# Shortest distance between points using a parameter and variational principle.

I know how to prove that the shortest distance between two points is a straight line by applying E-L equations to

$$L = \int_a^b ds = \int_a^b \sqrt{dx^2+dy^2} = \int_a^b \sqrt{1+(y')^2} \ dx$$

But this is considering $$y$$ as a function of $$x$$. If, instead, I look for both $$x$$ and $$y$$ as functions of a parameter $$t$$, I can't get the expected parametric solution.

Applying the E-L equations to

$$\int_a^b \sqrt{dx^2+dy^2} = \int_a^b \sqrt{(x')^2+(y')^2} \ dt$$

$$\frac{\partial \sqrt{(x')^2+(y')^2}}{\partial x'} = C_x\ \ \ \ \ \frac{\partial \sqrt{(x')^2+(y')^2}}{\partial y'} = C_y$$

$$\frac{x'}{\sqrt{(x')^2+(y')^2}} = C_x\ \ \ \ \ \ \frac{y'}{\sqrt{(x')^2+(y')^2}} = C_y$$

Which simplifies to

$$x' = A y'$$ $$y' = B x'$$

From here I can conclude that $$x$$ and $$y$$ trace a straight line, since any equation can be solved to give $$\frac{\Delta x}{\Delta y} = Const.$$

But I was hoping I'd get a parametric solution for a straight line

$$x' = Const.$$ $$y' = Const.$$

And I just can't see how I could get this or why this is not the straightforward solution.

Why should $$dx/dt$$ and $$dy/dt$$ be constant? A point can move along a line in any fashion. It can even move non-differentiably or non-continuously.
For example, the point $$(\sin t, \sin t)$$ oscillates sinusoidally along the line $$y=x$$ back and forth between the points $$(-1,-1)$$ and $$(1,1)$$. Neither the horizontal nor the vertical speed is constant.
A more extreme example is $$(\lfloor t\rfloor,\lfloor t\rfloor)$$ which jumps along discrete points, dwelling at each $$(n,n)$$ for one unit of time ($$n\in\mathbb Z$$). Its speed is zero at almost every point in time, and the speed is undefined at the moment of each jump (or if you like, it is $$+\infty$$).
Of course, I realize part of the hypothesis of your question is that $$x(t)$$ and $$y(t)$$ are at least differentiable (and so continuous), but there are other modes of movement.