Calculus of Variations and Euler's Elastica

This lecture covers variational methods, calculus of variations with constraints, equilibrium shape of a bent inextensible beam, Euler's Elastica, numerical methods like the shooting method, and analytical methods involving elliptic integrals. The instructor delves into the Brachistochrone problem, the birth of calculus of variations, and the analysis of large deformations of elastic curves. The lecture explores the buckling of slender structures, the determination of satisfying conditions, and the integration by parts. It concludes with the study of Euler's Elastica, dimensionless equations, and analytical solutions using scaling laws.