Suppose you are given a nowhere-vanishing exact 2-form B=dA on an open, connected domain D⊂R3. I'd like to think of B as a magnetic field.
Consider the product H(A)=A∧dA. At least in the plasma physics literature, H(A) is known as the magnetic helicity density.
How can one determine if there is a closed one-form s such that H(A+s) is non-zero at all points in D?
The reason I am interested in this question is that if you can find such an s, then A+s will define a contact structure on D whose Reeb vector field gives the magnetic field lines. Thus, the question is closely related to the Hamiltonian structure of magnetic field line dynamics.
I'll elaborate on this last point a bit. If there is a vector potential A such that A∧dA is non-zero everywhere, then the distribution ξ=ker(A) is nowhere integrable, meaning ξ defines a contact structure on D with a global contact 1-form A. The Reeb vector field of this contact structure relative to the contact form A is the unique vector field X that satisfies A(X)=1 and iXdA=0. Using the standard volume form μo, dA can be expressed as iBμo for a unique divergence-free vector field B (I'm having trouble typing B as a subscript). Thus, the second condition on the Reeb vector field can be expressed as B×X=0, which implies the integral curves of X coincide with the magnetic field lines.
More generally, suppose M is an orientable odd-dimensional manifold equipped with an exact 2-form ω of maximal rank. Also assume that the characteristic line bundle associated to ω admits a non-vanishing section b:M→ker(ω). What is the obstruction to the existence of a 1-form ϑ with dϑ=ω and ϑ(b)>0?
some observations/comments:
1) If A(B) is bounded above and below on D, then a sufficient condition for there to be an s that gives a nowhere-vanishing helicity density is the existence of a closed one-form α with α(B) nowhere vanishing. In that case, s=λα, where λ is some large real number (with appropriate sign), would work.
If there is such an α, then, being closed, it defines a foliation whose leaves are transverse to the divergence-free field B. I suspect the question that asks whether a given non-vanishing divergence-free vector field admits a transverse co-dimension one foliation has been studied before, but I am not familiar with any work of this type.
An example where D=3-ball and helicity density must have a zero:
Let D consist of those points in R3 with x2+y2<a2 for a real number a>1. Note that all closed 1-forms are exact in this case. Let f:[0,∞)→R be a smooth, non-decreasing function such that f(r)=0 for r<1/10 and f(r)=1 for r≥1/2. Let g:R→R be the polynomial g(r)=1−3r+2r2. Define the 2-form B using the divergence free vector field B(x,y,z)=f(√x2+y2)eϕ(x,y,z)+g(√x2+y2)ez. Here eϕ is the azimuthal unit vector and ez is the z-directed unit vector. It is easy to verify that B, thus defined, is an exact 2-form that is nowhere vanishing.
Because g(1)=0 and f(1)=1, the circle, C, in the z=0-plane, x2+y2=1, is an integral curve for the vector field B. I will use this fact to prove that the helicity density must have a zero for any choice of gauge. Let A satisfy dA=B and suppose A∧B is non-zero at all points in D. Note that A∧B=A(B)μo, meaning h=A(B) is a nowhere vanishing function. Without loss of generality, I will assume h>0. Thus, the line integral I=∮Chdl|B| satisfies I>0. But, by Stoke's theorem, I=2π∫10g(r)rdr=0, as is readily verified by directly evaluating the integral. Thus, there can be no such A.
An example where D=T2×(0,2π) and helicity density must have a zero:
Set D=S1×S1×(0,2π) and let (θ,ζ,r) be the obvious coordinate system. Set B=f(r)dr∧dθ+g(r)dr∧dζ where f(r)=cos(2r),
and
g(r)=sin(r).
Clearly,
A=12sin(2r)dθ−cos(r)dζ satisfies
B=dA and
B is nowhere vanishing. A quick calculation shows that
∫DA∧B=0.
Now suppose that s is an arbitrary closed 1-form. Either by using Stoke's theorem or by direct calculation, the fact that the total toroidal and poloidal fluxes, 2π∫2π0f(r)dr and 2π∫2π0g(r)dr, are zero implies that ∫D(A+s)∧B=0. Thus, the helicity density must always have a zero.
This post imported from StackExchange MathOverflow at 2015-05-27 22:10 (UTC), posted by SE-user Josh Burby