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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 frompotential 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. Generalizedcoordinates 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 theexact 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 thedistance 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.

22

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 thehorizontal through the support as the datum for gravitational potential energy,

k V= -mgr cos O + ~ (r_a)2
SO

= T-

V =7 m

[t:2+

(rb)2

]+mgrcosO--~
_

k (r - a) 2

Applying Lagrange's equation with ql = r we have

07~ OF
SO

=

mr:

-aT
and

0fs ar = m r

02

+ m g cos O -

k(r

-

a)

9

From equation (2.1)
d--r -bTmi-

mr

6

-

m g cos O + k ( r-

a) = 0

(i)

The generalized force Q~ = 0 because there is no externally applied radial force that is not included in V.

Generalized co-ordinates

23

Taking 0 as the next generalized co-ordinate
O~ = mr20

ao
SO

---d (8-~-~l : 2mr1:0+ mr~O
dt \ O0 ]

and
aTL = mgr sin 0 80

Thus the equation of motion in 0 is d--}= Q0 = 0 (ii)

2mreO + mr20 - mgr sin 0 = 0

Thegeneralized force in this ease would be a torque b e c a u s e the c o r r e s p o n d i n g generalized co-ordinate is an angle. Generalized forces will be discussed later in m o r e detail. Dividing equation (ii) by r gives
2meO + mrO - mg sin 0 = 0

(iia)

and rearranging equations (i) and (ii)leads to
mg cos O - k ( r a)= m(i;-

r02)

(ia)

and
- m g sin 0 = m(2el~ + r(~)

(iib)which are the equations obtained directly from Newton's laws plus a knowledge of the c o m p o n e n t s of acceleration in polar co-ordinates. In this example there is not much saving of labour except that there is no requirement to know the components of acceleration, only the components of velocity.

2.2

Generalized co-ordinates

A set of generalized co-ordinates is one in which eachco-ordinate is independent and the number of co-ordinates is just sufficient to specify completely the configuration of the system. A system of N particles, each free to move in a three-dimensional space, will require 3N coordinates to specify the configuration. If Cartesian co-ordinates are used then the set could be
{xl Yl zl x2 Y2
or {Xl X2 X3 X4 X 5 X 6 9 9 .
Z2" " " XN

YN ZN}

Xn_2...
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