Rotations of Earth and other Rigid Bodies

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Discussion Overview

The discussion revolves around the mathematical modeling of rotations of the Earth and other rigid bodies, specifically through the lens of one-parameter group actions. Participants explore the complexities of these rotations, including the effects of the Earth's non-spherical shape and the implications for modeling time-dependent rotations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant seeks references for using one-parameter group actions to model rotations of planets and rigid bodies.
  • Another participant notes that the Earth's non-spherical shape leads to additional complexities such as nutation and precession, suggesting the IERS as a relevant organization for reference frames.
  • A participant expresses interest in the geometry and mathematics of the problem, assuming a spherical Earth for simplification and seeking appropriate search terms related to kinematics and dynamical systems.
  • One participant distinguishes between the mathematical definition of rotation as an orientation-preserving isometry and the physical process of rotation as a time-dependent phenomenon, questioning the applicability of one-parameter actions of SO(3) in this context.
  • Another participant challenges the assumption that one-parameter actions of SO(3) apply universally, stating that this is only valid for bodies with spherical mass distributions and no external torques, which may not reflect realistic scenarios.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of one-parameter actions of SO(3) to real-world scenarios, indicating a lack of consensus on the assumptions regarding the Earth's shape and external influences on its rotation.

Contextual Notes

Participants acknowledge the limitations of their assumptions, particularly regarding the Earth's shape and the effects of external torques, which may impact the validity of their mathematical models.

Peaks Freak
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Does anyone know of a good mathematical reference covering the use of one-parameter group actions to model rotations of planets and/or other rigid bodies?

Thanks in advance!
 
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It's not that simple. The Earth is not a spherical body, so it undergoes nutation and precession as well as rotating.

That said, the International Earth Rotation and Reference System Service (IERS) is the official organization that defines reference frames used for navitation and astronomy. See http://www.iers.org/MainDisp.csl?pid=36-25787&prodversid=11221 .

I also suggest "Fundamentals of Astrodynamics and Applications", 3rd Edition, David Vallado.
 
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Thanks for the reply.

Since I'm mostly interested in the geometry and mathematics involved, I'm assuming that Earth is spherical. My main concern is the use of one-parameter group actions to model time-dependent rotations. As a mathematician, whose specialties are far afield from physics and astronomy, I'm not sure of the right search terms to use here. Kinematics? Dynamical Systems?

Any further help/insight would be appreciated!

Thanks.
 
OK, I think I've formulated a better question, one closer to my actual confusion.

In geometric terms, we define a rotation to be an orientation-preserving isometry that fixes some point p. Thus, a rotation is a map with properties.

In everyday terms, however, a rotation is a time-dependent, physical process. We observe rotations over time, the rotating rigid body passing through a continuum of orientations in between its starting and ending positions.

Whereas the former definition is of a particular kind of map, the second surely involves an action of the real numbers (acting as the passage of time). I'd like to conclude that this action is simply a one-parameter action of SO(3), but don't have experience with this physical setting. Do you know of a reference that addresses this scenario?

Also, does this conversation better belong in a different forum?

Thanks!
 
Peaks Freak said:
I'd like to conclude that this action is simply a one-parameter action of SO(3), but don't have experience with this physical setting.
That is only true for a body that has a spherical mass distribution and no external torques -- Not a particularly interesting or realistic situation.
 

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