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I'm about half way through this derivation of Thomas precession for simple circular motion, and have got stuck at the equation

[tex]\Delta \theta=-2 \pi \frac{v^2}{1-v^2}\left \langle \cos^2 \theta \right \rangle_{mean}[/tex]

I think this represents the total change in angle of a line segment that's moved once around the whole circumference an n-sided polygon at constant speed.

[itex]\theta[/itex] is defined as the "limiting value of both [itex]\theta_1[/itex] and [itex]\theta_2[/itex] as [itex]\alpha[/itex] goes to zero". I think this represents the change in angle of the line segment as it changes direction at the first vertex, or maybe at each vertex, I'm not sure.

The angles [itex]\theta_1[/itex], [itex]\theta_2[/itex] and [itex]\alpha[/itex] are defined in the second diagram on the page.

I don't understand the expression [itex]\left \langle \cos^2 \theta \right \rangle_{mean}[/itex].

[itex]\theta[/itex] was defined as a constant, but this seems to be treating it as a variable. Is theta being used in this equation as a variable for the limiting values of each pair of adjacent angles? But then if [itex]\theta_1[/itex] and [itex]\theta_2[/itex] approach the same limit as each other, and [itex]\theta_2[/itex] and [itex]\theta_3[/itex] also approach the same limit as each other, won't all of these limiting values be the same? Or is it allowing the possibility that alpha varies, i.e. that this is not a regular polygon, and would it be necessary to treat theta as a variable in that case, even if the equation is only being used to derive the limit as n goes to infinity and [itex]\alpha[/itex] to 1?

(I'm still hazy about the principles according to which some distinctions can be "neglected" while others must be observed in such proofs involving limits.)

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# Brown's derivation of Thomas rotation

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