=y , (1.30)

‚y dx ‚y

and this is zero if the Euler-Lagrange equation is satis¬ed. The quantity

‚f

I =f ’y (1.31)

‚y

is thus a ¬rst integral of the Euler-Lagrange equation. In the soap-¬lm case

y(y )2

‚f y

= y 1 + (y )2 ’

f ’y = . (1.32)

‚y )2 )2

1 + (y 1 + (y

When there are a number of dependent variable yi , so that we have

J[y1 , y2 , . . . yn ] = dxf (y1 , y2 , . . . yn ; y1 , y2 , . . . yn ) (1.33)

then the ¬rst integral becomes

‚f

I=f’ yi . (1.34)

‚yi

i

Again

dI d ‚f

f’

= y

dx dx ‚yi

i

‚f ‚f ‚f d ‚f

’ yi ’ yi

= yi + yi

‚yi ‚yi ‚yi dx ‚yi

i

‚f d ‚f

’

= yi , (1.35)

‚yi dx ‚yi

i

and this zero if the Euler-Lagrange equation is satis¬ed for each yi .

Note that there is only one ¬rst integral, no matter how many y™s there

are.

1.3 Lagrangian Mechanics

In his M´canique Analytique (1788) Joseph-Louis de La Grange, following

e

d™Alembert (1742) and Maupertuis (1744), showed that most of classical

10 CHAPTER 1. CALCULUS OF VARIATIONS

mechanics can be recast as a variational principle: the principle of least

action. The idea is to introduce the Lagrangian function L = T ’ V where T

is the kinetic energy of the system and V the potential energy, both expressed

in terms of generalized coordinates q i and their time derivatives qi . Then

™

Lagrange showed that the multitude of Newton™s F = ma equations, one for

each particle in the system, could be reduced to

d ‚L ‚L

’ = 0, (1.36)

‚ qi ‚q i

dt ™

one equation for each generalized coordinate q. Quite remarkably ” given

that Lagrange™s derivation contains no mention of maxima or minima ” we

observe that this is the precisely the condition that the action integral

tf inal

i

L(q i ; q ) dt

S= (1.37)

tinitial

be stationary with respect to variations of the trajectory q i (t) which leave the

initial and ¬nal points ¬xed. This fact so impressed its discoverers that they

believed they had uncovered the unifying principle of the universe. Mauper-

tuis, for one, tried to base a proof of the existence of God on it. Today the

action integral, through its starring role in the Feynman path integral for-

mulation of quantum mechanics, remains at the heart of theoretical physics.

1.3.1 One Degree of Freedom

We will not attempt to derive Lagrange from Newton and D™Alembert™s

extension of the principle of virtual work “ leaving this task to a mechanics

course ” but will satisfy ourselves with some examples which illustrate the

computational advantages of Lagrange™s approach, as well as a subtle pitfall.

Example: Atwood™s Machine. This device, invented in 1784 but still a fa-

miliar sight in undergraduate laboratories, is used to demonstrate Newton™s

laws of motion and to measure g. It consists of two weights connected by a

light string which passes over a light and frictionless pulley.

1.3. LAGRANGIAN MECHANICS 11

x1 g

T

x

T 2

m1

m2

The elementary approach is to write an equation of motion for each of the

two weights

m1 x1 = m1 g ’ T,

¨

m2 x2 = m2 g ’ T.

¨ (1.38)

We then take into account the constraint x1 = ’x2 to get

™ ™

m1 x1 = m1 g ’ T,

¨

’m2 x1 = m2 g ’ T.

¨ (1.39)

Finally we eliminate the constraint force, the tension T , to get the accelera-

tion

(m1 + m2 )¨1 = (m1 ’ m2 )g.

x (1.40)

The Lagrangian solution takes the constraint into account from the very

beginning by introducing a single generalized coordinate q = x1 = ’x2 , and

writing

1

L = T ’ V = (m1 + m2 )q 2 ’ (m2 ’ m1 )gq.

™ (1.41)

2

From this we obtain a single equation of motion

d ‚L ‚L

’ ’ (m1 + m2 )¨ = (m1 ’ m2 )g.

=0 q (1.42)

‚ qi ‚q i

dt ™

12 CHAPTER 1. CALCULUS OF VARIATIONS