ABSTRACT
As we will see in Chapter 3, the action functional given by a Hamiltonian system has the form
/ ( x) = (^Soce, x) + b{x), (2 .1)
where x varies in a real Hilbert space E, with inner product (•,•), So is an invertible self-adjoint linear operator on E and the nonlinearity b has compact gradient (it maps bounded sets into relatively compact sets). Both the positive and the negative eigenspaces of So turn out to be infinite di mensional, so critical points of functional / have infinite Morse index and co-index. However, the special form (2.1) allows us to define a relative Morse index, which will share many of the properties of the usual index. Relevant concepts here are the notion of commensurable subspaces and of Fredholm pairs. We will also compare our definition of the relative Morse index with another one, involving finite dimensional reductions. Function als of the above form appear quite often in variational calculus, so it seems useful to develop these concept in full generality, postponing until Chapter 3 the applications to Hamiltonian systems.
