1.3. Geodesics 7 where g is the magnitude of the gravitational acceleration at the sur- face of the earth. For a particle starting at rest at the surface of the earth, conser- vation of energy now implies that 1 2 mv2 + 1 2 mg R r2 = 1 2 mgR (1.10) so that v = g(R2 − r2) √ R . (1.11) It follows that the total travel time is T = R g θb θa ( dr dθ )2 + r2 R2 − r2 dθ . (1.12) We will look at this problem in greater detail later. We shall see that the terrestrial brachistochrone is a hypocycloid, the curve traced by a point on the circumference of a circle of radius either [R − (SAB/π)] (see Figure 1.4) or of radius SAB/π (see Figure 1.5), where SAB is the arc length along the surface of the earth between A and B, as it rolls inside a circle of radius R. The fastest Amtrak train makes the 400 mile trip between Boston and Washington, D.C., in six and a half hours. A tube train moving along a straight-line chord between Boston and Washington would penetrate 5 miles into the earth and take 42 minutes. The fastest tube train along a hypocycloid would, in turn, penetrate 125 miles into the earth and take 10.7 minutes. 1.3. Geodesics I do not want to give the impression that the calculus of variations is only brachistochrones. In this and the next section, we will look at two other classic problems. A line is the shortest path between two points in a plane. We also wish to find shortest paths between pairs of points on other, more general, surfaces. To find these geodesics, we must minimize arc length.
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