Variational problems without having any non-trivial Lie variational symmetries

TL;DR

This paper explores variational problems lacking traditional Lie symmetries and introduces a novel approach using $$-symmetries, based on a horizontal one-form on jet space, to solve these problems.

Contribution

It presents a new class of symmetries ($$-symmetries) for solving variational problems without non-trivial Lie symmetries, expanding the methods available in the field.

Findings

Constructed variational problems without Lie symmetries.

Developed a solution method using $$-symmetries.

Demonstrated the application of $$-symmetries to specific problems.

Abstract

In this paper we construct variational problems without Lie non-trivial variational symmetry and solving them using new class of symmetries ($\mu$-symmetry) which introduced by Guiseppe Gaeta and Paola Morando (2004). The central object in this paper is horizontal one-form $\mu$ on first order jet space $J^1 M$.