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  • Print publication year: 2002
  • Online publication date: June 2012

Appendix: Lagrange's Equation

Summary

Introduction

When we wish to use Newton's Laws to write the equations of motion of a particle or a system of particles we must be careful to include all the forces of the system. The Lagrangean form of the equations of motion that we shall proceed to derive has the advantage that we can ignore all forces that do no work (e.g., forces at frictionless pins, forces at a point of rolling contact, forces at frictionless guides, and forces in inextensible connections). In the case of conservative systems (systems for which the total energy remains constant) the Lagrangean method gives us an automatic procedure for obtaining the equations of motion provided only that we can write the kinetic and potential energies of the system.

Degrees of Freedom

Before proceeding to develop the Lagrange equations we must characterize our dynamical systems on some systematic way. The most important property of this sort for our present purpose is the number of independent coordinates that we must know to completely specify the position or configuration of our system. We say that a system has n degrees of freedom if exactly n coordinates serve to completely define its configuration.

Example 1

A free particle in space has three degrees of freedom since we must know three coordinates, x, y, z, for example, to locate it.

Example 2

A wheel that rolls without slipping on a straight track has one degree of freedom since either the distance from some base point or the total angle of rotation will enable us to locate it completely.