r time derivative in the rotating frame r position in rotating frame (short vector) v vr velocity in rotating frame a ar acceleration in rotating frame r' r position in fixed frame (long vector) v' vf velocity in fixed frame a' as acceleration in fixed frame F' F force in fixed frame = true force in rotating frame Feff Feff total effective force in the rotating frame Ffict total fictitious force in the rotating frame b R location of the rotating frame origin (measured in the fixed frame) S' f,V velocity of the rotating frame origin (measured in the fixed frame) S' f acceleration of the rotating frame origin (measured in the fixed frame)