I Is there a negative sign in front of the centrifugal potential energy?

AI Thread Summary
The discussion revolves around the effective potential in orbital mechanics, specifically the confusion between the positive and negative expressions for centrifugal potential energy. The effective potential is derived from angular momentum conservation, leading to the formula U_eff(r) = L^2/(2mr^2) - GmM/r, which indicates that potential energy decreases with increasing radius r. There is debate over whether centrifugal potential energy should be classified as potential or kinetic energy, with some arguing that it behaves like kinetic energy since it depends on velocity, which increases with r. The conversation highlights the importance of context, as different systems (circular vs. elliptical orbits) can lead to different interpretations of potential energy. Ultimately, the effective potential assumes constant angular momentum, but the introduction of centrifugal potential can vary based on the frame of reference.
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I am not sure if there is a negative sign in front of the formula for the centrifugal potential energy.
In orbital mechanics, the effective potential is given by ##\frac {1} {2} m r^2 w^2##, which can be expressed in terms of angular momentum ##L## which is conserved.

Yet, https://web.njit.edu/~gary/321/Lecture17.html apparently shows the centrifugal potential as the negative of the above. Deriving the paraboloid shape of some spinning water's surface also involves ## - \frac {1} {2} m r^2 w^2##, instead of ##\frac {1} {2} m r^2 w^2##.

So which is it? What is the difference?
 
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Does potential energy increase or decrease as ##r## increases?
 
PeroK said:
Does potential energy increase or decrease as ##r## increases?
I am not sure if centrifugal potential energy is really a potential energy. To me, it is a kinetic energy that just behaves like a potential energy (since v is only dependent on r). In that sense, I would expect 'potential energy' to increase as r increases, given that v increases. As for the gravitational potential energy, it increases with r too. So potential energy should increase?

Then how would this agree with the ## - \frac {1} {2} m r^2 w^2## formula, which shows it decreasing?
 
phantomvommand said:
I am not sure if centrifugal potential energy is really a potential energy. To me, it is a kinetic energy that just behaves like a potential energy (since v is only dependent on r). In that sense, I would expect 'potential energy' to increase as r increases, given that v increases. As for the gravitational potential energy, it increases with r too. So potential energy should increase?

Then how would this agree with the ## - \frac {1} {2} m r^2 w^2## formula, which shows it decreasing?
How could PE be KE? What normally happens ti PE as KE increases?
 
PeroK said:
How could PE be KE? What normally happens ti PE as KE increases?
In the orbital 'Effective potential energy' case, the tangential KE is treated as a potential energy? Hence PE can be KE?
from conservation of energy point of view, PE should decrease as KE increases.
 
phantomvommand said:
In the orbital 'Effective potential energy' case, the tangential KE is treated as a potential energy? Hence PE can be KE?
I've no idea what you mean by that.
phantomvommand said:
from conservation of energy point of view, PE should decrease as KE increases.
Precisely!
 
I do not understand the Lecture notes. The effective centrifugal potential assumes the angular momentum to be constant. Thus $$U_{eff}(r)=\frac {L^2}{2mr ^2}-\frac {GmM} r $$ and that takes care of the signs for the derivative...
 
hutchphd said:
I do not understand the Lecture notes. The effective centrifugal potential assumes the angular momentum to be constant. Thus $$U_{eff}(r)=\frac {L^2}{2mr ^2}-\frac {GmM} r $$ and that takes care of the signs for the derivative...
I think the lecture notes is referring to a circular orbit.

I guess I am asking about the derivation of the centrifugal potential energies. In the case of elliptical orbits, the effective potential energy (ignore gravitational potential energy) given by ##\frac {L^2} {2mr^2}## can be obtained from ##\int mrw^2 \, dr##, while in the case of water spinning/circular orbit, we use ##- \int mrw^2 \, dr##, essentially ##- \int F \, dr##.

@PeroK thanks for the replies. But in both forms ##- \int mrw^2 \, dr = - \frac {1} {2} mr^2w^2## and ##\frac {L^2} {2mr^2}##, the potential energy decreases with distance. I meant that ##\frac {L^2} {2mr^2}## is equal to ##\frac {1} {2} mv^2##, hence my 'KE is a PE' comment.

What is the difference between these 2 systems, that resulted in 2 different methods of potential energy derivation?

For reference to the lecture notes in question: https://web.njit.edu/~gary/321/Lecture17.html

Please do point out any conceptual errors I have.
 
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hutchphd said:
The effective centrifugal potential assumes the angular momentum to be constant.
This isn't correct. In a rotating frame (with angular velocity ##\boldsymbol{\Omega}## relative to an inertial frame) one may introduce a centrifugal potential energy term regardless of the dynamics of the system, i.e. whether its angular momentum is constant or not.

For simplicity, consider a unit mass moving in a potential ##U(\mathbf{r})## but otherwise arbitrarily (that is, arbitrary initial conditions). Recall the equation of motion of a particle in a uniformly rotating frame of reference ##K## in which ##\boldsymbol{v} = \dfrac{d\mathbf{r}}{dt} \bigg{|}_K## is\begin{align*}
\dfrac{d\boldsymbol{v}}{dt} &= -\dfrac{\partial U}{\partial \mathbf{r}} + 2\boldsymbol{v} \times \boldsymbol{\Omega} + \boldsymbol{\Omega} \times (\mathbf{r} \times \boldsymbol{\Omega}) \\

\boldsymbol{v} \cdot \dfrac{d\boldsymbol{v}}{dt} &= - \boldsymbol{v} \cdot \dfrac{\partial U}{\partial \mathbf{r}} + 2 \underbrace{\boldsymbol{v} \cdot \boldsymbol{v} \times \boldsymbol{\Omega}}_{=0} + \boldsymbol{v} \cdot \boldsymbol{\Omega} \times (\mathbf{r} \times \boldsymbol{\Omega}) \\

\dfrac{d}{dt} \left( \frac{1}{2}v^2 \right) &= - \frac{d}{dt} \left( U - \frac{1}{2} (\boldsymbol{\Omega} \times \mathbf{r})^2 \right)
\end{align*}therefore ##E \equiv \frac{1}{2}v^2 + U - \dfrac{1}{2} (\boldsymbol{\Omega} \times \mathbf{r})^2## is a constant and the term ##- \dfrac{1}{2} (\boldsymbol{\Omega} \times \mathbf{r})^2## is identified as a rotational contribution.
 
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hutchphd said:
I do not understand the Lecture notes. The effective centrifugal potential assumes the angular momentum to be constant. Thus $$U_{eff}(r)=\frac {L^2}{2mr ^2}-\frac {GmM} r $$ and that takes care of the signs for the derivative...
ergospherical said:
This isn't correct. In a rotating frame (with angular velocity ##\boldsymbol{\Omega}## relative to an inertial frame) one may introduce a centrifugal potential energy term regardless of the dynamics of the system, i.e. whether its angular momentum is constant or not.
You are talking about different potentials.

@hutchphd means this one with constant ##L## & ##E##:
https://en.wikipedia.org/wiki/Effective_potential#Gravitational_potential

While @ergospherical assumes constant ##\boldsymbol{\Omega}##
 
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Yes the link posted by @phantomvommand analyses the motion of a test mass ##m## in the background potential ##U(\mathbf{r})## using a co-rotating reference system rigidly attached to the binary star system (with [approximately] constant angular velocity ##\mathbf{w}##).
 
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