# Lagrangian formalism and Lie group approach for commutative semigroup of differential equations

## TL;DR

This paper explores how Lagrangian formalism and Lie group methods can generate and analyze a semigroup of second-order differential equations, highlighting their respective advantages and limitations in mathematical physics.

## Contribution

It introduces a novel semigroup structure for differential equations and compares two methods for deriving these equations, emphasizing the broader applicability of the Lagrangian formalism.

## Key findings

- Lagrangian formalism applies to all equations in the semigroup.
- Lie group approach is limited to a sub-semigroup.
- The methods have distinct advantages and disadvantages.

## Abstract

A set of linear second-order differential equations is converted into a semigroup, whose algebraic structure is used to generate many novel equations. Two independent methods that can be used to derive the equations of the semigroup are considered, namely, the Lagrangian formalism and the Lie group approach. The advantages and disadvantages of each method are discussed, and it is shown that the Lagrangian formalism can be established for all equations of the semigroup, however, the Lie group approach is only limited to a certain sub-semigroup . The obtained results are discussed in the context of their applications in mathematical physics.