Lagrangian formalism and Noether-type theorems for second-order delay ordinary differential equations

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