Last night's Cosmos & Isaac Newton's Principia

[pokerfan]

An elegant proof, no calculus needed....

The great physicist Richard Feynman once tried to read the section on the elliptical orbits of the planets, and eventually gave up, saying that Newton’s use of obscure properties of ellipses made the going too difficult. (Feynman’s response was to do what few others could have done: he created his own geometric proof. This proof is described in the book “Feynman’s Lost Lecture”).

However, there is at least one result that Newton derived in the Principia that is fairly easy to understand, and I will describe it in this post. It also happens to be one of the important theorems in the Principia: a proof that Kepler’s Second Law is a consequence of mechanics.

As a reminder, Kepler’s Second Law says that the planets sweep out equal areas in equal times as they go around their elliptical orbits. The following diagram illustrates this principle.



Read the rest: http://brightstartutors.com/blog/2011/a-gem-from-newtons-principia/

[Hissyspit]

1. "The Royal Society had blown pretty much their whole annual budget on the publishing of 'The History

of Fish.'"

I'm loving 'Cosmos.'

[Manifestor_of_Light]

2. And Hooke was busy trying to destroy Newton and generally be a bastard to all.

[muriel_volestrangler]

11. That's hard on Hooke

see eg http://io9.com/5877660/was-robert-hooke-really-sciences-greatest-asshole

Hooke was a very important figure, who was dismissed by those who put Newton on a pedestal.

[exboyfil]

3. I am watching the original Cosmos

at the same time as the new one. They seem to be tracking in theme. Sagan's 3rd episode talked extensively about Kepler (who was excommunicated from the Lutheran church and his mother was held in captivity as a witch - they tried to burn her like several other older women were burned at the time).

Sagan's final point that once Kepler's cherished ideal about the planets fitting in with the principal shapes did not fit the data, that he moved to a mathematical model which did work and predicted the motion of the planets (an observational model - it was for Newton to later develop a theoretical framework with explained the effect of gravity and was applicable to any non-relativistic bodies).

The two shows nest well together. I still much prefer Sagan's Cosmos, but I thought this was Tyson's strongest episode so far. I did not remember hearing about the story of Newton throwing Hooke's portrait on the fire - that was funny.

[Cartoonist]

4. Neal Stephenson

He wrote a fictional account of Newton's time that I thought was very entertaining. I had read a previous biography of Kepler, and Stephenson's books made me seek out the "real" story of Newton, Copernicus and the Church. Had I been doing Cosmos, I would have given the church a beatdown.

[Geoff R. Casavant]

5. I love the Baroque Trilogy

I love fiction that mixes historical characters into the cast, and Stephenson weaves so many threads together, without really altering "real" history. The Monmouth Rebellion, the Bloody Assizes, all the European intrigue, and the beginnings of modern finance.

[byronius]

7. One of the best series I've ever read. I'll probably read it a fifth time.

Who else could make the history of mathematics into a heart-pounding thriller?

[caraher]

6. "Hell's Bells"

Any documentation that Halley was fond of saying this? They had him say it at least twice in the cartoons...

[Hissyspit]

8. I was wondering that, too!

The earliest references I can find date back to the early 19th century. It most assuredly predates that, but how far back?

[pokerfan]

10. also... "Put up or shut up, Mr. Hooke!"

though I'm sure the meaning was the same despite the anachronistic language,

[Gothmog]

9. I love this show

Cosmos and NDT are amazing

[muriel_volestrangler]

12. It all drops out of 'angular momentum is conserved'

When there is no tangential force on an object, ie the only for is towards or away from a central point, its angular momentum, m.r.v (where v is the tangential velocity), is conserved. But v=r/t, so m.r.r/t is constant - so r.r/t is constant. That's proportional to the area swept out in a given time period - pi.r.r/t.

[pokerfan]

14. That's true in uniform circular motion

but in any other conic section, say an ellipse, the force is usually not orthogonal to the motion. Velocity (and momentum) is constantly changing.

[muriel_volestrangler]

15. That doesn't matter - the tangential velocity is proportional to the radius

multiplied by the angular velocity, so the angular momentum is proportional to the radius squared times the angular velocity - ie the rate area is swept out (or, for a particle, the moment of inertia I=mr^2, and angular momentum is I? ) .

[pokerfan]

16. I see

I thought you were only describing a situation where there was only an orthogonal force, i.e. uniform circular motion.

[muriel_volestrangler]

17. Not orthogonal, but radial

ie there is no tangential force. So for circular motion it would be at right angles to the motion, but in the other conic sections, it isn't, but the 'equal area per unit time' still holds.

[pokerfan]

18. Right.