Adapting the Euler-Lagrange equation to study one-dimensional motions under the action of a constant force

TL;DR

This paper introduces a simplified approach using mean delta operators to solve the Euler-Lagrange equations for one-dimensional motions under constant forces, facilitating teaching and understanding of classical mechanics.

Contribution

It proposes a new simplified method with mean delta operators to solve Euler-Lagrange equations, enhancing pedagogical approaches in classical mechanics.

Findings

Successfully applied to free fall, Atwood's machine, and inclined plane problems

Simplifies the solution process for Euler-Lagrange equations in basic mechanics

Potentially useful for introductory physics education

Abstract

The Euler-Lagrange equations (EL) are very important in the theoretical description of several physical systems. In this work we have used a simplified form of EL to study one-dimensional motions under the action of a constant force. From using the definition of partial derivative, we have proposed two operators, here called \textit{mean delta operators}, which may be used to solve the EL in a simplest way. We have applied this simplification to solve three simple mechanical problems under the action of the gravitational field: a free fall body, the Atwood's machine and the inclined plan. The proposed simplification can be used to introducing the lagrangian formalism to teach classical mechanics in introductory physics courses.