Setting up a extremization problem

[grandpa2390]

Homework Statement

I am having trouble setting up problems that require me to extremize. I know how to use Euler's equation, but I don't understand how to get what it is that I want to plug in.

what I need to show is that the the correct path between two points (x,t) and (x2,t2) maximizes the proper time. and that the correct path has a constant velocity v=dx/dt

Homework Equations

instead of distance, I am supposed to use spacetime interval dτ^2 = dt^2 − dx^2,
τ is proper time

The Attempt at a Solution

I tried to treat this as if it were the same as the distance equation, and the constant velocity v=dx/dt was just a constraint. but I don't know. I don't think that works. I haven't a clue how to set this up.

 

[RUber]

Have you solved $d\tau$ in terms of dx to remove t from the equations?

 

[grandpa2390]

Have you solved $d\tau$ in terms of dx to remove t from the equations?

I don't know what you mean. I solved $d\tau$ = sqrt (dt^2 - dx^2) and then simplified it to $d\tau$ = sqrt( 1 - x'^2)dt is that what you are saying?

 

[grandpa2390]

Let me try a different tack

I am trying to understand how to determine this. If I can figure out what equation to put into dy and dg, then I can solve it.

Maybe if you can explain why other equations were chosen.
If I want to maximize the area, the equation is ∫dA = ∫y*dx why? I understand that the width varies. that I am integrating between x1 and x2. but isn't y dependent on x? why wouldn't it just be the integral of dx?
or maybe we are looking at it as ∫dA = ∫y(x)dx ? I guess that would make sense?

or for length. we said that ds = √((dx)^2 + (dy)^2) that makes sense to me because that is the equation of the hypotenuse. So I feel like I should do the same with dτ = sqrt (dt^2 - dx^2)
I want to say that f( x' ; t ) = sqrt( 1 - x'^2 ) dt

that makes sense, but what about the other part? show that the correct path has a constant velocity v=dx/dt . is that a constraint, or is that the result I am trying to prove. how do I handle this?

do I say v = dx/dt and say g( x' , t ) = x' ?edit: I found an example problem that said for a fixed perimeter length and it used k= ∫dl = ∫sqrt (1+y'^2)
so I guess maybe for a fixed velocity k= ∫dv = ∫ something... maybe dv/dt? dv/dt = x"

edit: or maybe ∫dv = x'dt

then maybe I can factor out the dx/dt and solve it that way?

 

[grandpa2390]

tried that. Got x' = (t + C) / \sqrt{1 + (t + C)^2)}

now to integrate

edit: I got sqrt ( 1 + (t + C)^2)
is that right? does that make sense?