2017/01/27 by Andreas Fischle, Fischle, Andreas, Patrizio Neff +1
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.1701.08150
This is an overview paper collecting results distributed over three preceding papers (without proofs)
arxiv created 2017/01/27 · openalex publication_date 2017/01/27 · arxiv updated 2017/01/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let F ∈ \rm GL+(3) and consider the right polar decomposition F = Rp(F)⋅ U into an orthogonal factor Rp(F) ∈ \rm SO(3) and a symmetric, positive definite factor U(F) = √(FTF) ∈ \rm Psym(3). In 1940 Giuseppe Grioli proved that \rm argmin_R ∈ \rm SO(3) ||RTF - 1||2 = \ Rp(F) \ = \rm argmin_R ∈ \rm SO(3) ||F - R||2 . This variational characterization of the orthogonal factor Rp(F) ∈ \rm SO(n) holds in any dimension n ≥ 2 (a result due to Martins and Podio-Guidugli). In a similar spirit, we characterize the optimal rotations \rm rpolarμ,μc(F) := \rm argmin_R ∈ \rm SO(n) \lbrace μ ||\rm sym(RTF - 1)||2 + μc ||\rm skew(RTF - 1)||2 \rbrace for given weights μ> 0 and μc ≥ 0. We identify a classical parameter range μc ≥ μ> 0 for which Grioli's Theorem is recovered and a non-classical parameter range μ> μc ≥ 0 giving rise to a new type of globally energy-minimizing rotations which can substantially deviate from Rp(F). In mechanics, the weighted energy subject to minimization appears as the shear-stretch contribution in any geometrically nonlinear, quadratic, and isotropic Cosserat theory.