Rotation Matrix - Infinitesimal Rotations

Infinitesimal Rotations

The matrices in the Lie algebra are not themselves rotations; the skew-symmetric matrices are derivatives, proportional differences of rotations. An actual "differential rotation", or infinitesimal rotation matrix has the form

where dθ is vanishingly small. These matrices do not satisfy all the same properties as ordinary finite rotation matrices under the usual treatment of infinitesimals (Goldstein, Poole & Safko 2002, §4.8). To understand what this means, consider

We first test the orthogonality condition, QTQ = I. The product is

differing from an identity matrix by second order infinitesimals, which we discard. So to first order, an infinitesimal rotation matrix is an orthogonal matrix. Next we examine the square of the matrix.

Again discarding second order effects, we see that the angle simply doubles. This hints at the most essential difference in behavior, which we can exhibit with the assistance of a second infinitesimal rotation,

Compare the products dA_xdA_y and dA_ydA_x.

$\begin{align} dA_{\bold{x}}\,dA_{\bold{y}} &{}= \begin{bmatrix} 1 & 0 & d\phi \\ d\theta\,d\phi & 1 & -d\theta \\ -d\phi & d\theta & 1 \end{bmatrix} \\ dA_{\bold{y}}\,dA_{\bold{x}} &{}= \begin{bmatrix} 1 & d\theta\,d\phi & d\phi \\ 0 & 1 & -d\theta \\ -d\phi & d\theta & 1 \end{bmatrix}. \\ \end{align}$

Since dθ dφ is second order, we discard it; thus, to first order, multiplication of infinitesimal rotation matrices is commutative. In fact,

again to first order. Put in other words, the order in which infinitesimal rotations are applied is irrelevant, this useful fact makes, for example, derivation of rigid body rotation relatively simple.

But we must always be careful to distinguish (the first order treatment of) these infinitesimal rotation matrices from both finite rotation matrices and from derivatives of rotation matrices (namely skew-symmetric matrices). Contrast the behavior of finite rotation matrices in the BCH formula with that of infinitesimal rotation matrices, where all the commutator terms will be second order infinitesimals so we do have a vector space.

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