Variational Calculus and the Euler-Lagrange equation

Euler and Lagrange

Euler (1707-1783) was one of the most brilliant mathematicians of all time and he and Lagrange (1736-1813), a student of his and another great, invented variational calculus and the Euler-Lagrange equation.

Carroll keeps using variational calculus and I think I understand it now. Along the way I found a proof of the Euler-Lagrange equation in Youtube by the Faculty of Khan. It does it in under eight minutes!  That must be a world record. It took me about three hours to understand everything and I wrote it out in this little article at Commentary 3.3 Calculus of Variations and Euler-LaGrange equation.pdf. The proof takes about four pages and then I used it find the equation of a straight line on the plane. I then tried to use it to find a geodesic on a sphere as recommended by Orodruin. I failed. This all happens in section 3.3 and again at the start of section 3.5 'Expanding Universe Revisited' which uses calculus of variations to find geodesic equations and get Christoffel symbols from them. I have more to say on the advantages or not of this technique.

Seven months later videofountain answered my comment on the YouTube video and and pointed out a small error in Commentary 3.3. Now Corrected 😊