Do time-invariant Hamiltonians define closed systems? 
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*In classical mechanics, every time-invariant Hamiltonian represents a closed dynamical system? 

*Can every closed dynamical system be represented as a time-invariant Hamiltonian? Or are there closed dynamical systems that can't be described by a time-invariant Hamiltonian? 

*Are those that can't be described by a time-invariant Hamiltonian "unphysical"?
 A: You are confusing two definitions - closed system and conservation of energy. I'll clear them up for you.
In classical dynamics a closed system is one where no force external to the system acts. In a closed system, the total energy, total momentum and total angular momentum must be conserved. This follows from Noether's theorem. If a has no interaction with the outside world, then we expect it to obey spacetime symmetries. Noether's theorem then guarantees the conserved quantities above.
A2. Yes, every closed dynamical system can be represented as a time-invariant Hamiltonian. 
Note that this definition of closed system is not the same as the definition of closed system in thermodynamics. In fact, a closed system in classical dynamics is the same as an isolated system in thermodynamics. It's rather irritating that there's this abuse of nomenclature - I assume it's historical!
Our best fundamental theories assume that we can describe the universe as a closed system. Therefore in a sense, all of physics reduces to this case. However, just describing things in terms of closed systems is quite impractical. After all most experiments we perform are very much not in a closed system. At the moment, we're all sitting in the Earth's gravitational field, for instance. Therefore it's convenient (and completely physical) to describe the world in terms of more general non-closed systems. 
A3. No. Fundamentally we believe all physics is described by closed systems, which would have time-invariant Hamiltonians. But it is sometimes convenient to describe parts of the system on their own. That's a completely physical thing to do! There's no reason why parts of the system should have time-invariant Hamiltonians, or indeed obey any symmetries at all.
Finally we come to conservation of energy. Total energy is conserved if and only if the Hamiltonian is time-invariant. But notice that this is a less stringent condition than having a classical dynamics closed system. For example for a particle moving in a central field, energy is conserved but momentum is not. We can't describe this situation as a closed system (because there's an external force) but we can say its Hamiltonian is time invariant.
A1. No, there are time-invariant Hamiltonians which don't obey other spacetime symmetries, so don't describe closed systems! 
