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Table 2 Elements of the Lie algebra \(\tilde{\mathfrak g}\) and the associated Lie group \(\tilde{G}\)

From: Introducing the Logarithmic finite element method: a geometrically exact planar Bernoulli beam element

Element \(\varvec{{\tilde{\mathbf {Z}}}}_{\varvec{I,r}} \varvec{\in \tilde{\mathfrak g}}\) Element \({\varvec{\exp \left( {\tilde{{\mathbf {Z}}}_{I,r}}\right) \in \tilde{G}}}\) Subalgebra in \(\varvec{\tilde{\mathfrak g}}\)
\(\tilde{\mathbf {Z}}_{1,r} := \begin{pmatrix} z_{1,r} &{} -z_{1,r} &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \end{pmatrix}\) \(\exp \left( {\tilde{\mathbf {Z}}_{1,r}}\right) = \begin{pmatrix} \exp \left( {z_{1,r}}\right) &{} 1- \exp \left( {z_{1,r}}\right) &{} 0\\ 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 \end{pmatrix}\) \(\tilde{\mathfrak g}_1 = \langle \tilde{\mathbf {Z}}_{1,r} \rangle \), \(r \in \{1,2\}\)
\(\tilde{\mathbf {Z}}_{2,r} := \begin{pmatrix} z_{2,r} &{} 0 &{} -z_{2,r} \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0\end{pmatrix}\) \(\exp \left( {\tilde{\mathbf {Z}}_{2,r}}\right) = \begin{pmatrix} \exp \left( {z_{2,r}}\right) &{} 0 &{} 1-\exp \left( {z_{2,r}}\right) \\ 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 \end{pmatrix}\) \(\tilde{\mathfrak g}_2 = \langle \tilde{\mathbf {Z}}_{2,r} \rangle \), \(r \in \{1,2\}\)
  1. \(z_{I,r}\) assumes the following values: \(z_{1,1} = z_{2,1} = 1\), \(z_{1,2} = z_{2,2} = \mathrm{i}\)