Energy can just as well be E = mgh (potential energy) or rotational energy E = 1/2Iw^2, or elastic energy E=1/2kx^2, chemical energy, nuclear energy etc. with momentum nowhere to be found in those formulas. When we talk about energy conservation we mean the conservation across all the forms of energy that the system can take on - that is what gets conserved.
It just happens that in one particular manifestation of the energy, the kinetic energy, can also be expressed with a squared momentum in the formula - but does not mean that momentum "is" energy by any interpretation.
As I said before momentum and energy are completely different concepts altogether - energy can be stored and transformed. Momentum cannot be stored nor can you transform a linear momentum into another kind of momentum.
If not convinced, consider for a moment (pun intended) that momentum is a vector and it conserves (in each dimension) as a vector! - whereas energy is a scalar and conserves as a scalar.
> does not mean that momentum "is" energy by any interpretation
I never said that. You literally said "momentum has nothing to do with energy", and I gave you one example where they are directly related.
> can you transform a linear momentum into another kind of momentum
Yes, you can. This is exactly why I mentioned yo-yo.
> momentum is a vector and ... energy is a scalar and conserves as a scalar.
That's a good point, it's 100% correct and I'm not arguing with that. But I insist that saying that they are unrelated is still wrong.
Let's go back for a second to where we started. My claim was (and still is) that the only source of aerodynamic lift is the kinetic energy of the air molecules acting on the airfoil. There is just nothing else, after all. This is not conceptually different from how [solar] sail works. Now, in this specific case, the energy of particles acting on the air/solar foil is directly related to their momenta.
Not sure what could be unclear there at all. Nuclear energy can be turned into any other energy and vice versa. As long as something has mass it has energy - whether or not we can readily transform that is beside the point.
Your recurring yo-yo example only demonstrates that you don't understand the physical phenomena in the first place. The linear momentum is conserved when the yo-yo pulls on your hand and through that your body, you are either pushing or pulling on Earth via gravitational force, the Earth wobbles opposite of the yo-yo (albeit infinitesimally, thankfully). That's the conservation of the momentum.
The rotational energy of the yo-yo has nothing to do with the linear momentum, that rotation comes from the chemical energy of your muscles that have first lifted, pulled or tossed the yo-yo. With that, you have transformed chemical energy stored in your muscles into the rotational energy of the yo-yo. Momentum has nothing to do with energy. Just because both exist and both get conserved. It is quite profound actually that they are not related at all.
As for this discussion we were talking about you conflating momentum with energy, a common misconception actually, your reticence of even remotely entertaining the idea that you did indeed misuse these concepts diverted into a lengthy discussion that slowly drifted away from the actual points to flawed analogies and yo-yos - also not surprising and a common predicament
Why am I still replying? Because it demonstrates why it is so hard to discuss flying (the very point of the original post) the majority of participants conflate and misuse scientific concepts - then go onto lengthy roundabouts to avoid owning up to these mistakes.
It feels that you've got some idea that I'm not understanding the difference between energy and momentum because it's a common misconception, and hold on it. I do understand that they are different. My objection is your insistence on them being "unrelated".
As of the yo-yo, let's remove the muscle power and the wobbling earth out of the picture and consider a it a closed system.
We've got a fully wound-up yo-yo, not rotating. It has a certain amount of potential energy. When you release it, its potential energy starts transforming into kinetic energy of linear motion and of rotation. This kinetic energy can be measured at any moment via observing the linear and rotational momenta of the yo-yo, which are the functions of its mass and torque, and the both velocities. Speaking of which, there is no other way of measuring the energy of this system. As it reaches the end of the line, and starts winding up again, its potential energy is zero, its kinetic energy is at its maximum, and its linear momentum changes the direction to upwards.
I'm telling you this to demonstrate that I understand the difference and your main objection is not exactly applicable here.
And, of course, momentum and kinetic and potential energy are intimately related in such a system. I don't understand how one could deny that.
> Momentum has nothing to do with energy
That's what I meant. Kinetic energy is a function of momentum and you are insisting it is not!
Perhaps we are being confused by each other's different ways of using the word Energy. When I use it (in the mechanical context), I mean strictly kinetic energy or potential energy, but nothing else. I've been taught to use it that way and was quite harshly slapped on the wrist (verbally) for failing to stick to it (that is, for magical thinking).
You seem to be using it in a broader sense (e.g. "nuclear energy". I don't know what nuclear energy is -- it s what multiplied by what, specifically?).
Correction: it will go up given an ignorably small jerk on the line at the lowest point. Not the best example, right... What I was trying to say is that one can trade rotational momentum to linear (and vice versa) as long as the kinetic energy of the system stays the same.
momentum has nothing to do with energy and vice versa.
For example, energy can be stored (transformed into different forms, short or long term) and released later, momentum cannot.