   Chapter 12.5, Problem 92E

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# Kepler's Laws In Exercises 87-94, you are asked to verify Kepler's Laws of Planetary Motion. For these exercises, assume that each planet moves in an orbit given by the vector-valued function r. Let r = ‖ r ‖ , let G represent the universal gravitational constant, let M represent the mass of the sun, and let m represent the mass of the planet.Show that r ' G M × L − r r = e is a constant vector.

To determine

To prove: The expression rGM×Lrr=e is a constant vector.

Explanation

Given:

The specified expression:

rGM×Lrr

Proof:

Consider the left side of the specified expression

rGM×Lrr

Differentiate the above equation as follows:

ddt(rGM×Lrr)=ddt(rGM×L)ddt(rr)

Since ddt[rr]=1r3[(r×r')×r], so,

ddt(rGM×Lrr)=1GM(r×0+r×L)1r3[(r×r)×r]=1GM(0+(GMrr3)×(r×r))1r3[(r×r)×r]=<

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