Question

3. (20 points.) The distance between two points in a plane is characterized by the infinitesimal statement d s^{2}=d x^{2}+d y^{2} Let us prove the intuitively obvious statement that the curve of shortest distance going through two points (x1, Y1) and (x2; Y2) in a plane, the geodesics of a plane, is a straight line passing through the two points. This amounts to finding the extremal of the functional l[y]=\int_{\left(x_{1}, y_{1}\right)}^{\left(x_{2}, y_{2}\right)} d s=\int_{\left(x_{1}, y_{1}\right)}^{\left(x_{2}, y_{2}\right)} d x \sqrt{1+\left(\frac{d y}{d x}\right)^{2}} Using the fact that we and (x2, Y2), show that require the curve to necessarily pass through the points (x1, Y1) \frac{\delta l[y]}{\delta y(x)}=-\frac{d}{d x}\left[\frac{\frac{d y}{d x}}{\sqrt{1+\left(\frac{d y}{d x}\right)^{2}}}\right] Then, using the condition that the functional derivative is zero for the extremal curve,derive the equation of the geodesic to be given by \frac{d y}{d x}=c where c is a contant. Identify this as the equation of a straight line in a plane. Find c.

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