Symmetry Approach to Integration of Ordinary Differential Equations with Retarded Argument
Vladimir Dorodnitsyn, Roman Kozlov, Sergey Meleshko
TL;DR
This paper surveys symmetry-based methods for integrating ODEs with retarded arguments, focusing on Lie group classifications for DODEs with one delay and extending to two-delay variational frameworks. It develops both Lagrangian and Hamiltonian formalisms for delay problems, including a delay Legendre transformation and Noether-type theorems that yield first integrals and invariant solutions. Through concrete examples such as delay oscillators, it demonstrates how invariants and reduced equations enable recursive or explicit solution construction. The work identifies open challenges, including full two-delay classifications and extensions of direct methods, and situates delayed dynamics within a unified symmetry-variational framework with practical implications for nonlinear systems with memory.
Abstract
We review studies on the application of Lie group methods to delay ordinary differential equations (DODEs). For first- and second-order DODEs with a single delay parameter that depends on independent and dependent variables, the group classifications are performed. Classes of invariant DODEs for each Lie subgroup are written out. The symmetries allow us to construct invariant solutions to such equations. The application of variational methods to functionals with one delay yields DODEs with two delays. The Lagrangian and Hamiltonian approaches are reviewed. The delay analog of the Legendre transformation, which relates the Lagrangian and Hamiltonian approaches, is also analysed. Noether-type operator identities relate the invariance of delay functionals with the appropriate variational equations and their conserved quantities. These identities are used to formulate Noether-type theorems that give first integrals of second-order DODEs with symmetries. Finally, several open problems are formulated in the Conclusion.