This is the final chapter! At the same time, it is the beginning of a new geometric adventure into dimension four— the hyperspace. Arguably, the most natural differential equations residing in dimension four are the Hamiltonian systems with two degrees of freedom. Hence, we have chosen them as the subject of this chapter. Following a rapid introduction to the setting of Hamiltonian systems, we outline a topological program for the study of a small class of Hamiltonians—completely integrable systems—that can be analyzed successfully. From this contemporary viewpoint, we then study the flow of a pair of linear harmonic oscillators. Here, the term bifurcation gains yet another meaning in the context of level sets of the energy-momentum mapping. Our success with completely integrable systems is somewhat overshadowed by their rarity. Indeed, a satisfactory analysis of a general Hamiltonian system in four dimensions—unlike the case of the plane, one degree of freedom—is currently beyond reach. To hint at this complexity, we conclude the chapter with an example of a Hamiltonian that, in all likelihood, is nonintegrable.
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