Is lunar elevation at a given location for a given day unimodal? Is lunar elevation at a given location for a given day unimodal: 

Unimodal function

In other words, once the moon's elevation reaches a minimum (which may 
be above or below the horizon depending on if the moon is 
circumpolar), does it increase continuously to its maximum, or can it 
decrease briefly and then increase again (resulting in a local minimum 
that's not a global minimum) before reaching the maximum. 

And also similar to the above as it goes from the maximum elevation 
(which may also be above or below the horizon depending on whether the 
moon rises at all on a given day). 

Why this is important: I think I found a bug in libnova that assumes 
the moon is circumpolar if it is above the horizon when it's due 
north. This is untrue

My fix
does not assume this, but does assume that lunar elevation is 
unimodal. 

If that's untrue, I'll need to further fix my fix.

EDIT: Here's how far I've gotten: 


The moon's declination and hour angle (at a given location) are 
both functions of time. Thus, we can implicitly consider the moon's 
declination to be a function of its hour angle. This only works 
because the moon never experiences retrograde motion. Bodies that 
experience retrograde motion could have two declinations for a given 
hour angle, so the declination would no longer be a function of the 
hour angle. 
Using the standard formula, we see the moon's elevation is: 


$$ 
   \sin ^{-1}(\sin (\text{lat}) \sin (\text{dec}(\text{ha}))+\cos (\text{ha}) 
    \cos (\text{lat}) \cos (\text{dec}(\text{ha}))) 
$$ 

where dec(ha) is the declination at hour angle, ha is the hour angle, 
and lat is the latitude. 


Since the moon's elevation is a differentiable function (except 
potentially at the poles and equator), we find the derivative with 
respect to the hour angle: 


$$
   \frac{\text{dec}'(\text{ha}) (\sin (\text{lat}) \cos 
    (\text{dec}(\text{ha}))-\cos (\text{ha}) \cos (\text{lat}) \sin 
    (\text{dec}(\text{ha})))-\sin (\text{ha}) \cos (\text{lat}) \cos 
    (\text{dec}(\text{ha}))}{\sqrt{1-(\sin (\text{lat}) \sin 
    (\text{dec}(\text{ha}))+\cos (\text{ha}) \cos (\text{lat}) \cos 
    (\text{dec}(\text{ha})))^2}} 
$$


The moon's elevation will reach a min/max (in the sense of 
unimodality) when this derivative is 0. That occurs when: 


$ 
   \sin (\text{ha})\text{ = } \text{dec}'(\text{ha}) (\tan (\text{lat})-\cos 
    (\text{ha}) \tan (\text{dec}(\text{ha}))) 
$ 

So the question becomes: can this equation have more than two 
solutions for 0 &lt; ha &lt; 2*Pi 

Which I hope someone answers at: https://math.stackexchange.com/questions/587136/
Yes, it is unimodal.

Earth's movement is a simple rotation, thus if the Moon were fixed on sky it will describe a circumference like the fixed stars.

From that basic movement we need to introduce Moon's own movement on the fixed stars sphere, which makes Moon not to complete the circumference in 1 day. Instead, Moon performs an arc similar to that of the fixed stars but shorter.

This circumference-like movement does not wobble, so it has just one maximun and one minimum in elevation.