## The geometry of the maximum principle for affine connection control systems

##### Abstract

The maximum principle of Pontryagin is applied to systems where the drift
vector field is the geodesic spray corresponding to an affine connection.
The result is a second-order differential equation whose right-hand side is
the ``adjoint Jacobi equation.'' This latter relates the Hamiltonian
geometry of the maximum principle to the affine differential geometry of the
system's affine connection. The resulting version of the maximum principle
is then be applied to the situation where the cost function is the norm
squared of the input force.