Instability of periodic minimals (Q1943277)

The author considers second-order Euler-Lagrange systems of the form \[ {d\over dt} [\nabla_{\dot x} L(t,x(t),\dot x(t))]= \nabla_x L(t,x(t),\dot x(t)), \] where the Lagrangian \(L\), which may depend on time, position, and velocity, is given. If the dependence on time is \(T\)-periodic, then the \(T\)-periodic solutions of the system can be viewed as critical points of an appropriately defined action functional. The author explores two questions: the nature of periodic solutions that minimize the action, and the dynamics of action-minimal periodic solutions. The questions go back to Poincaré and Carathéodory. Of interest are ``periodic minimals'', i.e., \(T\)-periodic solutions that have a lower action when modified on any compact interval. Periodic minimals are always periodic minimizers of the action, and the converse is true in one dimension. The author's main result is as follows: If the Lagrangian \(L\) is continuous on an open set \(\Omega\subset (\mathbb{R}/T\mathbb{Z},\mathbb{R}^d,\mathbb{R}^d)\), \({\partial^2 L\over\partial\dot x^2} (t,x,\dot x)\) is positive definite everywhere in \(\Omega\), then every periodic minimal is unstable.