# Re: Laplace Transform vs Fourier transform

robert bristow-johnson wrote:
...

how about net force. pick any single dimension (so the vectors are
scalers) and for a body with two equal and opposite forces applied, it
does not accelerate and the net force is zero. picking which
direction is a convention, but some positive force got something added
to it, and the result is zero. what would you call that other force?
non-real?

That's an interesting one. Consider the classic "tug-of-war" exercise, with two equally matched teams. Sure, assuming they pull equally, the net motion of the rope is zero, but it is hardly zero "force" - there is massive tension in the rope, and in the limit it will break, with a corresponding quasi-explosive release of energy. A positive number and negative number can "collide" and result in a zero, but such things generally do not happen in the physical world, as other laws come into play, not least those dealing with conservation.

There is some game being played here with the difference between a quantity and a direction. Wrap that rope round a tree, so that the two halves are now virtually coincident, and both (identical) forces become "positive", add constructively, and the tree gets pulled down (or not).

Is a negative force one pulling "in the opposite direction", or one that is pushing?

...
might be a convention and the aliens on the planet Zog might have been
smart enough to adopt a convention that those muck less massive little
things "flying about" the nucleus are the positive ones, making what
we would call protons negatively charged. at least they would have to
learn that the electron flow is in the opposite direction of the
current they solved for with Kirchoff's (or Zog's) laws. but either
way, there are physical quantities that can only be represented as
quantitatively negative.

The real world offers plenty of examples of opposites, for which we need a vocabulary. I have had to consider this very recently, in designing and documenting programs to do Ambisonic panning. By arithmetic convention, the "positive" direction of rotation is anti-clockwise (using sine and cosine functions, needless to say); but this is counter-intuitive for the typical non-mathematical user for whom the natural analogy is the motion of clock hands. So in the end I decided to invert things so that a positive argument means clockwise rotation. It's a tough call - I don't want to mis-educate users, but I want the program to be as easy and intuitive to use as possible, too.

Even the number line has (by convention) a direction - positive to the right. A negative number is simply a "leftwards" one. So on the complex plane starting at X=1, the angle ~increases~ such that the point becomes increasingly ~negative~, towards x = -1. The names are a convention - we have to call these quasi-opposite (or complementrary) states ~something~!

An interesting question would be, how would those aliens see "the world" if their equivalent to Newton had happened upon quantum mechanics first, so that they would regard any notion of a continuous quantity as an illusion, or at least a counter-intuitive mystery?

Richard Dobson
.

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