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### Interesting Hamiltonian System [duplicate]

The definition of a Hamiltonian system I am working with is a triple $(X,\omega, H)$ where $(X,\omega)$ is a symplectic manifold and $H\in C^\infty(X)$ is the Hamiltonian function. I am wondering if ...
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### Does topology have any role in classical physics?

I've seen many applications of topology in Quantum Mechanics (topological insulators, quantum Hall effects, TQFT, etc.) Does any of these phenomena have anything in common? Is there any intuitive ...
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### What kind of manifold can be the phase space of a Hamiltonian system?

Of course it should have dimension $2n$. But any more conditions? For example, can a genus-2 surface be the phase space of a Hamiltonian system?
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### Why is the phase space of a simple pendulum defined on a cylinder and not $\mathbb{T}^{2}$?

Let's take the pendulum equation $\ddot{x} = -\sin x$. Here $x \in \mathbb{T}^{1}$. Now rewrite it as a coupled first order system $$\dot{y} = -\sin x, \quad \dot{x}=y.$$ Intuitively we know that $y$ ...
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