Gravitation and the conics

[whitefork]

Way back, maybe 1962 or so, I had a subscription to something called the "Science Series", paperback magazines with big colorful stamp-like thing to paste in. They covered all sorts of interesting areas of science, and didn't condescend to the 10-year old audience much at all.

I remember the Astronomy one quite well - fanciful renderings of the planets, asteroids, comets, etc, but what really struck me at the time was an illustration of a sectioned cone relating the orbits to the conic sections. Where the heck did THAT come from I ask my father? I don't think he had a good answer.

anyway:

1900 years before Newton Apollonius of Perga derived a large number of properties of the conic sections (circle, parabola, ellipse, hyperbola). That's a lot of the mathematics required for Newtonian physics, and all of it mathematically necessary.

It's possible that Apollonius drew on observations made while cutting up cones, but who's to say - in any event the mathematics as presented does not depend on physical measurement. It's not contingent on the material of the which the cone is made or the tools used to section it.

Now, the observation that bodies accelerate while falling is a contingent one - there's no reason to expect a priori that they would have this property. The fact that the path taken by an accelerated body is described by the same equations that describe a conic section is quite remarkable to me.

So, what if anything should we conclude from this coincidence, and is it a coincidence at all, or an indication of an underlying property of space, time, and matter?

Opinions?

[arcticpenguin]

That t he Universe we created by an intelligent race of aliens, the coneheads?

[whitefork]

coNics, not coMics.... geez, everybody's a conic these days.

[Thumper]

Quote:

1900 years before Newton Apollonius of Perga derived a large number of properties of the conic sections (circle, parabola, ellipse, hyperbola). That's a lot of the mathematics required for Newtonian physics, and all of it mathematically necessary.

[...]

Now, the observation that bodies accelerate while falling is a contingent one - there's no reason to expect a priori that they would have this property. The fact that the path taken by an accelerated body is described by the same equations that describe a conic section is quite remarkable to me.

So, what if anything should we conclude from this coincidence, and is it a coincidence at all, or an indication of an underlying property of space, time, and matter?

Opinions?

Well,... There is a great difference between the geometric properties (Greek) and the algebraic properties of the conic sections. Each conic section has two ways of defining it, one geometric and one algebraic. A parabola can be defined as either: the locus of points equidistant from a point (focus) and a line (directrix); or as f(x) = a(x-h)^2 - k. The first is a geometric description of it, while the second is the algebraic definition. All geometric descriptions have algebraic descriptions (and vice-versa).

The fact that it doesn't surprise me that a thrown body follows the path of a parabola doesn't come from the geometry of the situation, but from the algebra (specifically the calculus) of it. A constant linear acceleration always produces a parabolic path. An object constrained to a perfectly stiff string will always travel in a circular path. If that string is not perfectly strong (i.e. it has some 'give'), then the path is a sine wave coupled with a circle. Under proper conditions, i.e. the period of the orbit is an integral multiple of the period of the sine wave , an ellipse is produced. Upon closer examination, the orbits of the planets are not perfect ellipses, either, the period of the perturbation and of the orbit (of the sine wave and the circle, respectively) are not integer multiples.

Dannng. Another tangent. =)

So, to answer your question succinctly (if possible for me), it is a coincidence that gravitational motion follows conic sections in a geometrical sense. In an algebraic sense, it makes perfect sense.

Did that help, or did it make things more confusing?

[boooeee]

I think it's less of a coincidence, and more an indication of the underlying simplicity of nature's laws (or at least the approximations we've been able to discover so far).

There are numerous examples of old math having particular relevance to new physical problems (group thoery, matrix algebra, non-Euclidean geometries, etc.).

So, I wouldn't call it a coincidence since it happens very often in physics. I think it just lets us know that the subject under study is very fundamental in nature.

Although, I think we are still waiting for number theory to make its contribution to physics (unless someone knows of a specific example?)

[whitefork]

Quote:

Originally posted by Thumper
Did that help, or did it make things more confusing?

[/frankenstein monster]Confusion .... good....[/fm]

I did a lot of work on the philosophical history of science from the point of view of astronomy once upon a time, but this coincidence continues to vex me. You have Descartes' invention of analytical geometry in the early 1600's and then - bam - Newton's Principia opens with this truncated explanation of the calculus and a set of theorems about ellipses (don't have a copy here sadly). Then universal gravitation - "see - conic sections everywhere". Quite the amazing explosion in the 1600's.

I don't know about the algebraic/geometic split. It seems to me that Apollonius' talking about the construction of squares is not fundamentally different from Descartes/Newton's analysis, but I'm not in a position to offer good arguments.

Had the Greeks found a neat method for computation, who knows what would have transpired. I've always believe that Ptolemy actually saw that his geocentric model of the planetary system was actually a transformation of a heliocentric one, but that he was more concerned with predicting the apparent motions, and the math was easier to work out with the earth at the center. The Almagest is a nice piece of analysis with a lot of good geometry.

Just some more rambling.... Any historians of science out there?

It is my understanding from reading a little here and there from biographies on Newton and from the Principia itself that Newton believed very much that the universe was a geometric one, and that a study of geometry and the principles of it would give us knowledge about the ordering and structure of the universe. What we know of as Newtonian mechanics has more to do with such individuals as Laplace, Lagrange, Euler, Coulomb, and many others.

[xouper]

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