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# kepler - Math 32A Lecture 3(Rogawski Notes on Keplers Laws...

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Math 32A, Lecture 3 (Rogawski): Notes on Kepler’s Laws These notes contain some challenging mathematics. Read them care- fully and try to put all the pieces together. Here are the headings: 1. Newton’s Second Law 2. A Simple but Important Result 3. Angular Momentum 4. Area Swept Out by the Radial Vector 5. The Planetary Differential Equation 6. Law of Equal Areas in Equal Times 7. The Velocity Circle 8. Grand Finale: The Law of Ellipses 9. Summary 1. Newton’s Second Law Let r ( t ) denote the path of a particle of mass m in three-dimensional space. The velocity and acceleration vectors are v ( t ) = r ( t ) , a ( t ) = r ′′ ( t ) When convenient, we drop the reference to t and write r , v , a , but it is understood these are vector-valued functions of time. Newton’s Second Law of Motion is often stated in scalar form: F = ma (force equals mass times acceleration). However, both force and acceleration are vectors, and the Law of Motion is true as a vector equation: F = m a 2. A simple but important result Theorem. For any vector-valued function h ( t ) , d dt h ( t ) × h ( t ) = h ( t ) × h ′′ ( t ) Proof. By the Product Rule for cross products, d dt h ( t ) × h ( t ) = h ( t ) × h ( t ) bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright this is zero + h ( t ) × h ′′ ( t ) = h ( t ) × h ′′ ( t ) Note that h ( t ) × h ( t ) = 0 because v × v = 0 for any vector v . square 1

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2 Actually, this result is not quite as simple as it seems. Suppose that h ( t ) = ( x ( t ) , y ( t ) , 0 ) . Then h ( t ) × h ( t ) = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle i j k x y 0 x y 0 vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = ( xy x y ) k and the derivative simplifies due to cancellation: d dt h ( t ) × h ( t ) = d dt ( xy x y ) k = ( xy ′′ + x y x y xy ′′ ) k = ( xy ′′ xy ′′ ) k This is equal to h ( t ) × h ′′ ( t ) as claimed since h ( t ) × h ′′ ( t ) = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle i j k x y 0 x ′′ y ′′ 0 vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = ( xy ′′ x ′′ y ) k 3. Angular Momentum The angular momentum of a moving particle is the vector m J where J ( t ) = r ( t ) × r ( t ) Since m is fixed in our case, we focus on J
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kepler - Math 32A Lecture 3(Rogawski Notes on Keplers Laws...

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