- #1

mataleo

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## Homework Statement

Find the shortest distance between two points using polar coordinates, ie, using them as a line element:

ds^2 = dr^2 + r^2 dθ^2

## Homework Equations

For an integral

I = ∫f

Euler-Lagrange Eq must hold

df/dθ - d/dr(df/dθ') = 0

## The Attempt at a Solution

f = ds = √(1 + (r * θ')^2)

df/dθ = 0

df/dθ' = r^2 * θ' / √(1 + (r * θ')^2) = C

where C is a constant

Now I want to show this is a straight line so the form should be

y = m*x + b ==> r * cos(θ) = m * r * sin(theta) + b

but I'm struggling with how to prove this. I rearranged the terms and solved the integral

θ = ∫(dr / r √(r^2 - C^2))

but I get a piecewise solution so it seems like I've gone in the wrong direction or am missing something. How do I show that this is equivalent to a straight line?

Thanks

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