Shortest Distance Path Between Two Points On A Plane

[@adamya1639]

A simple yet joyful example, nothing makes you feel like diving deeper than small yet continuous understanding:)

[@spandansreyanshupadhee]

You actually deserve views in millions sir , the concept really helped me to understand the Fermat's principal

[@JohnSmith-kj2od]

I kid you not, I got a handout(about 4 days ago) after learning Newtonian mechanics and out of the 6 problems, two of them use concepts that were recommended to me by this channel on the same day, I just have incredible luck

[@ha15224]

Thank you for these videos sir, I am currently reviewing/brushing up on my studies to prepare for graduate school applications (aerospace engineering major) and you have been an immense help. This channel is on the levels of 3b1b and Eugene Khutoryansky and deserves millions of views.

[@davidbrisbane7206]

Great video. You deserve to get a large subscriber base.

[@claudefazio]

Very well explained! I enjoy your calm style of presentation.

[@amukelaniebenezer406]

Very simply explained

[@faheemgulzar1]

Excellent videos

[@andrerossa8553]

Thanks for such a simple and clear explanation. By the way which application do you use to write on the screen?

[@Freeball99]

The app is called "Paper" by WeTransfer running on an iPad Pro 13 inch and using an Apple Pencil. Screen capture is recorded by attaching the iPad to my Mac and using Quicktime.

[@andrerossa8553]

Thank you very much for your prompt reply. You're the best!!

[@Rafael-c9c]

Great video! You got +1 sub. Which hardware do you use to illustrate this video?

[@Freeball99]

The app is called "Paper" by WeTransfer running on an iPad Pro 13 inch and using an Apple Pencil. Screen capture is recorded by attaching the iPad to my Mac and using Quicktime.

[@manmis007]

Waiting for some time before I see you again with some grt stuff again.......

[@jmsolangi]

Thanks, very helpful and clear

[@ozgurartvin]

tHANK YOU VERY MUCH. gREAT LECTURE

[@claudiogomes2493]

Thank you so much

[@hadsonwafula3751]

Thank you it is very much useful

[@prerakvaishnav7512]

Thank you this explains quite bit of minimization approach 😀

Sir, since I discover your channel some months ago, I've been following you! You're doing a huge, beautiful and awsome worke with this videos. I would like to ask for recommendations to practice this theory. Maybe if you know an excerse book or something like that. Thanks in advance!

[@Freeball99]

If you're talking about books the explain Variational methods, these are very good: 1. "Solid Mechanics A Variational Approach" By Clive L. Dym, Irving H. Shames 2. "Energy And Finite Element Methods In Structural Mechanics" By Irving Shames and Clive Dym

@@Freeball99 thanks for your quick answer. But more than theory, I'm looking for problems to apply what I have learned from you. I guess books will have good problems for that! Thanks!

[@Freeball99]

For the introductory stuff, "Mechanical Vibrations" by SS Rao is useful. It has some errors, but lots of problems for you to work.

[@Pedritox0953]

Great video!

[@abcd-gp6is]

Thank you ❤️❤️❤️

[@workerpowernow]

very helpful. Thanks

[@spencergee6948]

Your verba presentation is first class. Do I detect a South African origin?

[@Freeball99]

Yup. Grew up in Durban!

[@alonshamay5163]

graet job!!!

[@shevek5934]

Am I understanding it correctly that this argument for the path being a straight line doesn't depend at all on the specific Lagrangian for the distance, but would be true for any functional that depends only on either y or y' but not both?

[@Freeball99]

I'm not sure I understand the part of your question about the functional depending only on y. I think just about any path integral would have a y' component to it - perhaps you have an example.

[@thevegg3275]

If delta I needs to go to 0 to give the shortest distance, didn't you already have the shortest distance before you added the delta S? I am not seeing delta S being forced to 0 in your example. To me, that's like saying add the squiggly line then remove it and voila, you've got the straight line! Seems like one can end up with the answer of a straight line in flat space without using the variation squiggly technique.

[@Freeball99]

Let's clarify: 1. We don't start with the shortest path. We start with an unknown path y(x). 2. δI going to 0 doesn't mean δS goes to 0. It means the variation of the integral is 0 for the optimal path (as I showed in the original video where I derived the Euler-Lagrange eqn). 3. The "squiggly line" (variation) isn't added then removed. It's a mathematical tool to explore nearby paths. 4. The process derives conditions any optimal path must satisfy, without assuming we know it's a straight line. 5. Solving these conditions reveals that only a straight line satisfies them in flat space. The power of this method is it works for more complex problems where the optimal path isn't intuitively obvious. We use the same technique to find optimal paths in curved spaces or with constraints.

[@dogeness]

Thanks this was really helpful. I just have one small question. You didn’t use del(I)=0 (equation 3) in the solution, did you? So why did you write it down?

[@Freeball99]

The result of setting δI = 0 yields the Euler-Langrange Equation. By setting δI = 0, I was showing that we could use the results of the previous video (the E-L Equation) to give us the solution.

[@cana5312]

Could you cover the shortest distance between the two points on the arbitrary plane, not the 2D plane?

[@Freeball99]

Yes, I'll get there eventually. I have this on list a video to make on the shortest distance between two points on a sphere (a geodesic). However, I have a few others to make before that.

[@jaafars.mahdawi6911]

I can guarantee that your final hope (that this video was useful) has been granted. Only a minor detail seems in place, namely, in eq.7, the constant cannot be 1. Homework for fellow viewers: can you tell why, like, intuitively?

[@haric_]

How do you proof that y' has to be a constant in the first place?

[@NoName-rn8uk]

Brilliant

[@ccdavis94303]

Thanks.

[@GreenMeansGOF]

Not boring at all. I am curious however, if there is a more elementary proof. Say a Euclidean/Coordinate Geometry proof.

[@Freeball99]

I'm not familiar with any other proofs, but I am sure they exist.

[@phymacillustrator]

I gree