On the Essence of Lagrange's Equations

TL;DR

This paper offers a new derivation of Lagrange's equations by linking energy conservation, the momentum theorem, and the kinetic energy theorem through the chain rule, providing deeper insight into their fundamental relationship.

Contribution

It presents a novel perspective that derives Lagrange's equations from energy conservation principles and the chain rule, clarifying their intrinsic connection.

Findings

Establishes the relationship between momentum and kinetic energy theorems.

Derives Lagrange's equations using differential forms and coordinate transformations.

Reveals energy conservation as the foundation for momentum conservation.

Abstract

From a new perspective, this paper rederives Lagrange's equations. By applying the chain rule of differentiation, the intrinsic relationship between the momentum theorem and the kinetic energy theorem is first established. Subsequently, expressing the differential form of energy conservation in an arbitrary coordinate system and performing suitable differential operations yields Lagrange's equations. Generalized forces and generalized displacements are shown to be component representations of forces and displacements in a chosen coordinate system. Consequently, the essence of Lagrange's equations is identified as the transformation of the kinetic energy theorem into the momentum theorem via the chain rule for composite functions, thereby revealing how energy conservation constructs momentum conservation.