On strain measures and the geodesic distance to $SO_n$  in the general linear group

Raz  Kupferman; Asaf  Shachar

doi:10.3934/jgm.2016015

Article Contents

2016, Volume 8, Issue 4: 437-460. Doi: 10.3934/jgm.2016015

This issue Previous Article The Tulczyjew triple in mechanics on a Lie group Next Article Kirchhoff's equations of motion via a constrained Zakharov system

On strain measures and the geodesic distance to $SO_n$ in the general linear group

Raz Kupferman^1, and
Asaf Shachar^1,

1.
Institute of Mathematics, The Hebrew University, Jerusalem 91904

Received: February 29, 2016

Revised: July 31, 2016

Published: November 2016

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Abstract

We consider various notions of strains---quantitative measures for the deviation of a linear transformation from an isometry. The main approach, which is motivated by physical applications and follows the work of [12], is to select a Riemannian metric on $GL_n$, and use its induced geodesic distance to measure the distance of a linear transformation from the set of isometries. We give a short geometric derivation of the formula for the strain measure for the case where the metric is left-$GL_n$-invariant and right-$O_n$-invariant. We proceed to investigate alternative distance functions on $GL_n$, and the properties of their induced strain measures. We start by analyzing Euclidean distances, both intrinsic and extrinsic. Next, we prove that there are no bi-invariant distances on $GL_n$. Lastly, we investigate strain measures induced by inverse-invariant distances.

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Mathematics Subject Classification: Primary: 53Zxx; Secondary: 74Bxx.

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