On the consistency of the Lagrange multiplier method in classical mechanics

TL;DR

This paper investigates the conditions under which the Lagrange multiplier method yields consistent constraint forces in classical mechanics with velocity-dependent constraints, ensuring reliable application of the method.

Contribution

It establishes the specific conditions on the Lagrangian that guarantee consistent computation of constraint forces using Lagrange multipliers.

Findings

Derived conditions for the Lagrangian to ensure force consistency

Clarified the independence of constraint forces from equations of motion

Enhanced understanding of velocity-dependent constraints in mechanics

Abstract

Problems involving rolling without slipping or no sideways skidding, to name a few, introduce velocity-dependent constraints that can be efficiently treated by the method of Lagrange multipliers in the Lagrangian formulation of the classical equations of motion. In doing so one finds, as a bonus, the constraint forces, which must be independent of the solution of the equations of motion, and can only depend on the generalized coordinates and velocities, as well as time. In this paper we establish the conditions the Lagrangian should obey that guarantee that the constraint forces can be obtained consistently.