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-loop: minimum speed re...   Elliptic coordinates:    • Introducing elliptic coordinates   Constrained Lagrangian mechanics:    • Constrained Lagrangian mechanics: understa...   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: https://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