Introducing elliptic coordinates

[@shlokdave6360]

Your selection of problems to share is simply superb! I was about to ask you for this after the previous video. Thank you so much doing all this and starting such a wonderful channel.

[@DrBenYelverton]

Thanks for all your kind comments! This coordinate system was new to me as well - I discovered it just a few weeks ago after trying unsuccessfully to solve the elliptical loop problem with Lagrangian mechanics in both Cartesian and polar coordinates.

[@shlokdave6360]

@@DrBenYelverton Fascinating! Sir, would it be possible for you to share your calculations in cartesian and polar coordinates as well? I am trying to figure out the equation of motion using christoffel symbols for elliptical coordinate systems. Your calculations in the other systems will be helpful. Thanks!

[@DrBenYelverton]

I didn't keep my attempts at solving it in those coordinates, but the video on the elliptic coordinate solution should be up within a few hours!

[@MissPiggyM976]

Interesting !

[@matthewsarsam8920]

So since the constant curves of mu and nu intersect at more than one point, we need to choose the quadrant of the point that corresponds to the value of nu that we chose. This way any point mu,nu will describe a single point in space. I also did this by looking at x and y with a fixed mu, and that just gives you a cos and sin parametrization with different scaling factors. Let me know if this is a correct interpretation please. Thank you!

[@mastershooter64]

Are there ellipsoidal coordinates for 3D?

[@DrBenYelverton]

Apparently, yes! en.wikipedia.org/wiki/Ellipsoidal_coordinates

[@rahulrohilla1742]

Thanks, this video helped a lot. Can you tell me any reference book for the same. Actually I want to study where the ellipse semi major axis is along the y axis. (c sinh mu cos nu, c cosh mu sin nu)

[@CarlosBonin]

How should I go about to finding the unit vectors in the directions of mu and nu, Dr. Yelverton?

[@DrBenYelverton]

Write the position vector as r = xi + yj, where i and j are the Cartesian unit vectors, and replace x and y by their definitions in elliptic coordinates. You can then get the unit vectors in elliptic coordinates by differentiating r with respect to each elliptic coordinate and then normalising the results. See e.g. planetmath.org/derivationofunitvectorsincurvilinearcoordinates

[@CarlosBonin]

@@DrBenYelverton thank you! 😊

[@thomasolson7447]

I think you need to be very careful when you square sinh. This is a mistake that everyone in the world is making. This is what is messing up relativity and a bunch of other things.