Proof that the shortest curve joining two points is a straight line.

Let the two points be labelled \(A\) and \(B\) and have coordinates \((a,y(a))\) and \((b,y(b))\). The length of an element of path \(ds\) is given by \[ds=\sqrt{dx^2+dy^2} = \sqrt{1+y'^2}\;dx \] and hence the total path length is given by \[ L=\int_a^b{ \sqrt{1+y'^2}} \;dx. \] The Euler-Lagrange equation \[\frac{\partial F}{\partial y} = \frac{d}{dx}\left(\frac{\partial F}{\partial y'} \right)\] can be rearranged to \[\frac{\partial F}{\partial y'} = \mathrm{const.} \] since \( F= \sqrt{1+y'^2} \) does not contain \(y\) explicity. We can now differentiate \(F\) and set it to a constant value. \[ k=\frac{\partial F}{\partial y'} = \frac{y'}{\sqrt{1+y'^2}} \] \[ k^2(1+y'^2) = y'^2\] \[ k^2 = y'^2-k^2y'^2 = y'^2(1-k^2) \] \[ \Rightarrow y'=\frac{k}{\sqrt{1-k^2}} \] Integration gives \[ y(x) = \frac{k}{\sqrt{1-k^2}}x+c\ \] which, as expected, is the equation of a straight line in the form \( y=mx+c \). \[\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\Box.\]

Note by Samuel Braun
10 months, 3 weeks ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

Sort by:

Top Newest

You should mention that the curve is confined to a plane. Such curves are called geodesics, and have different shapes on different surfaces. On the surface of a sphere for example, it is the great circle of the sphere. As you are including xx and yy only, it refers to a plane curve.

Alak Bhattacharya - 10 months, 3 weeks ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...