Upon the completion of my book on geometric control theory, I realized that this subject matter, which was traditionally regarded as a domain of applied mathematics connected with the problems of engineering, made important contributions to mathematics beyond the boundaries of its original intent. The fundamental questions of space control, starting with the possibility of navigating a dynamical system from an initial state to a given final state, all the way to finding the best path of transfer, inspired an original theory of differential systems based on Lie theoretic methods, and the quest for the best path led to the Maximum Principle of optimality. This theory, apart from its relevance for the subject within which it was conceived, infuses the calculus of variations with new and fresh insights: controllability theory provides information about the existence of optimal solutions and the Maximum Principle leads to the solutions via the appropriate Hamiltonians. The new subject, a synthesis of the calculus of variations, modern symplectic geometry and control theory, provides a rich foundation indispensable for problems of applied mathematics.

This recognition forms the philosophical underpinning for the book. The bias towards 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 of the subject. This bias also reveals a much wider relevance of Hamiltonian systems for problems of geometry and applied mathematics than previously understood, and, at the same time, it offers a distinctive look at the theory of integrable Hamiltonian systems.

This book is inspired by several mathematical discoveries in the theory of integrable systems. The starting point was the discovery that the mathematical formalism initiated by G. Kirchhoff to model the equilibrium configurations of a thin elastic bar subjected to twisting and bending torques at its ends can be reformulated as an optimal control problem on the orthonormal frame bundle of ℝ3, with obvious generalizations to any Riemannian manifold.

Let us now return to the symmetric Riemannian pairs (G,K) and the Cartan decompositions. In the previous chapter we investigated the relevance of the left-invariant distributions with values in p for the structure of G and the associated quotient space G/H. In particular, we showed that the controllability assumption singled out Lie group pairs (G,K) in which the Lie algebraic conditions of Cartan took the strong form, namely,

In this chapter we will consider complementary variational problems on G defined by a positive-definite quadratic form Q(u, v) in k and an element. More precisely, we will consider the left-invariant affine distributions D(g) = {g(A + X) : X ∈ k} defined by an element A ∈ p. Each affine distribution defines a natural control problem in G,

with control functions u(t) taking values in.

We will be interested in the conditions on A that guarantee that any two points of G can be connected by a solution of (9.1), and secondly, we will be interested in the solutions of (9.1) which transfer an initial point g0 to a given terminal point g1 for which the energy functional is minimal. We will refer to this problem as the affine-quadratic problem.

In what follows, we will use ⟨u, v⟩ to denote the negative of the Killing form Kl(u, v) = Tr(ad(u) ∈ ad(v)) for any u and v in g. Since the Killing form is negative-definite on, the restriction of ⟨, ⟩ to is positive-definite, and can be used to define a bi-invariant metric on. This metric will be used as a bench mark for the affine-quadratic problems. For that reason we will express the quadratic form Q(u, v) as ⟨Q(u), v⟩ for some self-adjoint linear mapping Q on k which satisfies ⟨Q(u), u ⟩> 0 for all u = 0 in k. Then, Q = I yields the negative of the Killing form, i.e., the bi-invariant metric on.