With 2 different anonymii it's hard to respond. however, i have to agree with the one which mentioned that a plane turning is different from a car turning. The plane banks to the right so that there is a component of lift from the wings is to the right.
A car turns to the right through the friction on the tyres, this friction supplies a force to the right. (bloody physics again!!)
Sway...here's my version
Kinetic energy of body travel is converted into potential energy (hence rise). But there is more...We stop the motion by planting a foot (as in natural turn waltz - left foot plant). But initially there is what'#s called a turning 'moment' because the energy of the movement acts from the centre of gravity of the body (which, in general, is higher than the foot). It is this turning 'moment' which must be cancelled with sway. In a human body, it's very complicated in detail because it's not a rigid object, and the muscle actions absorb energy. One thing is clear - more body flight will give both more rise, and will need more sway because of the increased turning moment.
If you just plant the foot and don't sway, things are fine for small turning moments (i.e. less body flight), but as a dancer improves, he would simply topple over without sway.
Sorry about the physics (i love it!

Phil, you are getting closer to figuring it out now. However, keep in mind that the sway precedes the turn by quite a bit, and will still occur even if there is not going to be turn. Sway was not used in the very small book actions for non-turning figures, but today it is a part of almost everything.

Except for tango of course. Tango, with it's rapid turns and no sway. Some folks (not you) have some explaining to match that with their theories!

What is the difference in swing and sway? I'm confuse. I checked the American Heritage Dictionary of the English Language and its definition seem very alike. So, let me give it a shot at swing. Let me know if I'm wrong. Firstly, a swinging motion has the following components: 1. a central/focal point that the moving object cycles around, 2(a). the swinging motion is circular or curvaliner; 2(b). the path of inertia is tangential to the circular movement of said object; 2(c). the object starts to accelerate from a starting position and achieves its maximum speed midway through its travel distance and then starts to decelerate until it stops to reverse its direction, 3. a centrifugal force pulling the moving object away from the central/focal point, 4. a centripetal force pulling said object inward towards the central/focal point, 5. the centrifugal force will alway equal the centripetal force, otherwise the motion will change from a swing into a tragetory, 6. in a vertical plane, the centrifugal force is gravity on the bottom half of the plane. So having said that, a swing in waltz is on a vertical plane, right? and there is an accelerationa and deceleration of speed of motion, right? Then what the heck is sway?

Also, is the term sway used to describe forces on the horizontal plane?

"I checked the American Heritage Dictionary 3. a centrifugal force pulling the moving object away from the central/focal point, 4. a centripetal force pulling said object inward towards the central/focal point, 5. the centrifugal force will alway equal the centripetal force, otherwise the motion will change from a swing into a tragetory,"

The American Heritage dictionary is wrong. There is no centrifugal force, only a force acting inwards (centripital) to bend the trajectory. The object's inertia provides the outwards "force" only in reaction to a real inwards force trying to pull it inwards.

"Then what the heck is sway?"

Deviation of the body axis from the vertical. Many people mistakenly think that sway means the body axis points towards the center point of the swing, but that is not really true. In some situations it might point in that direction, but in others it will not - we never sway into the movement on a downswing for example, despite the center point of a downswing being ahead of us.

Body Sway in ballroom dancing are used chiefly for effect. Sway should be made by inclining the body to the left or right,
Sway can be used on nearly all turns. Exceptions are, all Spins, where the turn is too quick to permit Sway. Sway to be used with comfort. Sway is also used on figures that curve or wave and in some figures such as a Cross Chasse. The main principle of sway is to incline the body towards the centre of the turn. Thus the practicle value of Sway would be found in its assisance to prevent the dancer from overbalancing or overturning . The greatest value of sway is purly decorative.
To put something right. The Sway on the first three of the Natural in the Waltz and thereafter on other steps is . Sway to the Right on 2 and 3. on the Natural Turn. This combines with Swing.

"Body Sway in ballroom dancing are used chiefly for effect."

Nope, though if you haven't yet learned to use it to increase your overall movement, you might be forgiven for thinking so.

"The main principle of sway is to incline the body towards the centre of the turn."

Not really. The true direction of incline is much closer to "against the movement". Or, at the start of some actions, very briefly into it.
There may be a bias towards the inside of the turn (especially in the later case) but that is not the primary direction of the sway.

"The greatest value of sway is purly decorative."

Again, if you don't understand what it's really for, you can be forgiven for thinking so. That goes for numerous othe aspects of dancing, of the hold, etc - people who don't understand their function think they are just for show. People who do understand their function know otherwise.

"The main principle of sway is to incline the body towards the centre of the turn."

Already explained why this is not so, but here is one you will be quite likely to get wrong.

Steps 1-6 of a natural turn, commenced DW. Where is the center of this turn?

Bet you were going to say, DW against LOD.

But actually, the center of this turn is to the center.

Remember, natural turns rotate right to turn LEFT.

Which dance. Waltz, Foxtrot or Quickstep. If its Waltz we have two turns over six beats and six steps. Centre is in the middle of something that is an equal distance between two points. I think Alex Moore worked out that the centre of the turn was at the end of step three with another at the end of step six. Some might argue that the centre of the first turn is the end of one and the other the end of four. Which makes the first backing diagnal to centre and the other backing diagnal to wall. But do you know what. It doesn't really matter.

Anonymous. Where do you get your weird ideas from. You might like to know I copied my contibution exactly as it is written in the technique book. If you want to disagree with the Technique Book be my guest.