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For a wide class of curvature energy functionals defined for planar curvesunder the fixed-length constraint, we obtain optimal necessary conditions forglobal and local minimizers. Our results extend Maddocks' and Sachkov'srigidity principles for Euler's elastica by a totally different approach, andin particular lead to complete classification of stable closed $p$-elasticaefor all $p\in(1,\infty)$ and of stable pinned $p$-elasticae for $p\in(1,2]$.Our proof is based on a simple but robust `cut-and-paste' trick withoutcomputing the energy nor its second variation, which works well for planarperiodic curves but also extend to some non-periodic or non-planar cases.

General rigidity principles for stable and minimal elastic curves