Integrating Hamiltonian systems defined on the Lie groups SO(4) and SO(1,3)

In this paper we study constrained optimal control problems on semi-simple Lie groups. These constrained optimal control problems include Riemannian, sub-Riemannian, elastic and mechanical problems. We begin by lifting these problems, through the Maximum Principle, to their associated Hamiltonian formalism. As the base manifold is a Lie group G the cotangent bundle is realized as the direct product G×g* where g* is the dual of the Lie algebra g of G. The solutions to these Hamiltonian vector fields l ∈ g*, are called extremal curves and the projections g(t) ∈ G are the corresponding optimal solutions. The main contribution of this paper is a method for deriving explicit expressions relating the extremal curves l ∈ g* to the optimal solutions g(t) ∈ G for the special cases of the Lie groups SO(4) and SO(1,3). This method uses the double cover property of these Lie groups to decouple them into lower dimensional systems. These lower dimensional systems are then solved in terms of the extremals using a coordinate representation and the systems dynamic constraints. This illustrates that the optimal solutions g(t) ∈ G are explicitly dependent on the extremal curves.