  A Lagrangian formalism for variational second-order delay ordinary
differential equations (DODEs) is developed. The Noether operator identity for
a DODE is established, which relates the invariance of a Lagrangian function
with the appropriate variational equations and the conserved quantities. The
identity is used to formulate Noether-type theorems that give the first
integrals for DODE with symmetries. Relations between the invariance of the
variational second-order DODEs and the invariance of the Lagrangian functions
are also analyzed. Several examples illustrate the theoretical results.
