Elastic instability

Remarks

Example: a rigid bar of length $L$ hinged at its foot, held upright by a torsional spring of stiffness $k$, with a vertical load $P = \lambda$ on top. With tilt angle $\theta$, $\Pi(\theta, P) = \tfrac{1}{2} k \theta^2 - P L (1 - \cos \theta)$. The upright state $\theta = 0$ is an equilibrium for every $P$, and $\Pi''(0) = k - P L$. So it is stable for $P < k/L$ and the critical load is $P_c = k/L$.

Non-example: the same bar hanging down from its hinge, with $\Pi(\theta, P) = \tfrac{1}{2} k \theta^2 + P L (1 - \cos \theta)$. Here $\Pi''(0) = k + P L > 0$ for every $P \ge 0$, so the second variation never loses positivity and there is no instability.

At $\lambda_c$ the second variation has a nontrivial kernel in the usual cases, and new branches of equilibria bifurcate there.