Reasons for the Arnold conjecture I am trying to understand the Arnold conjecture in Symplectic Geometry, which basically tells us the following: If $M$ is a compact symplectic manifold and $H_t$ be a 1-periodic Hamiltonian function, then we can consider the Hamiltonian equation of motion which defines us a family $\psi_t$ of symplectomorphisms of $M$. We then consider the fixed points of $\psi_1$ and call a fixed point $x$ non-degenerate, if $\det(1- d\psi_1(x)) \neq 0$. In the case that all the fied points are non-degenerate the Arnold conjecture then is: If every fixed point of $\psi_1$ is non-degenerate, then the number of fixed points is at least the sum off all Betti numbers of $M$.  \
I would now like to know the answer to the following questions: \
1. Why is such a result helpful for our understanding of Symplectic Geometry? Why would somebody like to know whether such a conjecture is true or not? \ß
2. Why could this result be true? Can you maybe give me an explanation or reference, why Vladimir Arnold conjectured this result?
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Every answer to the above questions would be appreciated.
 A: In a certain sense, symplectic geometry (or safer to say symplectic topology) as we know it now was not existing before Arnold formulated these conjectures. So many would say that Arnold conjectures gave birth to symplectic geometry. At the time Arnold made this conjecture only one non-trivial statement in this directions was known -  Poincaré's last geometric theorem http://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Birkhoff_theorem
A: The conjecture is listed as problem 1972-33 in the book Vladimir Arnold: Arnold's Problems, and in the Comments-section of this book you can you can find plenty of background information on it in a contribution by Mikhail B Sevryuk.
A: The Arnold conjecture is interesting to symplectic geometers because a lot of new math has resulted from people trying to prove it.  I think the most notable of which is that Hamiltonian Floer homology was developed as a means to obtaining a proof of the conjecture.  I would also argue (only on the grounds of ideology) that it is interesting that the topology of a symplectic manifold constrains the possible evolution of any dynamical system defined on it.
The motivation behind why the result could be true is that the Arnold conjecture closely resembles the Morse inequalities.  Floer homology degenerates to Morse homology in the proper limit, and so really the Arnold conjecture can be thought of as a generalization of the Morse inequalities.  
If you can read French, a great reference is Audin and Damian's book, "Theorie de Morse et homologie de Floer".  Alternatively, Salamon has some notes which are very thorough
http://www.math.ethz.ch/~salamon/PREPRINTS/floer.pdf
A: Here is a trivial  example that I read from a survey article written by Arnold in the late 80s.
Consider $T^*S^1$, the cotangent bundle of $S^1$  which we can identify with the product $\newcommand{\bR}{\mathbb{R}}$ $S^1\times\bR$. I will denote  the obvious coordinates on this cylinder by $(\theta, t)$.
Like any cotangent bundle, $T^*S^1$ carries a symplectic structure,  and in this case, any curve on this symplectic manifold is a lagrangian submanifold. However, there are curves, and there are curves.
Take for example  the curves $C_\tau:=\lbrace t=\tau\rbrace$, $\tau$ a nonzero constant,  which are disjoint  from the zero section  and  are  deformations of the zero section via the   symplectic  flow
$$ (\theta,t)\mapsto \Phi_\tau(\theta,t)=(\theta,t+\tau). $$
Consider next  a smooth function
$$ S^1\ni\theta\mapsto f(\theta). $$
Its differential is a section of $T^*S^1$, and its graph $\Gamma_{df}=(\theta,f'(\theta))$ intersects the zero section     along the critical points of $f$.
The lagrangian $\Gamma_{df}$ is a rather special deformation of the zero section:  it is a Hamiltonian deformation, the points of intersection of $\Gamma_{df}|$ correspond to the periodic  orbits of the   Hamiltonian deformation.
Why is this fascinating?   Certain pairs of lagrangian subspaces intersect in more points   than predicted by topology alone, which is in itself an indication that   symplectic topology is a bit more rigid  than smooth  topology alone.
How does the above trivial example  fit the general picture?
A lagrangian submanifold $L$ of a symplectic manifold has a tubular neighborhood  symplectomorphic  to $T^* L$. Thus  the case of cotangent bundles can be viewed as local situations of the more  general cases. of lagrangian  submanifolds  and their hamiltonian perturbations.
Given a   Hamiltonian flow $\Phi_t$ on a symplectic manifold $X$,  the graph of the time $1$-map  is  a lagrangian submanifold in $X\times X$.    Its fixed points correspond to the intersection of the graph with the diagonal in $X\times X$, which is another   lagrangian submanifold.  Thus the  problem  of intersection of lagrangian submanifolds    contains  as a special case the problem of existence of periodic solutions of hamiltonian systems.
Leaving aside  the  mysterious rigidity of  symplectic topology alluded to above, the problem of existence of periodic orbits of hamiltonian systems has fascinated    many classics, such as Poincare,  because of it's obvious connection to the many body problem and the philosophical question:    does the  history  of our planetary  system repeat itself?
