Are Angular Quantities Vectors? — Question 4

Question 4

Infinitesimal rotation addition works only to first order

For a very small rotation vector $\delta\vec\theta$, a position changes by $\delta\vec r=\delta\vec\theta\times\vec r$ to first order. Let $\delta\vec\theta_1=\epsilon\hat x$ and $\delta\vec\theta_2=\epsilon\hat y$, with $|\epsilon|\ll1$.

1. Show why their combined first-order effect is independent of order.

2. For initial $\vec r=R\hat z$, find that first-order change and identify the order of the omitted noncommuting correction.

Question 4 - Solution

Infinitesimal rotation addition works only to first order

(a) Derivation and calculation

Apply the two first-order operations successively: $$\vec r_{12}=\vec r+\delta\vec\theta_1\times\vec r +\delta\vec\theta_2\times\vec r+O(\epsilon^2).$$ $$\boxed{\delta\vec r=(\delta\vec\theta_1+\delta\vec\theta_2)\times\vec r+O(\epsilon^2)}.$$ The retained linear sum is independent of order.

(b) Result and interpretation

For $\vec r=R\hat z$, $$\delta\vec r=\epsilon R(\hat x\times\hat z+\hat y\times\hat z) =\boxed{\epsilon R(\hat x-\hat y)}\quad\text{to first order}.$$ The leading general order difference is $$\vec r_{12}-\vec r_{21}=(\delta\vec\theta_2\times\delta\vec\theta_1)\times\vec r+O(\epsilon^3).$$ It is generally $O(\epsilon^2)$, but this quadratic term vanishes for $\vec r\parallel\hat z$.

Physical check

The first-order increment has zero dot product with $R\hat z$, so it preserves the radius to first order.