Zero Curl Vector Fields - Does it Mean Potential Energy?

[jackiefrost]

1. If a vector field has zero curl, does it always mean that it is the gradient of some scalar (potential) field?

2. If the vector field is a force field and its curl is zero does that mean that the "potential" scalar field that it is the gradient of is always some form of "potential energy" (as in PE + KE = Constant). And, is that why they call vector fields like that (where the curl is zero) "conservative"? That is, are they called conservative fields because integrating F.ds around any closed path equals zero and therefore they obey (? - not the right word...) the conservation of energy? But... calculating the (closed loop) line integral of F.ds for some F vector field that has curl would not equal zero, would not be path independent, and would not be the gradient of some scalar potential field function; but it should NOT violate the conservation of energy either... right? Anyway, I'm kind of confused.

3. Is there any physical significance to the term "potential field" if the zero curl vector field in question is a velocity field rather than a force field?

Thanks for any insights - (BTW - this isn't any assignment. I study stuff on my own.)

jf

 

[StatusX]

1. Yes. Define the potential at a point x as the path integral of the vector field from some fixed reference point to x. This is well defined, since the difference between any two paths is a path integral over a closed loop, and so will always be zero. It is easy to show the gradient of this potential is the vector field you started with.

2. If the vector field represents the force at all points, then, since work done is F.ds, the net work done is zero over any closed path. You may define the negative of the scalar potential associated with this force field as the potential energy function, and in this case the total energy (KE+PE) would be conserved. A vector field with non-zero curl may represent a force, but then there is always some dissipation of energy that depends on the path taken. This force is not conservative, and so not derivable from a potential.

3. In fluid mechanics, when an incompressible fluid has no vorticity (the velocity field has zero curl), it is said to be irrotational. It can then be modeled as the gradient of some potential function obeying Laplace's equation. This is called potential flow.

 

[jackiefrost]

Thanks for the reply.
I'm studying Stokes' Theorem and circulation. If a force field exists that has curl it seems that there can be a circulation energy associated with it. Is this so? If so, can that energy produce work by the proper choice of loop path(s) for some conveying substance through such a field? [I'm not interested in "getting something for nothing" or anything like that - just trying to learn how energy relates to various field properties].

jf

 

[dextercioby]

1. Not really. Indicentally $\mathbb{R}^n$ is topologically trivial and the Poincare' lemma holds globally, but on another manifold, that thing is only valid locally...

Daniel.

 

[StatusX]

You can abstractly define any force field you want, and use it to make all kinds of loops with net energy changes. But unless these correspond to something physical, what's the use?

 

[jackiefrost]

Though, in my op I didn't state explicitly, I was mainly interested in fields that are related to physical realities - not mathematical abstractions.

 

[Physics Monkey]

As it happens, these topological questions are not merely academic. Consider the phase of a superfluid order parameter in two dimensions. One often writes the superfluid velocity as proportional to $$\vec{\nabla} \theta$$ but this gradient of the phase is a sketchy object. It doesn't behave quite like your ordinary gradient because the phase is undefined at the origin of coordinates. It's a simple exercise to verify that the curl of this quantity is zero everywhere except the origin where it reduces to a delta function. This "gradient" has the further strange property that when integrated around a closed loop that circles the origin the result need not be zero corresponding to vortices. In this case the phase is defined on a space with nontrivial topology (the plane with a hole) and even though the curl of the phase gradient is zero everywhere on the space, certain well known results for vector fields that are truly gradients do not hold. All these considerations are important, for example, in the study of vortices in superfluids.

The message to take home is that you shouldn't take dextercioby's advice too lightly. In simple applications one can usually get away with being careless, but you do have to be careful sometimes.

 

[jackiefrost]

As it happens, these topological questions are not merely academic. Consider the phase of a superfluid...
The message to take home is that you shouldn't take dextercioby's advice too lightly. In simple applications one can usually get away with being careless, but you do have to be careful sometimes.

Thanks. Message recieved. I should know better - whenever I use the word "always" I ALMOST always regret it Whenever I think I'm on to something that could be so, it's not long before the exceptions arrive at the party.

jf