Magnetic Field Lines: Proving c(T)=c(0) with T≠0

[jostpuur]

Suppose that a function $B:\mathbb{R}^n\to\mathbb{R}^n$ and $c:\mathbb{R}\to\mathbb{R}^n$ are defined such that $c$ is differentiable, and

$$\dot{c}(t) = B(c(t))$$

for all $t$. The question is that what must be assumed of $B$, so that it would become possible to prove that

$$c(T)=c(0)$$

with some $T\neq 0$?

 

[n1person]

Do you mean prove that c(dot)(T) = c(dot)(0)?

 

[jostpuur]

No. I mean that the curve comes back to where it started from. (Not that it would point in the same direction at least twice.)

 

[klondike]

Suppose that a function $B:\mathbb{R}^n\to\mathbb{R}^n$ and $c:\mathbb{R}\to\mathbb{R}^n$ are defined such that $c$ is differentiable, and

$$\dot{c}(t) = B(c(t))$$

for all $t$. The question is that what must be assumed of $B$, so that it would become possible to prove that

$$c(T)=c(0)$$

with some $T\neq 0$?

$$\oint _{\partial S}B \bullet ndS = 0$$ or B must be divergence free.

 

[jostpuur]

That answer is incorrect.

$n=2$, $B(x)=(x_1,-x_2)$, $c(t)=(e^t,e^{-t})$ give a counter example.