All periodic minimizers are unstable (Q943338)

The paper deals with the action functional associated to a Lagrangian acting on scalar-valued periodic functions. The problem of the instability of (local) minimizers goes back to Poincaré. In 1935, {C. Carathéodory} showed [cf. Calculus of variations and partial differential equations of the first order, Chelsea, New York (1982; Zbl 0505.49001)] that the nondegenerate periodic minimizers of this functional are always unstable. \textit{E. N. Dancer} and \textit{R. Ortega} [J. Dyn. Differ. Equ. 6, 631--637 (1994; Zbl 0811.34018)] proved that the same is true for all isolated periodic minimizers, even if they are degenerate. The author proves that every (local) periodic minimizer is unstable, as a solution of the Euler-Lagrange equation, a remarkable result.