Combining Symmetries and Helmholtz's Conditions to Construct Lagrangians

TL;DR

This paper introduces methods to construct Lagrangians with desired symmetries by combining Noether's identities, Helmholtz's conditions, and inverse problem techniques, demonstrated through illustrative examples.

Contribution

It develops two new approaches to incorporate symmetry constraints directly into the inverse problem of mechanics, enhancing Lagrangian construction methods.

Findings

Derived new relations from Noether's identity linking Hessian, symmetries, and constants of motion.

Developed two methods to embed symmetry requirements into Lagrangian inverse problems.

Validated methods with one- and two-dimensional mechanical examples.

Abstract

We present new relations derived from Noether's identity that reveal the compatibility between the components of the Hessian matrix of the Lagrangian, the infinitesimal symmetry transformation of the configuration variables and time, and a constant of motion. Using these relations, we develop two new methods to incorporate symmetry requirements directly into the inverse problem of mechanics, thereby restricting the set of acceptable Lagrangians. We accomplish this by combining these relations with Helmholtz's conditions, which allow us to construct Lagrangians whose actions exhibit specific symmetries from the outset. The theory is illustrated with one- and two-dimensional examples.