Work done by the gravitational force

[eoghan]

Hi there!
I'd like to calculate the work done by the gravitational force. I know the work is defined by the integration of a 1-form:
$$L=\int_\gamma \omega$$
where
$$\omega=F_xdx+F_ydy+F_zdz$$

This works fine in cartesian coordinates and I know how to integrate it, but what if I want to use spherical coordinates?
Then I'd have:
$$\omega=F_rdr+F_{\theta}d{\theta}+F_{\phi}{d\phi}=F_rdr$$
Suppose $$\gamma$$ is a curve defined in spherical coordinates (i.e. $$\vec\gamma=R(t)\hat r+\Theta(t)\hat\theta+\Phi(t)\hat\phi$$),
how do I integrate the 1-form along $$\gamma$$?

 

[jtbell]

$$\omega=F_rdr+F_{\theta}d{\theta}+F_{\phi}{d\phi}=F_rdr$$

No, you need to use the line element in spherical coordinates:

$$d \vec l = dr \hat r + r d\theta \hat \theta + r \sin \theta d\phi \hat \phi$$

so that

$$\omega = F_r dr + F_\theta r d\theta + F_\phi r \sin \theta d\phi$$

Now, what are $F_r$, $F_\theta$, and $F_\phi$?

 

[eoghan]

No, you need to use the line element in spherical coordinates:

$$d \vec l = dr \hat r + r d\theta \hat \theta + r \sin \theta d\phi \hat \phi$$

so that

$$\omega = F_r dr + F_\theta r d\theta + F_\phi r \sin \theta d\phi$$

Now, what are $F_r$, $F_\theta$, and $F_\phi$?

Then the integral is like this?

$$\int_\gamma \omega = \int_{t_0}^{t_1} \vec F \cdot\frac{d\vec l}{dt}dt=\int_{t_0}^{t_1} \left( F_r\frac{dr}{dt}+F_\theta r \frac{d\theta}{dt}+F_\phi rsin\theta\frac{d\phi}{dt}\right)dt$$

 

[MaxwellsDemon]

Wouldn't the work done when moving between two points in a gravitational field just be the difference between the potential energies at those two points? You'd really only need to worry about the up direction...or r in spherical polar coordinates...if the coordinate origin is the Earth's center.

 

[PJC01]

Hello! Great question. To calculate the work done by the gravitational force in spherical coordinates, we can use the same formula for work (L = ∫ω) but with the appropriate components of the force (F_r, F_θ, and F_φ) and differentials (dr, dθ, and dφ). So the integral would look like this:
L = ∫F_rdr + ∫F_θdθ + ∫F_φdφ
To evaluate each of these integrals, we would need to use the appropriate limits of integration for each coordinate (r, θ, and φ) along the curve γ. This may require some manipulation and substitution to get the limits in terms of t, the parameter along the curve. Once we have the limits, we can integrate each component separately and add them together to get the total work done by the gravitational force. I hope this helps! Let me know if you have any further questions.