Composition of rotations and parallel transport

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Abstract

This note provides the details and proofs of the results announced by Levi 1993 Fields Insitute Communications vol 1 pp 133-8. The main result of this note is a geometrical representation of the reconstruction problem for SO(3) in terms of parallel transport. It is, of course, well known that the solution of a linear equation ẋ = Ω(t)x in Rn cannot in general be expressed by x(t) = e∫0tΩ(τ)dτx(0) (false) because the coefficient matrices Ω(t1) and Ω(t2) may fail to commute for t1 ≠ t2. Nevertheless, when n = 3 and when Ω(t) is skew-symmetric, i.e. when it lies in the Lie algebra of the group of rigid rotations in R3, the above false formula is almost correct, as we will show here. The main result of this note is a geometrical expression for the matrix solution X(t) of matrix equations on TSO(3) of the form Ẋ = Ω(t)X, ΩT = -Ω (*) where X, Ω are 3 × 3 matrices with real coefficients. The argument relies on a theorem of Poinsot together with some observations on geodesic curvatures of moving curves.

Original languageEnglish (US)
Pages (from-to)413-419
Number of pages7
JournalNonlinearity
Volume9
Issue number2
DOIs
StatePublished - Dec 1 1996

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

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