HOFERS L(INFINITY)-GEOMETRY - ENERGY AND STABILITY OF HAMILTONIAN FLOWS .1.

Consider the group Ham(c)(M) of compactly supported Hamiltonian symple ctomorphisms of the symplectic manifold (M, omega) with the Hofer L(in finity)-norm. A path in Ham(c)(M) will be called a geodesic if all suf ficiently short pieces of it are local minima for the Hofer length fun ctional L. In this paper, we give a necessary condition for a path gam ma to be a geodesic. We also develop a necessary condition for a geode sic to be stable, that is, a local minimum for L. This condition is re lated to the existence of periodic orbits for the linearization of the path, and so extends Ustilovsky's work on the second variation formul a. Using it, we construct a symplectomorphism of S-2 which cannot be r eached from the identity by a shortest path. In later papers in this s eries, we will use holomorphic methods to prove the sufficiency of the condition given here for the characterisation of geodesics as well as the sufficiency of the condition for the stability of geodesics. We w ill also investigate conditions under which geodesics are absolutely l ength-minimizing.