Constrained Lagrangian mechanics in elliptic coordinates: loop-the-loop revisited

Revisiting the elliptical loop-the-loop problem using the more formal approach of Lagrangian mechanics. The geometry of the problem suggests that elliptic coordinates are the most natural choice, which makes this is a good example of working in a non-standard coordinate system!
Some related videos are linked below.
Easier method: • Elliptical loop-the-lo...
Elliptic coordinates: • Introducing elliptic c...
Constrained Lagrangian mechanics: • Constrained Lagrangian...
About me: I studied Physics at the University of Cambridge, then stayed on to get a PhD in Astronomy. During my PhD, I also spent four years teaching Physics undergraduates at the university. Now, I'm working as a private tutor, teaching Physics & Maths up to A Level standard.
My website: benyelverton.com/
#physics #mathematics #dynamics #lagrangian #lagrangianmechanics #constraints #ellipticcoordinates #coordinates #ellipse #conicsection #eccentricity #eccentric #particle #circularmotion #mechanics #looptheloop #centripetalforce #verticalcircle #geometry #velocity #speed #maths #math #science #education

[@shlokdave6360]

Standing Ovation!🙌 What a beautiful solution!

[@DrBenYelverton]

Thanks! It was satisfying to reach the same result as with the radius of curvature method after all that working.

[@MissPiggyM976]

Simply brilliant, indeed !

[@DrBenYelverton]

Thank you!

[@mingmiao364]

My goodness what a problem! Thanks for showing this example. Nonstandard coordinates are rarely encountered in university courses problems.

[@DrBenYelverton]

Indeed, I don't think I had ever worked in this coordinate system until a few weeks ago when it occurred to me to try solving this!

[@ayaan_maan]

Never thought about this before. Really goes beyond the standard sqrt(5gl) result in string pendulums 😅

[@CarlosBonin]

Very well explained! Thank you

[@r2k314]

Lagrange would be proud!

[@DrBenYelverton]

Ha, I hope so!