Equations of motion for general constrained systems in Lagrangian mechanics

Firdaus E Udwadia,Aaron D Schutte

doi:10.1007/s00707-009-0272-2

Abstract

This paper develops a new, simple, explicit equation of motion for general constrained mechanical systems that may have positive semi-definite mass matrices. This is done through the creation of an auxiliary mechanical system (derived from the actual system) that has a positive definite mass matrix and is subjected to the same set of constraints as the actual system. The acceleration of the actual system and the constraint force acting on it are then directly provided in closed form by the acceleration and the constraint force acting on the auxiliary system, which thus gives the equation of motion of the actual system. The results provide deeper insights into the fundamental character of constrained motion in general mechanical systems. The use of this new equation is illustrated through its application to the important and practical problem of finding the equation of motion for the rotational dynamics of a rigid body in terms of quaternions. This leads to a form for the equation describing rotational dynamics that has hereto been unavailable.

Highlights

Obtaining the explicit equation of motion for constrained mechanical systems with singular mass matrices has been a source of major difficulty in classical mechanics for some time
We show that under the conditions stipulated by Udwadia and Phohomsiri [2] for when the equation of motion of a constrained mechanical system becomes unique—a circumstance that must necessarily arise when one is modeling real-life physical systems in classical mechanics because the observed accelerations are always unique—a simpler set of equations can be obtained that again have the same form as the fundamental equation
Corollary 3 For the mechanical system D described in Result 4, which satisfies D’Alembert’s principle, as long as the positive semi-definite matrix M is such that Mhas rank n at each instant of time, the acceleration of the constrained system can be obtained explicitly by imagining that the matrix M is replaced by the matrix MA = M + A+ A, and using the fundamental equation

Summary

Introduction

Obtaining the explicit equation of motion for constrained mechanical systems with singular mass matrices has been a source of major difficulty in classical mechanics for some time now. We show that under the conditions stipulated by Udwadia and Phohomsiri [2] for when the equation of motion of a constrained mechanical system becomes unique—a circumstance that must necessarily arise when one is modeling real-life physical systems in classical mechanics because the observed accelerations are always unique—a simpler set of equations can be obtained that again have the same form as the fundamental equation. These equations are valid for systems whose mass matrices can be singular (positive semi-definite) and/or positive definite. An illustration of the use of our new equation is provided to this important application area of rotational dynamics, where we use a Lagrangian approach to directly and obtain a hereto unavailable form of the equations of rotational motion of a rigid body in terms of quaternions

System description and general constraints

1: For any mechanical system S

Result

Explicit equation for constraint force Qc

Application to rotational dynamics

Conclusions and remarks