Buckling load P_cr = π²EI/L², post-buckling elastica curve from Euler angle ODE
Euler (1744) derived the elastica — the exact shape of a buckled elastic beam — by solving EI·d²θ/ds² = −P·sin(θ), a nonlinear pendulum ODE expressible in elliptic integrals. The critical buckling loads are P_n = n²π²EI/L² (Euler loads), with n=1 being the lowest. Post-buckling shape has finite amplitude growing as √(P/P_cr − 1). The elastica also describes: free-standing DNA loops, surgical sutures, and fiber-reinforced composites under compression.