EPILOGUE

Summary

In its treatment of dynamical systems, this book has emphasized several themes, among them nonlinearity, Hamiltonian dynamics, symplectic geometry, and field equations. The last section on the Hamiltonian treatment of nonlinear field equations unites these themes in bringing the book to an end. To arrive at this point we have traveled along the roads laid out by the Newtonian, Lagrangian, Hamiltonian, and Hamilton–Jacobi formalisms, eventually replacing the traditional view by a more modern one. On the way we have roamed through many byways, old and new, achieving a broad understanding of the terrain.

The end of the book, however, is not the end of the journey. The road goes on: there is much more terrain to cover. In particular, statistical physics and quantum mechanics, for both of which the Hamiltonian formalism is crucial, lie close to the road we have taken. These fields contain still unsolved problems bordering on the matters we have discussed, in particular questions about ergodic theory in statistical physics and about chaos in quantum mechanics. As usual in physics, the progress being made in such fields builds on previous work. In this case it depends on subject matter of the kind treated in this book.

Although they are deterministic, classical dynamical systems are generically so sensitive to initial conditions that their motion is all but unpredictable. Even inherently, therefore, classical mechanics still offers a rich potential for future discoveries and applications.