Delay ordinary differential equations: from Lagrangian approach to Hamiltonian approach

TL;DR

This paper develops a Hamiltonian framework for delay ordinary differential equations, establishing a delay Legendre transformation, invariance properties, and a Noether-type theorem to find conserved quantities.

Contribution

It introduces a Hamiltonian formulation for delay ODEs, linking it with Lagrangian approaches and deriving a Noether-type theorem for these equations.

Findings

Established a delay Hamiltonian operator identity.

Formulated a Noether-type theorem for delay Hamiltonian systems.

Provided examples illustrating theoretical results.

Abstract

The paper suggests a Hamiltonian formulation for delay ordinary differential equations (DODEs). Such equations are related to DODEs with a Lagrangian formulation via a delay analog of the Legendre transformation. The Hamiltonian delay operator identity is established. It states the relationship for the invariance of a delay Hamiltonian functional, appropriate delay variational equations, and their conserved quantities. The identity is used to formulate a Noether-type theorem, which provides first integrals for Hamiltonian DODEs with symmetries. The relationship between the invariance of the delay Hamiltonian functional and the invariance of the delay variational equations is also examined. Several examples illustrate the theoretical results.