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Unformatted text preview: 1.4 Exercises 17 5. Prove that the shortest path between two points in R n is a straight line by using the techniques described below. Any path in R n between two points a and b can be mathematically rep- resented as a vector-valued function p : [0 , 1]-→ R n with p (0) = a and p (1) = b . Assume p is differentiable. p ( t ) can be thought of as the position at time t of a particle that moves along that path. The velocity vector is thus ˙ p ( t ) (see Figure 1.2). Assume that the particle never stops, i.e. ˙ p ( t ) = 0 for any t ∈ [0 , 1]. p(t) p(t) . b a x 1 x 2 1 p Fig. 1.2. There are now two ways to proceed, the first is a little messy, and the sec- ond utilizes an observation that significantly simplifies the calculations. a) The total path length traveled L is the integral of the speed L = 1 ˙ p * ( t ) ˙ p ( t ) dt. The problem is now in a form for which the calculus of variations is applicable. Find a differential equation for the optimal p and show that a straight path between...
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- Winter '09
- Derivative, shortest path, Geodesic