# Use Lie Sub-Groups of GL(3, R) for elastic deformation ?

I'm interested in representing elastic deformations (e.g. stretching) using Lie groups. There are a few references to using $GL(3,\mathbf{R})$ but I'm wondering if possible to use subgroups of $GL(3,\mathbf{R})$. For example, deformation gradient ($F$) can be decomposed into rotation $R$ and symmetric positive definite stretch $U$ as in $F = RU$. $U$ then decomposable by SVD into $U = PEP^T$ where $P =$ matrix of eigenvectors of $U$ and $E =$ diag. matrix of eigenvalues of $U$. Such diag. matrices are Lie subgroup of $GL(3,\mathbf{R})$ and represent pure stretches along orthogonal stretch axes. Similarly, 3x3 identity matrices with off-diag. positive entries represent shears.

So, my question is:

Can elastic deformations be represented by subgroups of $GL(3,\mathbf{R})$ like those mentioned above?

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I think the answer is that it depends on what you mean by "represented". Often there is a volume-conservation constraint, so you are stuck with the subgroup $SL_2(\mathbf{R})$. The decompositions you mention are well-known in the theory of Lie groups, e.g., $F=RU$ is called Iwasawa decomposition. I should note that shears may have negative entries off the diagonal. –  S. Carnahan Sep 29 '10 at 14:11
Scott, Thanks for the insights & context. My initial post should have mentioned my interest as homogeneous elastic deformation (F depending only on time, not position). By "represented", I mean representing these deformations as some combination of (sub)groups with physical meaning (i.e. pure stretch, shear, etc.) like done with rigid body motion (i.e. SO(3) X R^3). Thanks, John. –  John Craighead Sep 29 '10 at 14:47