  In this paper, we study simple splines on a Riemannian manifold $Q$ from the
point of view of the Pontryagin maximum principle (PMP) in optimal control
theory. The control problem consists in finding smooth curves matching two
given tangent vectors with the control being the curve's acceleration, while
minimizing a given cost functional. We focus on cubic splines (quadratic cost
function) and on time-minimal splines (constant cost function) under bounded
acceleration. We present a general strategy to solve for the optimal
hamiltonian within the PMP framework based on splitting the variables by means
of a linear connection. We write down the corresponding hamiltonian equations
in intrinsic form and study the corresponding hamiltonian dynamics in the case
$Q$ is the $2$-sphere. We also elaborate on possible applications, including
landmark cometrics in computational anatomy.
