planetary - Math 21a Supplement on Planetary Motion Suppose...

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Math 21a Supplement on Planetary Motion Suppose that an object (planet, asteroid, whatever) travels through space under the gravitational influence of a star. Newton’s laws allow one to describe trajectory of the planet. This supplement describes the situation. a) Newton’s Laws There is a natural coordinate system to describe the motion. This is the so called center of mass coordinate system. (See the final section, f.) In this coordinate system, the position vector r (t) for the object at time t (the vector from the origin to the planet’s position) evolves in time according to a differential equation of the following form: d dt 2 2 r = - κ r /r 3 . (1) Here, κ is a positive constant which is determined by the masses of the object and the star. Also, r = | r | is the distance of the object from the origin. To simplify the typing and notation, allow me to introduce the following shorthand: d dt r r ´ . d dt 2 2 r r ´´ . (2) For example, Equation (1) reads r ´´ = - κ r /r 3 . (3) b) The energy The standard strategy for solving (3) is to find so called “constants of the motion”. These are quantities which are constant along each trajectory. The energy, E = | r ´| 2 /2 - κ /r . (4) is one such constant. To show that E is constant, consider its time derivative under the assumption that r (t) obeys (3). One finds that E´ = r ´· r ´´ + κ r ´· r /r 3 . (5) The fact that the latter is zero follows by taking the dot product of both sides of (3) with the vector r ´. The derivation of (5) uses the fact that r = ( r · r ) 1/2 and that the time derivative of a dot product obeys ( A · B )´ = A ´· B + A · B ´ (6)
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when A and B are both vector valued functions of time. (Equation (6) can be verified by writing out A · B = a 1 b 1 + a 2 b 2 + a 3 b 3 and then differentiating.) You should think about what it means if the energy E is constant along a trajectory. For example, suppose E is negative.
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This note was uploaded on 02/18/2012 for the course MATH 310 taught by Professor Gregoryj.galloway during the Fall '11 term at University of Miami.

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planetary - Math 21a Supplement on Planetary Motion Suppose...

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