sorry - posted by accident - hadn't finished.
For example 1-2-3 Natural turn waltz. you've pushed off the standing leg and you've lowered, so your body has forward kinetic energy. somewhere during beat 3 (let's not digress by being too specific on this 'somewhere') the body has considerably less forward energy, maybe none at all. Plus somewhere along the line we've stopped out rotation. To do this the body has to 'absorb' the original forward energy or convert it into some other form. So we rise, sway and so on. These are all complex movements and impossible to specify 'scientifically'. A beginner would get this wrong by, for example toppling to the side on 3 - having to put a foot out to stop falling - because they haven't developed the skill to 'manage' and control all the forces needed to achieve the dynamic' balance which we all recognise when we execute that perfect natural turn. Or they may come out of a telemark turn with the wrong alignment because they haven't got the 'balance' correct.

"Forces are always balanced. If there is an accelerating force is there always an equal and opposite force produced by inertia."

The problem with that definition of balance is that it's impossible for a "bad" dancer to loose that sort of balance, so it's useless to talk about it...

"You clearly admit that dancers view the concept of balance differently so why do you insist on your definition in a forum for dancers where dance is being discussed. Dancers have very clearly chosen to use this word to convey a particular concept for good reason. When we dancers speak of balance we are describing a feeling."

The problem is that this feeling - which I summarize as never letting gravitiy force you to do something you didn't want to do - can only be judged subjectively. You can't tell if a dancer is balanced by looking at a picture of them, instead you have to know what they "wanted" to do. Though if they then appear alarmed by what happened, you can guess that it wasn't what they wanted.

"In dance you 'feel' balance. This is not a measurement or observation taken by an external observer outside the body itself as in a physics experiment."

Exactly - this sort of balance can't be judged by anyone not participating in it, because there is no objective standard for the balance.

However, within a given style of dance there are objective standards for the resulting movement - and so it is those to which we should turn out attention when we wish to make objective recommendations.

"If you want to discuss balance as a physicist then bugger-off to some forum for physicists and discuss it there."

Fine - but if you want to discuss dancing in a non-visual medium, shouldn't you confine your comments to things that can be literally expressed in words? Describing the quality of movement would be a good. Much better than making comments that are emotional valid but literally false, and attempting (as many others than you have) to claim that they are literally true.