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

<br /> \dot{c}(t) = B(c(t))<br />

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

<br /> c(T)=c(0)<br />

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

<br /> \dot{c}(t) = B(c(t))<br />

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

<br /> c(T)=c(0)<br />

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.