Euler (1744): derived the critical buckling load for a slender column using variational methods — one of the first applications of the calculus of variations. The critical load P_cr = π²EI/L² depends only on stiffness EI and length L, not material strength.
Elastica: above P_cr, the exact shape satisfies the nonlinear pendulum-like ODE θ'' = −(P/EI)sinθ. The solution involves elliptic integrals of the first kind K(k), giving the exact post-buckled profile. The maximum deflection grows as √(P/P_cr − 1) (supercritical pitchfork bifurcation).
Slenderness ratio: buckling occurs when L/r > π/√(σ_y/E), where r is radius of gyration. Most structural columns are designed to stay below P_cr by a safety factor of 2–3.