Bifurcations and Hamilton's principle

Abstract

An abstract bifurcation theory of nondegenerate critical manifolds for functions on Banach manifolds was developed by Reeken [R], who applied his theory to some eigenvalue problems in Hilbert space. In this paper, we show that the definition of nondegeneracy used by Reeken (and due originally to Bott [B]) can be weakened to allow for a "gain of differentiability" by the hessian operator. The resulting bifurcation theory is then strong enough to yield a new proof of a theorem by the author [W 1, W2], Moser [MO], and Bottkol [BK] on periodic orbits of hamiltonian systems. This proof is similar in spirit to those of Moser and Bottkol, but the added insight into the role of Hamilton's principle given by the abstract approach has allowed us to remove an unnecessary hypothesis (the exactness condition of [BK] and [W2]) from the statement of the periodic orbit theorem. Section 1 of this paper is devoted to Hamilton's principle. It contains a new derivation of this variational principle, which may be of some interest in itself, as well as an explanation why critical manifolds of the Hamilton functional cannot be nondegenerate in the usual sense. Section 2 is devoted to the abstract bifurcation theory of critical manifolds, which is applied to the periodic orbit problem in Section 3. I would like to thank R. Palais for his advice and encouragement. He and J. Morava, and independently J. Mather, have developed an approach (unpublished) to the Fuller index [F] which is a nonhamiltonian version of the method described in this paper.