Question:
Let $X:[0, 1] \rightarrow \mathbb{R}^3$ be a smooth curve, and $\mathbf{B}$ be the vector field in $\mathbb{R}^3$ defined as $$ \mathbf{B}(y) = \int_{\mathbb{R}^3} \frac{y-X(t)}{\|y-X(t)\|^3} \times X'(t) \ dt.$$ Considering the flow of $\mathbf{B}$, here are some possibilities for its orbits:
- singleton (where $\mathbf{B} = 0$)
- closed loop (periodic orbit)
- Not closed, but limits (including point at infinity) as $t \rightarrow \infty$ and $t \rightarrow -\infty$ both exist.
Can there exist other possibilites? (e.g. orbit which asymptotically approaches to a closed loop)
Details:
By Biot-Savart law, $\mathbf{B}$ is the magnetic field generated by current through the curve $X$ (I must have missed some constants), and the flow of $\mathbf{B}$ is called 'magnetic field lines.'
I could found affirmative answers like 'Yes, because there is no magnetic monopole', but I desire to find mathematical proof/counterexample.