Thanks. That's exactly the sort of explanation I was hoping for.

Being below maximum height certainly seems to correlate with steps that lead to horizontal motion, and is a possibly a prerequisite for that motion. But I am more inclined to think that the reason for this is the fact that the legs have to be off vertical in order for the leg muscles in question to generate any force in the horizontal plane.

If you look at both Luca and Marcus who are dancing to different tempos. In both but more by Luca after every second beat except in the middle of a Weave you will see the push up onto the next step. Both all the way through their Foxtrot. Which is backed by something Billy Irvine said whilst in Singapore. This is a direct quote from his lecture. A couple must display varying shades in the interpretation of the music. He gave two examples where the expression of the music and the accentuation of the movement must be shown very clearly. In the Waltz he stressed on the First accentuated beat. In the Foxtrot the accentuation of the THIRD beat. Printed by courtesy of Danceworld in Bras Bash Complex Singapore.]
Accentuate the third beat is this what you can see from both Marcus and Luca in their Foxtrot.. A push upwards

BILL Irvine

Sir Bill Irvine

"But the laws of physics say that a force acting in a particular direction can have no effect perpendicular to it."

Indeed, but in what directions are the forces involved? Remember, every force has it's equal and opposite reaction - either singular, or two (or more) vectors which together add to to equal the third.

Did you ever draw those triangular sum of forces diagrams in physics class? For a skier on a hill or a pendulum on a string?

If your body weight is not directly over your center of foot pressure, then your force diagram will be a triangle: gravity is down, the normal force supporting you through your leg follows the angle of the leg from the vertical, and the remaining vector of the force sum triangle is a forward or backward horizontal acceleration.

Re the triangle of vectors. I think that is more or less what I was saying.

I think I can accept the general idea of a trade off between the capacity for elevation and horizontal movement (linked to how off vertical you are) but its just an issue of the language for me and the idea of energy going from one form to another (which I would say is not correct).

Not to say there aren't other answers to the original question as well.

"and the idea of energy going from one form to another (which I would say is not correct)."

Well what do you think occurs when you use rise to slow down, such as on the upswing in waltz?

Sure, you could use muscles to stop yourself, and then use muscles to lift yourself. But that wouldn't be a very efficient, or a very elegant way to do it.

Gravity cannot slow down horizontal movement as it acts in the wrong direction. That is fact. My guess as to why you slow down when you rise is that the muscles involved become less well poised to generate force in the horizontal direction. This would happen quite naturally as the leg muscles at maximum rise would be aligned close to the vertical.

"Gravity cannot slow down horizontal movement as it acts in the wrong direction. That is fact."

Wrong.

Gravity can slow down horizontal movement in a situation in which geometry provides coupling between the horizontal and vertical. Specifically, any situation in which the normal force which is the floor's reaction to you pushing down on it, is not vertical but instead at an angle from the vertical. That's where those triangular force diagrams come in - gravity is vertical, but the normal force that opposed it is not, and the difference between them is a horizontal acceleration.

Consider that if you were right, pole vaulting would be impossible. As would swinging on playground swing. Or rolling down a hill. And so (though less obviously) would walking.

If you were to take a step towards and onto a stiff leg, that would be like pole vaulting - you'd convert horizontal movement energy to vertical gravitational potential energy.

The instant-by-instant situation in waltz is a bit more complicated, in that the transition from low and fast to high and slow follows a path that involves bending the knees and then re-straightening them, but the overall effect is much the same.

I am not sure the analogies are appropriate.

There is no elasticity in the leg cf a pole-vaulters pole.

The pendulum is a special case where gravity 'appears' to affect horizontal motion, but actually doesn't - the movement of the pendulum is not really horizontal - its circular - and it works through the vertical component.

Also when the pendulum is at maximum height the (pseudo)horizontal force is at it's maximum, the opposite situation to dancing.

To me the situation here (at the the top of swing in dancing) doesnt really have a geometry that couples the horizontal and the vertical.