A proof of instability of some Euler elasticas

• Analyzing stability of Euler elasticas by Sturm theory of differential equation.
• Proving theoretically instability of Euler elasticas of all higher modes in small deflection.
• Proving theoretically instability of some Euler elasticas with inflexions in large deflection.

Based on the conjugate point theory in calculus of variations, sufficiency of instability of some Euler elasticas with one clamped end and the other clamped in rotation in the post-buckling is presented. The conjugate points of Jacobi equation are investigated by Sturm's qualitative theory of ordinary differential equation. Without numerical methods, we proved that those Euler elasticas with more than one inflexion in small deflection and some Euler elasticas with more than one inflexion in large deflection are unstable.