Searching for a Solid Proof of Chasles/Mozzi/Cauchy's Theorem

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Discussion Overview

The discussion centers around the search for a solid proof of Chasles' Theorem, which asserts that any rigid body displacement can be expressed as a screw motion. Participants explore the theorem's implications, seek proofs, and share resources related to its understanding, particularly focusing on algebraic approaches.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant expresses frustration over the lack of convincing proofs for Chasles' Theorem, specifically seeking a linear algebraic proof.
  • Another participant shares a link to a paper by Dunham Jackson, claiming it provides a comprehensive explanation of rigid body motions and their decomposition into screw motions, referring to the theorem as Mozzi's theorem.
  • A participant mentions having developed a semi-proof for the 2D case using similar triangles and expresses interest in finding a proof for the 3D case.
  • One participant requests a free copy of the paper mentioned, indicating a desire for accessible resources.

Areas of Agreement / Disagreement

Participants generally agree on the interest in finding solid proofs of Chasles' Theorem and share resources, but there is no consensus on the existence of a definitive proof or the completeness of existing ones.

Contextual Notes

Some participants note the lack of diagrams in the recommended paper, which may limit understanding. There is also a distinction between the 2D and 3D cases that remains unresolved.

ObsessiveMathsFreak
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I've unsuccessfully been looking for a decent proof of Chasles' Theorem which states that any rigid body displacement whatsoever can be decomposed into a screw motion. In other words, no matter what the displacement is, you can consider it the result of the partiles having moved to their positions by following a circular helixical path about a common axis, with a common angular speed.

I suppose I'm mostly looking for a (linear) algebraic type proof. Most proofs I've encountered have been fairly loose and unconvincing. Does anyone know of a solid proof of this theorem?

Awkwardly, this theorem is also variously known as Mozzi's or Cauchy's screw theorem.
 
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In answer to my own question:

Shortly after making this post I stumbled across one of the best papers I've ever read.
http://www.jstor.org/view/00029890/di991259/99p1550p/0 by Dunham Jackson. It's old, but great, and completely and totally explains rigid body motions and their decompositions into rotations and translations and finally into screw motions. He refers to the theorem as Mozzi's theorem.

It's a great paper. I'd seriously recommend anyone to give it an hour of their time. It's an easy read, and the proofs are just brilliant. The only thing I could fault it on is a lack of diagrams.
 
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Thanks for the link, this is a problem I've been annoyed with for a while too. I've worked out a semi-proof for the 2d case using similar triangles, but I would like to see the 3d case as well. Do you have a link for a free copy of the paper?
 
EFuzzy said:
Thanks for the link, this is a problem I've been annoyed with for a while too. I've worked out a semi-proof for the 2d case using similar triangles, but I would like to see the 3d case as well. Do you have a link for a free copy of the paper?
Attached is a copy of the paper.
 

Attachments

Thanks!
 

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