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A useful mnemonic which always makes me smile! Having said that, I remain a dedicated fan and student of A&H and Arnold. But I'm sure others in the mechanics community cut their teeth on Goldstein. Ah how those (we) Ph106 students used to suffer at old Caltech! A great place to learn mechanics in a truly coordinate dependent notation!
For those who really like classical mechanics from a classical point of view, don't neglect Herbert Goldsteins's now-nearly ancient book, Classical Mechanics. There one finds what H.G. calls the "Jabberwockian statement", "The polhode rolls without slipping on the herpolhode lying in the invariable plane."
Maybe this comment is misguided, but bobye only stipulated $U$ and $V$ be diagonal. $A$ is apparently a scalar multiple of an orthogonal matrix, since $A^{T}A/a^{2} =I$.
Dear Regina, I think your problem is quite interesting and so I feel motivated to do something I normally wouldn't be so concerned with, and that is to ask you to add a few words addressing the origins of this problem. I'm pretty sure many would agree doing so would help people help you in your search for an answer.
