Differential, integral, and variational delta-embeddings of Lagrangian systems

TL;DR

This paper introduces new delta-embeddings for Lagrangian systems that unify differential, integral, and variational approaches on arbitrary time scales, ensuring consistency with the least action principle.

Contribution

It presents the first coherent delta-embedding for the discrete calculus of variations compatible with the least action principle.

Findings

Integral and variational delta-embeddings coincide on any time scale.

A new coherent embedding for discrete calculus of variations is established.

The embedding maintains compatibility with the least action principle.

Abstract

We introduce the differential, integral, and variational delta-embeddings. We prove that the integral delta-embedding of the Euler-Lagrange equations and the variational delta-embedding coincide on an arbitrary time scale. In particular, a new coherent embedding for the discrete calculus of variations that is compatible with the least action principle is obtained.