How to prove that the shortest distance between two points is a line?

[madhura2498]

Homework Statement

This is the given functional derivative of distance between the two points:
$$\frac{\partial L[y]}{\partial x_{i} } = \frac{d}{d \lambda } L[y(x) + \lambda \delta(x- x_{i} )] \Big|_{\lambda =0}$$
where $\delta(x-x_i)$ is the Dirac's delta function.

I know the Hamilton Variation method. Don't know how to use the Dirac's delta function in the derivation.

Relevant Equations

$$\frac{\partial L[y]}{\partial x_{i} } = \frac{d}{d \lambda } L[y(x) + \lambda \delta(x- x_{i} )] \Big|_{\lambda =0}$$
where $\delta(x-x_i)$ is the Dirac's delta function.

I tried using hamilton method but i don't think that's correct

 

[jedishrfu]

Are you trying to use Calculus of Variations to prove it?

It reminded me of the brachistochrone problem ie the time is timewise shortest downhill path in a constant g field is a catenary and not a straight line.

https://en.wikipedia.org/wiki/Brachistochrone_curve

so perhaps a similar math setup with g=0 would prove it.