Symmetry Approach to Integration of Ordinary Differential Equations with Retarded Argument

TL;DR

This paper reviews Lie group methods for delay ordinary differential equations, classifies invariant equations, constructs solutions, and explores variational principles and conservation laws for equations with delays.

Contribution

It provides a comprehensive classification of symmetries in delay ODEs and extends variational and Noether's theorem concepts to equations with delays, which is a novel approach.

Findings

Classification of invariant delay ODEs for first and second order.

Construction of invariant solutions using symmetries.

Extension of variational principles and Noether's theorem to delay equations.

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.