Summary: Any left-invariant optimal control problem (with quadratic cost) can be lifted, via the celebrated Maximum Principle, to a Hamiltonian system on the dual of the Lie algebra of the underlying state space \(\mathbf G\). The (minus) Lie-Poisson structure on the dual space \(\mathfrak g^*\) is used to describe the (normal) extremal curves. As an illustration, a typical left-invariant optimal control problem on the rotation group \(\mathbf{SO}(3)\) is investigated. The reduced Hamilton equations associated with an extremal curve are derived and then explicitly integrated by Jacobi elliptic functions.