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Symplectic twist maps are obtained from a Lagrangian variational principle. It is well known that nondegenerate minima of the action correspond to hyperbolic orbits of the map when the twist is negative definite and the map is two-dimensional. We show that for more than two dimensions, periodic orbits with minimal action in symplectic twist maps with negative definite twist are not necessarily hyperbolic. In the proof we show that in the neighborhood of a minimal periodic orbit of period n , the n th iterate of the map is again a twist map. This is true even though in general the composition of twist maps is not a twist map.

We calculate linear stability boundaries for natural symplectic maps, which are symplectic mappings derived from Lagrangian generating functions having positive definite kinetic energy. Simplified stability conditions are obtained in terms of the Hessian of the potential and applied to a four-dimensional pair of coupled standard maps.