Source
r/askmath•
Monory

Why does the presence of curl destroy conservation of energy?

I can see mathematically how you can get from conservation of energy to curl(f) = 0, but conceptually I am having trouble picturing why a nonzero curl would prevent conservation when putting it into physical terms. In two dimensions for example, how would the presence of rotation prevent energy from being conserved?

Edit: This explanation seems to clear it up. Thanks everyone who commented!

Posted Jul 19, 2018, 2:11 PM UTC · edited

725
MattAmoroso•

Trace out some random closed loop. In order to come back to where you started you must go away from your starting point just as much as you go towards. But you don't need to go clockwise just as much as you go counterclockwise. Your whole trip might be either. If the force in question has a curl then it is either helping or resisting the clockwise motion. In the other scenario, if it is helping you move away from the starting point then it is resisting the coming back towards the starting point and they cancel out.

6
MonoryOP•

Thank you! While the other comments made mathematical sense, this is exactly what I meant when I didn't understand conceptually what was happening. It makes perfect sense that you have to go along x and y the exact same amount forward and backward in a closed loop, but that that doesn't hold for rotational movement. This was the explanation I needed.

2
InSearchOfGoodPun•

A better way to describe it isn't that a nonzero curl prevents energy from being conserved, but rather I'd say that it prevents energy from being defined, since the relevant question is the following: Given a vector field V (btw, you probably shouldn't use the variable f for a vector field), is there a potential function E such that grad(E) = V?

Well, if there is such an E, that means V is the direction of steepest ascent for E. In particular, it can't have any "closed loops" as you follow the direction V. On an infinitesimal level, when curl(V) is nonzero at a point p, that intuitively means that V is "rotating around p" in some sense. But this goes against the "non-looping" behavior of gradient vector fields. To make this rigorous, we need to turn to the mathematical proof using Stokes Theorem.

4
MonoryOP•

So I was thinking of a vector field with curl as a field in which conservation of energy was broken because somehow the presence of curl provided "free energy", but the better interpretation would be that a field with curl is simply no longer related to potential energy?

1
InSearchOfGoodPun•

Yes, that's how I would put it, but actually, your way works, too. Your way is just more of a physicist's way of thinking, and I happen to be a mathematician.

1
MonoryOP•

I think I have a better understanding now, thanks!

1
MonoryOP•

Another commenter pointed out what I think is the heart of what I was getting at. In a closed loop, even though you you have to go exactly the same amount forward and back along the x and y directions, this isn't true for clockwise and counterclockwise movement since an entire closed loop could be clockwise. Thus, any clockwise or counterclockwise force (curl) would not be cancelled out along the closed loop.

1
Theowoll•

If curl(f) ≠ 0, the amount of work W required to move a body from A to B depends on the path from A to B. Take two paths p and q with W(q)>W(p). You gain the amount of energy W(q)-W(p) out of nowhere when you move the body first from A to B along p, and then back from B to A along q.

3
MonoryOP•

That is the part I can understand mathematically, the problem I have is why this is the case. What is it about curl that allows energy to come out of nowhere, if in say two dimensions curl is just a measure of rotation?

1
Theowoll•

From a physical point of view the simple answer is that there isn't more to it than the math you already understand. In the real world, energy is conserved, so you don't need to understand how energy comes out of nowhere. If energy wasn't conserved, it wouldn't be a very useful concept.

From a mathematical point of view, curl can be defined to measure the deviation from path-independence, so curl(f) ≠ 0 unsurprisingly implies path-dependence, which obviously (as seen above) violates conservation of energy.

3
MonoryOP•

So it is not that curl is some kind of property that provides free energy, but rather that a vector field with curl is just no longer a proper mathematical model of energy?

1
Theowoll•

With curl(f) ≠ 0 you can still define energy, but it becomes a function of paths between two points, not just a function of the two endpoints. It is still mathematical meaningful, but not that interesting for physics, where conserved quantities are used to understand the behavior of physical systems.

1
WheresMyElephant•

Consider this force field:

https://goo.gl/images/HW9RDt

Now, just as an example, imagine we have an object connected to the origin by a rod of unit length, so that it's constrained to follow a circular orbit. Let's say it starts out moving counterclockwise. At every point in that orbit the force is pushing the object counterclockwise, increasing its speed. At no time will the force ever cause its speed to decrease. When it gets back to its starting point, it will have more kinetic energy than before.

As we know, the work done by any force is the integral of the dot product F.dr where dr is the infinitesimal displacement vector. The rod does no work, because it's always perpendicular to the direction of motion (that is, to dr). But our force field is oriented in such a way that the dot product is always positive if the object is moving counterclockwise, or always negative if it's moving clockwise.

As other people said, this also means it's impossible to define a "potential energy" associated with this force. Potential energy is just a way of accounting for the work performed by a conservative force. When we say "the potential energy at point a is 0, and the potential energy at point b is 1J," what we really mean is that when you move from point b to point a, our force does 1J of work. This causes the kinetic energy to increase by 1J, unless perhaps there are other forces involved of course. The change in kinetic energy is equal to 1J; the change in "potential energy" is -1J, so therefore the sum of kinetic energy plus "potential energy" remains constant. A handy little bookkeeping technique!

But in this case that won't suffice. If we travel counterclockwise from a to b then the force will do positive work; but if we travel clockwise it will do negative work. Clearly it's impossible for us to determine how much work was done just by knowing a and b: we also need to know what path the object took.

And not coincidentally, this also means that we can get "free" work out of this force by travelling from a to b along one route and returning along a different route. In this example we can travel from (1,0) to (-1,0) by following the top half of the unit circle, and we can return to (1,0) by following the bottom half, and now we have more energy than we started with. See, if we returned along the same route--reversed course and travelled clockwise along the top semicircle--then we would precisely cancel out the work we did before. (Remember, the work is just the sum of the dot product F.dr along the path. If we take a path and reverse it, then at every point we have the same F but the exact opposite dr, so the dot product flips its +/- sign.) But if there exist other routes from a to b where the amount of work is different, then we can take one of those routes in reverse to return to a, and it won't cancel out!

1
Godivine•

whats the context, and what is f?

2
MonoryOP•

The context would be a two dimensional vector field, and f is the vector field. What is it about curl, conceptually, that allows it to destroy conservation of energy? Or I guess another way, what is it about curl that allows it to add free energy?

1
Godivine•

Maybe I'm the wrong person to try to answer this because I don't know what the energy of a function R^n -> R^n is.

The reason I ask is if v=v(x,t) is the velocity of a fluid solving the navies stokes equations, energy (defined as the L^2 norm of v in space) is dissipated (in the inviscid case, it is conserved). And the curl of the velocity, called the vorticity, is not 0 in general.

Maybe you are talking about flows defined via d_t u = F(u), where F is your vector field? I unfortunately don't know much about this. I do know its common to talk about gradient flows(= curl free).

Edit- on a google it seems you are indeed talking about these kinds of flows. In the special case of gradient flows / potential flows, i.e. x_t = grad(V(x)) it seems reasonable to define energy as

E(t) = |grad(V(x))|^2 + V(x)

And I think you are proposing to replace grad(V) with something else? What would be the correct definition of E?

1
MonoryOP•

I'm not sure about those specific gradient flows, but from talking to the other commenters I think the problem was that I was thinking about vector fields with curl as related to some potential energy function but with curl "breaking" conservation of energy, but the better way to think of it is that a vector field with curl is simply not a mathematical model of potential energy along a path.

2
Godivine•

Sure, there's no reasonable way (that I can think of) to assign a 'potential' V to an arbitrary field F, that makes some sense.

1
kcl97•

A nice way to see this is from the field of defect theory and the idea of a Burger's vector (not a food). The idea is to picture layers of crystals. Now, draw a closed curve around these layers. Walk along the curve from start to end. As you cross each layer of crystal add one or subtract one depending on whether you are crossing the layer bottom-up or up-bottom (the layers are oriented). Now if you do this for nice layered defectless crystal, you will get zero after completing a cycle. However imagine there is an extra layer that pop out of nowhere but does not extend across the crystal (a dislocation defect). Now depending on how you draw your path you may or may not get zero after completing the cycle. You can think of the layers of crystal as like a gradient field and curl (going around a path) is a test for the defects in the gradient field. As you may have guessed, we can have tons of dislications originate from one point, that would be a point defect, the curl around a point defect is huge and extends throughout unless it is cancelled by other defects. As an extra, point defects in the language of electromagnets is charges.

1
FunPackedShow•

Not directly related, but you might find Helmholtz' theorem(s) interesting. Roughly speaking, they say that reasonable vector fields can be decomposed into a curled part and a uncurled part.

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Godivine•

(uniquely!)

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