Potential at a point on Earth due to orbital motion and the Moon

Question

In "Physics of the Earth" by Frank Stacey and Paul Davis, they approximate the height of the tides on page 103.

They assume the Earth is perfectly spherical, covered in a uniform layer of water and that the Earth spins as fast as the Moon orbits around it, so as to present a constant face to the Moon. They use the below diagram to help with the upcoming derivation:

Now, they claim that the potential $W$ (their choice of variable, not mine) at the point $P$ is the sum of the potential due to the Moon's gravitational force at point $P$ and the Earth-Moon system's orbital motion about its barycentre. They state the following expression for $W$: $$W = -\frac{Gm}{R'} \color{red}{-} \frac{1}{2}\omega_L^2 r^2.\tag{1}$$

Could someone explain to me why the potential due to the orbital motion is negative? I did the following working to find it myself:

A point-like test mass at $P$ with mass $M$ would have total mechanical energy $$E = E_K + E_{rot} + E_{moon} = E_K + \frac{1}{2}I\omega_L^2 - \frac{GmM}{R'}.\tag{2}$$

Thus, $$E_K = E - \left(\frac{1}{2}I\omega_L^2 - \frac{GmM}{R'}\right).\tag{3}$$

I can call the right-hand expression the 'effective potential energy' of the system, which means that potential $W$ is $$W = \frac{U_{eff}}{M} = -\frac{Gm}{R'} \color{red}{+} \frac{1}{2}\omega_L^2 r^2.\tag{4}$$

As you can see, my 'rotational potential' is positive, as opposed to their negative value. Could someone please explain what I did wrong (or if their expression is a typo)?

Answer 1

The centrifugal force is known to be $$\vec{F}_\text{centrifugal} = M\ \omega_L^2 r\ \vec{e}_r$$ where $r$ is the distance from the axis of rotation and $\vec{e}_r$ is the unit vector pointing away from the axis.

A force (both gravitational, and centrifugal as well) is defined by the negative gradient of a potential. $$\vec{F}=-M\ \vec{\nabla}\ W$$

So with the centrifugal force from above we have $$\vec{F}_\text{centrifugal}=-M\ \vec{\nabla}\ W_\text{centrifugal}$$ or $$M\ \omega_L^2 r\ \vec{e}_r=-M\ \vec{\nabla}\ W_\text{centrifugal}$$

The solution of this equation is $$W_\text{centrifugal} = - \frac{1}{2}\omega_L^2 r^2$$ because $\vec{\nabla}(\frac{1}{2}r^2)=r\ \vec{e}_r$.

Answer 2

lets look at the reduce "two body" equations at the center of mass coordinate

I ) non rotate CM coordinate system

with

$$T=\frac{\mu}{2}\,\dot r^2\\ U=-\frac{G\,m\,M}{r}$$ and the Lagrange $~L=T-U~$ you obtain the EOM

$$ \mu\,\ddot r=\underbrace{-\frac{G\,m\,M}{r^2}}_{F_G}\quad\Rightarrow\\ U=-\int F_G\,dr$$

II ) rotate CM coordinate system

if the CM coordinate system rotate you obtain additional fictitious forces . the EOM is now

$$ \mu\,\ddot r=\underbrace{~-\frac{G\,m\,M}{r^2}~}_{F_G}\underbrace{~+\mu\omega^2\,r~}_{F_Z}\,\quad\Rightarrow\\ U=-\int (F_G+F_Z)\,dr=-\frac{G\,m\,M}{r}-\frac{\mu}{2}\,\omega^2\,r^2$$

so your potential energy is not correct !!

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the EOM with EL

$$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot r}\right)=\left(\frac{\partial L}{\partial r}\right)$$

Edit:

$$\vec F_Z=-m\,(\vec\omega\times\,(\vec\omega\times \vec R))$$

with

$$\vec\omega=\begin{bmatrix} 0 \\ 0 \\ \Omega \\ \end{bmatrix}\quad, \vec R=\begin{bmatrix} r \\ 0 \\ 0 \\ \end{bmatrix}\\ \vec F_Z= \begin{bmatrix} m\,\Omega^2\,r \\ 0 \\ 0 \\ \end{bmatrix}$$

Answer 3

I haven't done Lagrangians or Hamiltonians yet, so I didn't actually understand much of the mathematics of the posts, but these two questions answered mine as well:

Lagrangian of an effective potential

Effective potential of a two-body system