Conservative force in spherical coordinates

In summary, the question is whether or not the force vector F, with components F_r, F_θ, and F_φ, is a conservative force. The equations given for F_r, F_θ, and F_φ are written in terms of spherical coordinates, but the force can also be described in terms of Cartesian coordinates. To find the curl of this force, the chain rule of differentiation must be used.
  • #1
Felipe Lincoln
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Homework Statement


Is ##F=(F_r, F_\theta, F_\varphi)## a conservative force?
##F_r=ar\sin\theta\sin\varphi##
##F_\theta=ar\cos\theta\sin\varphi##
##F_\varphi=ar\cos\varphi##

Homework Equations


##\nabla\times F=0##

The Attempt at a Solution


In this case we have to use the curl for spherical coordinates, but since it's clear that every component of this force can be described by cartesian coordinates I'm wondering if there's a way of doing the curl of cartesian coordinates, can we do this conversion ?
 
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  • #2
What you have listed is not the spherical components of the vector, but the representation of the cartesian components using the spherical angles. For example ##F_x=ar \sin\theta \cos \phi##, etc. So to find the curl, just take the appropriate derivatives and be sure to apply the chain rule of differentiation.

On edit: It looks like you have swapped the symbols and used ##\varphi## for the polar angle and ##\theta## for the azimuthal angle.
 
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  • #3
Right!
But why should I use the chain rule? I can just do some substitution like ##x=r\sin\varphi\cos\theta##
And yes, I swapped the symbols, I'm used to do this way.
Thank you for the answer
 
  • #4
Wait, I din't understand why you can say that ##F_x=ar\sin\varphi\cos\theta## (please use the symbols I introduced, it's my book convension). What the problem gave me ins't the radial componente of the force, in the case of ##F_r##?
 
  • #5
What I'm saying is that the same vector ##\vec F## is written in spherical coordinates as
##\vec F = F_r \hat r~+~F_{\varphi}\hat {\varphi}~+~F_{\theta}\hat {\theta}##
and in Cartesian coordinates as
##\vec F = F_x ~\hat x~+~F_{y}~\hat {y}~+~F_{z}~\hat {z}##
the last expression can be written in terms of the spherical angles as
##\vec F = F \sin\varphi \cos \theta~\hat x~+~F \sin\varphi \sin\theta~\hat {y}~+~F \cos\varphi~\hat {z}##
where ##F=\sqrt{F_x^2+F_y^2+F_z^2}=ar##.
I assumed you wanted to calculate the curl in the Cartesian representation, in which case you would have to find terms like ##\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}## which is where the chain rule comes in if you use the expressions of ##F_x## and ##F_y## that involve the spherical angles.

Note: Your book's convention is unconventional. Bear that in might if you use other books in the future.
 
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  • #6
I got it.
Thank you very much !
 

1. What is a conservative force in spherical coordinates?

A conservative force in spherical coordinates is a type of force that depends only on the position of an object and not on its path. This means that the work done by the force on an object moving from one point to another is independent of the path taken, and only depends on the starting and ending points.

2. How is a conservative force different from a non-conservative force?

A non-conservative force, such as friction or air resistance, depends on the path taken by an object and not just on its starting and ending points. This means that the work done by a non-conservative force on an object moving from one point to another can vary depending on the path taken.

3. What are some examples of conservative forces in spherical coordinates?

Examples of conservative forces in spherical coordinates include gravitational force, electric force, and elastic force. These forces only depend on the position of an object and not on the path taken, making them conservative.

4. How is the work done by a conservative force calculated in spherical coordinates?

The work done by a conservative force in spherical coordinates can be calculated using the line integral of the force over the path taken by the object. This is represented by the equation W = ∫F·ds, where F is the force and ds is the infinitesimal displacement along the path.

5. What is the significance of conservative forces in physics?

Conservative forces play a crucial role in many physical phenomena and are important in the study of energy and work. They help us understand and predict the motion of objects in various systems, such as celestial bodies in space or particles in a chemical reaction.

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