# Show that the motion is a circular path

Question:

The position $$\mathbf r$$ of a particle satisfies

$$\ddot {\mathbf r} = f\dot {\mathbf r} \times \mathbf k$$

where $$f$$ is a known constant and $$\mathbf k$$ is the unit vector in the $$z$$ direction.

Given that $$\dot {\mathbf r} \cdot \mathbf k \equiv 0$$, show that the path of the particle is a circle.

Attempt:

Firstly, there are a few quantities that I know are constant:

$$\frac{d}{dt}\big(|\dot{\mathbf r}|^2 \big) = 2\ddot {\mathbf r} \cdot \dot {\mathbf r} = 2f(\dot{\mathbf r} \times \mathbf k) \cdot \dot{\mathbf r} = 0 \implies |\dot{\mathbf r}|^2 \equiv \text{constant}$$

$$\frac{d}{dt}\big(\mathbf r \cdot \mathbf k\big) = \dot{\mathbf r} \cdot \mathbf k =0 \implies {\mathbf r} \cdot \mathbf k \equiv \text{constant}$$

Moreover, I noticed that

$$\frac{d^2\dot{\mathbf r}}{dt^2} = f^2\dot{\mathbf r}$$

and solving this gives

$$\dot{\mathbf r}(t) = \mathbf c_1 \cos(ft)+\mathbf c_2 \sin (ft)$$

but I am unsure how to deduce from this that $$\mathbf r$$ traces out a circle.

Let $$\mathbf u = \mathbf r + \frac1f (\dot{\mathbf r} \times \mathbf k)$$, we have $$\dot{\mathbf u} = \dot{\mathbf r} + \frac1f(\ddot{\mathbf r} \times \mathbf k) = \dot{\mathbf r} + ((\dot{\mathbf r} \times \mathbf k) \times \mathbf k) = \dot{\mathbf r} + ((\dot{\mathbf r}\cdot \mathbf k) \mathbf k - \dot{\mathbf r}) = \mathbf 0$$ This means $$\mathbf u$$ is a constant. Notice $$\frac{d}{dt}|\mathbf r - \mathbf u|^2 = 2(\mathbf r-\mathbf u)\cdot \dot{\mathbf r} = -\frac{2}{f} (\dot{\mathbf r} \times \mathbf k)\cdot \dot{\mathbf r} = \mathbf 0$$ The expression $$|\mathbf r-\mathbf u|^2 = R^2$$ is a constant. The particle is moving on a sphere centered at $$\mathbf u$$ with radius $$R$$. Together with what you know $$\mathbf r \cdot \mathbf k =$$ constant. The particle is moving along the intersection of a sphere and a plane, i.e. a circle.

Insert the solution you found of the derived equations into the original equations. $$-f\vec c_1\sin(ft)+f\vec c_2\cos(ft)=f(\vec c_1×\vec k)\cos(ft)+f(\vec c_2×\vec k)\sin(ft).$$ Comparing coefficients, $$\vec c_2=~~\vec c_1×\vec k\\ \vec c_1=-\vec c_2×\vec k\\$$ is only possible if both vectors are orthogonal to $$\vec k$$, of the same length and orthogonal to each other. With $$\vec k=(0,0,1)$$ you get $$\vec c_1=(a,b,0)$$ and $$\vec c_2=(b,-a,0)$$. This means that $$\dot{\vec r}$$ follows a circular curve, and that remains preserved under integration, the integration constant is the midpoint around which $$\vec r$$ circulates.

One way is to show that the curvature is constant. Recall that $$\kappa = \frac{||\mathbf{v}(t) \times \mathbf{a}(t)||}{||\mathbf{v}(t)||^3} = \frac{||\dot{\mathbf{r}}(t) \times \ddot{\mathbf{r}}(t)||}{||\dot{\mathbf{r}}(t)||^3}$$ Which should be straightforward since you have $$\dot{\mathbf{r}}(t)$$.

It might help to write: \begin{align} \dot{\mathbf{r}}(t) &= \mathbf{c_1}\cos(ft) + \mathbf{c_2}\sin(ft)\\ &= \left[\begin{array}{c}c_{11}\\c_{12}\\c_{13}\end{array}\right]\cos(ft) + \left[\begin{array}{c}c_{21}\\c_{22}\\c_{23}\end{array}\right]\sin(ft)\\ &=(c_1\cos(ft) + c_2\sin(ft))\hat{\mathbf{i}} + (c_3\cos(ft) + c_4\sin(ft))\hat{\mathbf{j}} + (c_5\cos(ft) + c_6\sin(ft))\hat{\mathbf{k}} \end{align} Then just do the grinding to get the final result.

EDIT: I just noticed that it might be easier to replace quantities like $$\ddot{\mathbf{r}}(t) = f\dot{\mathbf{r}}(t) \times \mathbf{k}$$ into the curvature expression as well.

EDIT2: It's been a while since I"ve dealt with this... Showing that torsion is $$0$$ is important too.

• I am not quite familiar with the concept of "curvature", so could you please kindly explain why this quantity being constant implies that $\mathbf r$ traces out a circle? Nov 20 '18 at 2:14
• @glowstonetrees It actually takes a little while to get there if you haven't seen curvature before! This answer can get you started on that path though if you want to head down that way: math.stackexchange.com/questions/1505638/… Nov 20 '18 at 2:28