Prove a rotationally symmetric central force field is conservative

In summary, the problem is to prove that the function F:ℝd\{0} → ℝd, where F(x) = xg(|x|) and g:(0, +∞) → ℝ is continuous, is conservative. This can be done by showing that F is equal to the gradient of a C1 function f:ℝd\{0} → ℝd. By using the anti-derivative G(|x|) of |x|g(x) and differentiating it, it can be shown that F is indeed grad(f), where f is just G.
  • #1
Ryker
1,086
2

Homework Statement


Suppose g:(0, +∞) → ℝ is continuous, and consider F:ℝd\{0} → ℝd, where F(x) = xg(|x|). Prove F is conservative.

Homework Equations


F is conservative iff there exists a C1 function f:ℝd\{0} → ℝd, s.t. F = grad(f). (edit: Or is the codomain of f actually ℝ, so that it's a scalar-valued function? I namely may have put it down wrong in class.)

F is conservative if, for every closed rectifiable path ∫Fdx = 0.

The Attempt at a Solution


I tried approaching this problem from both angles, i.e. "relevant equations", and came to the conclusion (might be wrong, of course) that if I'm to prove this the second one won't be of particular help. So I went for the first one and tried to construct the function f.

Here's where I run into problems, though. Since g is continuous, we know it has an anti-derivative, call it G. But...
[tex]\frac{\partial G}{\partial x_{i}} = \frac{x_{i} h(|x|)}{|x|} =: H[/tex]
So the numerator is fine, but I can't get rid of |x|. I then also though of saying that H is conservative, and then showing |x|H is also such. But I can't get anywhere with that, either.

Any help here would be greatly appreciated!
 
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  • #2
If g(|x|) is continuous then so is |x|g(x). Call the antiderivative of that G(|x|).
 
  • #3
And then just differentiate G(|x|) to show that F is indeed grad(f), where f is just G? Is that it or am I missing something? Because if that IS it, then damn I'm slow, I can't believe I didn't arrive to that conclusion myself.
 
  • #4
Ryker said:
And then just differentiate G(|x|) to show that F is indeed grad(f), where f is just G? Is that it or am I missing something? Because if that IS it, then damn I'm slow, I can't believe I didn't arrive to that conclusion myself.

No, that's all there is to it. And f is a scalar. Force laws are usually written in terms of a unit vector times magnitude, (x/|x|)*m(|x|). Then you can just integrate m. Your form is missing that extra absolute value so you need to put it back in. x*g(|x|)=(x/|x|)*|x|*g(|x|).
 
  • #5
Thanks a lot!
 

FAQ: Prove a rotationally symmetric central force field is conservative

1. What is a rotationally symmetric central force field?

A rotationally symmetric central force field is a type of force field where the magnitude and direction of the force acting on an object only depend on the distance between the object and a fixed point, and not on the direction in which the object is moving. This type of force field is also symmetric, meaning that it remains unchanged when rotated around the fixed point.

2. How do you prove that a rotationally symmetric central force field is conservative?

To prove that a rotationally symmetric central force field is conservative, we must show that the work done by the force on an object moving along any closed path is zero. This can be done by using the fundamental theorem of calculus to evaluate the line integral of the force along the closed path and showing that it equals zero.

3. What is the significance of a conservative force field?

A conservative force field has the property that the work done by the force on an object is independent of the path taken by the object. This means that the total mechanical energy of the object (kinetic energy + potential energy) remains constant, and the work done by the force can be converted into potential energy. This is important in understanding the behavior of objects in many physical systems, such as celestial bodies in orbit.

4. Can a non-rotationally symmetric central force field be conservative?

Yes, a force field can be conservative even if it is not rotationally symmetric. For example, a force field that follows an inverse square law, such as the gravitational force, is conservative even though it is not rotationally symmetric. However, a rotationally symmetric force field is always conservative.

5. How is a rotationally symmetric central force field different from a spherically symmetric central force field?

A rotationally symmetric central force field is symmetric only when rotated around a fixed point, while a spherically symmetric central force field is symmetric in all directions. This means that a spherically symmetric force field remains unchanged when the entire system is rotated, not just a single point. However, both types of force fields are conservative and have similar properties.

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