This paper presents two viewpoints of optimal systems that are alternatives to the familiar Eulor-Lngrange method. In the first the use of mode theory aids the conceptual understanding of an optimal system by expressing the conditions of optimality in terms of controllability and observability. In the second the conditions and associated numerical techniques are derived from a viewpoint of mathematical physics by the use of adjoint operators ; the results are approached directly rather than indirectly through multiplier functions. The results are well known, but these two different approaches contribute to a better understanding of the subject.