Lec16-KeplerAndMultivariable

# Lec16-KeplerAndMultivariable - Math 32 A Wednesday November...

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Math 32 A Wednesday, November 4, 2009

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Office Hours Today: 12 – 1 PM EXAM 2: Next week, Friday Nov. 13 REVIEW SESSION Monday Nov. 9, Kinsey Pavilion 1200B Google: UCLA map
The Law of Ellipses

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Newton’s demonstration of the law of ellipses is a watershed that separates the ancient world from the modern world .
It is one of the crowning achievements of the human mind, From: Feynman’s Lost Lecture David Goodstein, Judith Goodstein (Physicist, Vice Provost, CalTech)

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Comparable to Beethoven’s symphonies, or Shakespeare’s plays or Michelangelo’s Sistine Chapel
It is a conclusive demonstration of the astonishing fact: Nature obeys mathematics .

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Published 1687 Newton’s Geometric Proof of the Second Law
Second Law when G = 0 (gravitational force is zero

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Elementary Geometry: Triangle have SAME AREA
Newton observes: under a central force, The planet is displaced parallel to the radial vector. THE AREA SWEPT DOES NOT CHANGE by elementary geometry!

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Conservation of angular momentum The vector J = r x r' is constant

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Conservation of Angular Momentum
Energy r = distance from sun, v = speed Kinetic Energy: (1/2) mv 2 Potential Energy: -GMm /r TOTAL ENERGY = (1/2)mv 2 - GMm / r

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Conservation of Energy E = (1/2)v 2 - GM / r dE / dt = 0 E is a constant of motion.
Two Conservation Laws (1/2)v 2 - GM / r = E r v sin( Ψ ) = J

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ELIMINATE v GET EQUATION LINKING r and Ψ. (1/2) v
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## This note was uploaded on 10/26/2011 for the course MATH 32A taught by Professor Gangliu during the Spring '08 term at UCLA.

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Lec16-KeplerAndMultivariable - Math 32 A Wednesday November...

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