Prove the shortest distance between two points is a line

[MxwllsPersuasns]

Homework Statement

Let γ : [0, L] → Rⁿ be arclength parametrized. Show that the distance between the endpoints of the curve can at most be L, and equality can only hold when γ is a straight line segment. Thus, the shortest path between two points is the straight line segment connecting them.

Homework Equations

I guess maybe arclength: s(t) = ∫ ||γ'(t')||dt' from 0 to t

The Attempt at a Solution

So my attempt would be to include the properties of an arclength parameterized curve; Namely that the length of such a curve is 1. Such that when you compute the arc length you get something like ∫1dt' from 0 to t (or in this specific case 0 to L) this tells us the length is L but I don't know how to prove that the equality holds only when its a straight line. Any suggestions?

 

[pasmith]

*cough* Calculus of variations *cough*

 

[FactChecker]

See the example in https://en.wikipedia.org/wiki/Calculus_of_variations