Nonseparable Hamiltonian

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I’m trying to decipher this particular passage from a paper.

Because the direct sum is used in classical mechanics to define the states of a composite system in terms of its components, rather than the tensor product operation as in quantum mechanics, there are no nonseparable states in classical mechanics. There are nonseparable Hamiltonians in classical mechanics – the Hamiltonian corresponds to the total energy of the system and is related to the time evolution of the system. This type of nonseparability is the result of nonlinear terms in the equations of motion. Perhaps a kind of emergence can be associated with it. Some measure of plausibility is given to this claim since a classical system can exhibit chaotic behavior only if its Hamiltonian is nonseparable.
Ref: FM Kronz, JT Tiehen - Philosophy of Science, 2002 “Emergence and Quantum Mechanics”

Can you provide an example of a physical system which corresponds to a classical system with a nonseparable Hamiltonian (ex: the Milky Way galaxy, flow of water)? What measurable characteristics (ex: motion or potential/kinetic energy of planets, local velocity of fluid) is Kronz refering to when he claims such things are nonseparable? What equations are not separable and what are the equations describing?
 

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