This book offers a new theoretical perspective for optimal control theory that incorporates a synthesis of the calculus of variations and symplectic geometry. The author finds that control theoretic interpretations of variational problems lead directly to a new interpretation in terms of Hamiltonian systems.

In many respects this is a sequel of sorts to the author’s *Geometric Control Theory*. That earlier book focused on more applied questions of space control and the problems of steering a dynamical system from an initial state to a final state and finding the best path of transfer. The author found that the earlier book “made important contributions to mathematics beyond … the original intent”. His intention in the current book is to show how “the bias toward control theoretic interpretations of variational problems provides a direct path to Hamiltonian systems and reorients our understanding of Hamiltonian systems inherited from the classical calculus of variations in which the Euler-Lagrange equation was the focal point …”.

The first six chapters of the book develop the theoretical background. These begin with accessibility theory for control systems based on Lie theoretic methods and continue with discussions of Lie groups, homogeneous and symmetric spaces, symplectic and Poisson manifolds. An especially important aspect is the Maximum Principle; it states that each extremal trajectory of a control system is the projection of an integral curve of a certain Hamiltonian system on the cotangent bundle of the manifold on which the control system resides. It provides a necessary condition for a trajectory to be on the boundary of the reachable set.

The remaining chapters are directed to problems of geometry and applied mathematics that are amenable to the methods the author has described. They begin with a discussion of the author’s perspective on the Hamiltonian view of classical geometry and then go on symmetric spaces and sub-Riemannian problems. A few applications tending a little bit more toward the concrete are considered as well. One of them looks at the equilibrium configurations of an elastic rod subject to bending and twisting torques at its ends; another takes a new look at rigid body rotations and the motion of the heavy top.

The prerequisites for this book are pretty formidable. Although the author provides some limited background material, readers should be comfortable with Lie groups, Lie algebras and symplectic geometry in the context of the theory of Hamiltonian systems, and have at least a nodding appreciation of the basic ideas of control theory and the calculus of variations. Much of the discussion is at a very theoretical level. There are relatively few examples and no exercises.

Bill Satzer (bsatzer@gmail.com) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.