TITLE: Why not develop a Hamiltonian-based Morse theory?
QUESTION [9 upvotes]: I have begun to learn the basics of Morse theory and Floer homology. I understand that Floer homology is the natural theory for symplectic manifolds, but from my preliminary knowledge of Morse theory am curious why the seemingly more natural analogue of this can not be applied to the study of symplectic manifolds (or indeed maybe it can but is of no use?). By this I mean a sort of finite-dimensional "Hamiltonian Morse theory" where the gradient-like vector field is replaced with a Hamiltonian-gradient-like vector field $X_H:i_{X_H} ω=dH$, and it is the critical points of H that are relevant rather than those of the action functional (as in Floer homology). Any comments would be welcomed.

REPLY [11 votes]: To any continuous  dynamical system $\Phi_t$ on a reasonable space $X$ (say a compact metric space)  there is an associated Morse like theory, namely  the Conley index theory,

C. Conley: Isolated Invariant Sets and the Morse Index, CBMS Regional Conf. Series, vol.
  38, Amer. Math. Soc., 1978.

Conley and Zehnder, in Sect. 3 in their paper Morse type index theory for flows ..., Comm. Pure Appl. Math. (37) (1984),  have shown how to extend the Morse inequalities to this more general case.  This extension is based on  a certain  increasing filtration of the space $X$  compatible in a certain  way with the flow $\Phi_t$.    Such a filtration produces a spectral sequence in  homology  and the  Morse inequalities  relate the Betti numbers of the $E_1$ page  of this spectral sequence to the Betti numbers of $X$. $\newcommand{\bZ}{\mathbb{Z}}$
In the  very special case when $X$ is a compact smooth manifold, and the flow in question is the  (negative) gradient flow of a  self-indexing Morse function satisfying the Smale transversality conditions  several nice things happen.

The above Conley-Zehnder filtration    can be identified with the filtration by the sublevel sets $\bigl\lbrace f\leq k+\frac{1}{2}\bigr\rbrace$, $k\in\bZ_{\geq 0}$. (I recall that the critical values of $f$ are nonnegative integers.) 
The $E_1$ page  of the  canonical spectral sequence of this filtration is the Floer complex; see sec. 2.5 of these lectures.

I have not seen    applications of the Conley index applied directly to  Hamiltonian flows as   suggested in your question, though there might exist. At a first sight this seems difficult but, as Arnold liked to say, you never know how hard  a problem is until you try to solve it.