Dinamica

2
Lagrange's Equations

2.1

Introduction

The dynamical equations of J.L. Lagrange were published in the eighteenth century some one hundred years after Newton's Principia. They represent a powerful alternative to the Newton--Euler equations and are particularly useful for systems having many degrees of freedom and are even more advantageous when most of the forces are derivable from potential functions. The equations are d 0~ 0~ ai 1 < i < n (2.1)
N

where ~. is the Lagrangian defined to be T-V, T is the kinetic energy (relative to inertial axes), V is the potential energy, n is the number of degrees of freedom, q~ to q, are the generalized co-ordinates, Q~ to Q~ are the generalized forces and d/dt means differentiation of the scalar terms with respect to time. Generalized coordinates and generalized forces are described below. Partial differentiation with respect to tji is carried out assuming that all the other cj, all the q and time are held fixed. Similarly for differentiation with respect to qg all the other q, all tj and time are held fixed. We shall proceed to prove the above equations, starting from Newton's laws and D'Alembert's principle, during which the exact meaning of the definitions and statements will be illuminated. But prior to this a simple application will show the ease of use. EXAMPLE

A mass is suspended from a point by a spring of natural length a and stiffness k, as shown in Fig. 2.1. The mass is constrained to move in a vertical plane in which the gravitational field strength is g. Determine the equations of motion in terms of the distance r from the support to the mass and the angle 0 which is the angle the spring makes with the vertical through the support point.

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Lagrange's equations

Fig. 2.1
The system has two degrees of freedom and r and O, which are independent, can serve as generalized co-ordinates. The expression for kinetic energy is

T = ~ - m [t:2 + (rOli and for potential energy, taking the