 As discussed in Section 1.2.2, linear systems always obey the principle of linear superposition. This implies that the Hamiltonians for such systems are always separable. A separable Hamiltonian can always be transformed into a sum of separate Hamiltonians with one element in the sum corresponding to each subsystem. In effect, a separable system is one where the interactions among subsystems can be transformed away leaving the subsystems independent of each other. The whole is the sum of the parts, as it were. Chaos is impossible for separable Hamiltonians. For a nonlinear systems, by contrast, Hamiltonians are never separable. There are no transformation techniques that can turn a nonseparable Hamiltonian into the sum of separate Hamiltonians. In other words, the interactions in a nonlinear system cannot be decomposed into individual independent subsystems, nor can the whole system and its environment be ignored (Bishop 2010a). Nonseparable classical systems are the kinds of systems where chaotic behavior can manifest itself. So alternatively one could say that nonseparability of a Hamiltonian is a necessary condition for stretching and folding mechanisms and, hence, for chaos (e.g., Kronz 1998). 