Using Griffith's formalism for constrained variational problems in one independent variable [Exterior differential systems and the calculus of variations. Progress in Mathematics, Vol. 25 Boston-Basel-Stuttgart: Birkhäuser (1983; Zbl 0512.49003)], the Delaunay variational problem defined by the arclength functional acting on the space of curves \(\gamma \in \mathbb{R}^3\) with constant torsion \(\tau =1\) is studied. A main characterization is that a biregular curve \(\gamma \) is a critical point of the Delaunay functional if and only if the associated binormal curve is an elastic curve in the 2D sphere \(S^2\). So, a presentation of elastic curves on \(S^2\) in terms of elliptic functions is included. An important conclusion of this paper is that the Euler-Lagrange system associated to the Delaunay problem is Liouville-integrable, in the sense that the symplectification of the characteristic vector field is a Liouville-integrable Hamiltonian system.