1. In this problem you will prove that the shortest distance between two points is a line using the calculus of variations. Consider a curve y(x) whose endpoints are fixed: y(x = 0) = 0, y(x = d) = h. That is, the curve goes from (0,0) to (d, h). Consider an infinitesimal variation of the curve y(x) + y(x) + dy(x) that preserves the endpoints. This means that dy(x = 0) = 0, dy(x = d) = 0. Write down a differential equation for y(x) that results from requiring that ol = 0 under this variation. Show that the only solution is a straight line.
1. In this problem you will prove that the shortest distance between two points is a line using the calculus of variations. Consider a curve y(x) whose endpoints are fixed: y(x = 0) = 0, y(x = d) = h. That is, the curve goes from (0,0) to (d, h). Consider an infinitesimal variation of the curve y(x) + y(x) + dy(x) that preserves the endpoints. This means that dy(x = 0) = 0, dy(x = d) = 0. Write down a differential equation for y(x) that results from requiring that ol = 0 under this variation. Show that the only solution is a straight line.
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Transcribed Image Text:1. In this problem you will prove that the shortest distance between two points is a line
using the calculus of variations. Consider a curve y(x) whose endpoints are fixed:
y(x = 0) = 0,
y(x = d) = h.
That is, the curve goes from (0,0) to (d, h).
Consider an infinitesimal variation of the curve
y(x) + y(x) + dy(x)
that preserves the endpoints. This means that
dy(x = 0) = 0,
dy(x = d) = 0.
Write down a differential equation for y(x) that results from requiring that ol = 0 under
this variation. Show that the only solution is a straight line.
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