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Ha

m

t

o

ns

Pr

nci

p

I

e

3.1

Introduction

In

the previous chapters the equations of motion have been presented

as

differential equa-

tions.

In

this chapter we shall express the equations in the form of stationary values of

a

time

integral. The idea of zero variation of a quantity was seen in the method of virtual work and

extended to dynamics by means of DAlemberts p rinciple. It has long been considered that

nature works

so as

to minimize some quantity often called action. One of the first statements

was made by Maupertuis in 1744. The most commonly used form is that devised by Sir

William Rowan Ham ilton around 1834.

Ham iltons principle could be considered to be a basic statement of mechanics, especially

as it has wide applications in other areas of physics, but we shall develop the principle

directly from Newtonian laws. For the case with conservative forces the principle states that

the time integral of the Lagrangian is stationary with respect to variations in the path in

configuration space. That is, the correct displacement-time relationships give a m inimum

(or maximum ) value of the integral.

In the usual notation

61;.

dt

= 0

or

61 = 0

where

This integral is som etimes referred to

as

the action integral. There are several different inte-

grals which are also

known

as ac tion integrals.

The calculus of variations has an interesting history with many applications but we shall

develop only the techniques necessary for the problem in hand.

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Derivation

of

Hamilton Sprinciple

47

3.2

Derivation

of

Hamiltons principle

Consider a single particle acted upon by non-conservative forces

F,, F

Fk nd conservative

forcesf;, J , which are derivable from a position-dependent potential function. Referring to

Fig.

3.1

we see that, with p designating momentum, in the x direction

d

F ,

+ f ;

= PI)

with similar expressions for the y and z directions.

For a system having

N

particles DAlemberts principle gives

F, f; t p,)

6xl =

0,

1 5

i S

3N

?(

l

1

?(

l f;

;il

l

P I )

6 4

dt

=

0

?( Fl dt

-

1

t

-

[Pl6X11+ 1

PI)

6x1) dt ) = 0

1 E

16xl-

6V +E1 6 x , ) dt

= 0

We

may

now integrate this expression over the time interval t , to t2

Nowf; = -

av

and the third term can be integrated by parts.

So

interchanging the order of

summ ation and integration and then integrating the third term we obtain

x1

t 2 I2 d

3.3)

t 2 t2 av

tl

t, axl

tl tl

We now impose a restriction on the variation such that

it

is zero at the extreme points

t ,

and

tz;

therefore the third term in the above equation vanishes. Reversing the order of summa-

tion and integration again, equation 3.3) becomes

3.4)

I 1

Let us assume that the momentum is a function ofvelocity but not necessarily a

lin-

ear one. With reference to Fig.

3.2

if P is the resultant force acting on a particle then

by definition

Fig. 3.1

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4 8

Hamilton

s

principle

Fig. 3.2

dP i

pi =

dt

so the work done over an elemental displacement is

d p .

P, ; = - dr = xidpi

dt

The kinetic energy of the particle is equal to the work done,

so

T

=

$ x i d p i

Let the complementary kinetic energy, or co-kinetic energy, be defmed by

Tc = Jp,&

It follows that

6P

= pi6& so substitution into equation 3.4) leads to

1 6 T *

-

V) +?j 6 x j ) dt = 0

or

(T* V) dt = - (Z F ; S x , )d t = 6 1'2(-W)dt

ti

I t , I t ,

;

t,

1:

3.5)

where 6

W

is the virtual work done by non-conservative forces. This is Hamilton sprinciple.

If mom entum is

a

linear function of velocity then

T* =

T. It is seen

in

section

3.4

that the

quantity T* V) is in fact the Lagrangian.

If all the forces are derivable from potential functions then Ham ilton's p rinciple reduces

to

6 X d t = O 3.6)

All the comments made in the previous chapter regarding generalized cosrdinates apply

equally well here

so

that Z is independent of the co-ordinate system.

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Application

of

HamiltonS rinciple 49

3.3 Application

of

Hamilton s principle

In

order to establish a general method for seeking a stationary value of the action integral

we shall consider the simple madspring system with a single degree of freedom shown in

Fig. 3.3. Figure 3.4 shows a plot o fx versus t between two arbitrary times. The solid line is

the actual plot, or path, and the dashed line is a varied path. The difference between the two

paths is

6x.

This is made equal to Eq t), where

q

is an arbitrary k c t i o n

of

time except that

it is zero at the extremes. The factor E is such that when it equals zero the two paths coin-

cide. We can establish the conditions for a stationary value of

the

integralI by setting dlldc

= 0

andthenputtingE=O.

From Fig. 3.4we see that

6

x + dx) = 6x + d 6x)

Therefore

6 dr)

= d 6x) and dividing by dt gives

d x d

dt dt

6 -

=

6x)

m i 2 kx2

3

-7)

For the problem at hand the Lagrangian is

E =

2 2

Fig.

3.4

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50 Hamilton

S

rinciple

Thus

the

integral to be minim ized is

The varied integral with x replaced by =

x

+ ~q is

+ ET^) x + ~ q i t

2

Therefore

Integrating the first term in the integral by

parts

gives

By the definition of

q

the first term vanishes on account of

q

being zero at t and at t2,

so

P

12

Now q is an arbitrary fimction of time and can be chosen to be zero except for time = t

when it is non-zero. This means that the term in parentheses must be zero fo r any value of

t , that

is

m , f + k x =

0 3 -9

A quicker method, now that the exact meaning of variation is

known,

is as follows

k

2

S I t , ;X2 T ~ 2 )

r

= 0

Making use of equation 3.7), equation

3.10)

becomes

Again, integrating by parts,

h x

-

t mi 6x dt kx 6x dt = 0

4

(3.10)

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Lagrange

3

equations d er iv ed jk m Hamilton S principle

5

or

-

t : m2

+ la

6x dt = 0

and because

6r

s arbitrary

it

follows that

+ l a =

0

3.1 1

3.4 Lagrange s equations derived from Hamilton s principle

For a system having n degrees of freedom the Lagrangian can be expressed in terms of the

generalized co-ordinates, the generalized velocities and time, that is

P =

P qi

,qi

t .Thus

with

t

tl

I = / X d t

(3.12)

we have

Note that there is no partial differentiation with respect to time since the variation applies

only to the co-ordinates and their derivatives. Because the variations are arbitrary we can

consider the case for all q to be zero except for

q .

Thus

Integrating the second term by parts gives

Because

6qj =

0 at t , and at

t2

Owing to the arbitrary nature of 6qj we have

3.13)

These are Lagrange's equations for conservative systems. It should be noted that = T*

- V

because, with reference to Fig.

3.2,

it is the variation

of

co-kinetic energy which is

related to the momentum. But, as already stated, when the momentum is a linear function

of velocity the co-kinetic energy T* = T , he k inetic energy. The use of co-kinetic energy

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52 Hamilton

S rinciple

becomes important when particle speeds approach that of light and the non-linearity

becomes apparent.

3.5 Illustrative example

One of the areas in which Hamilton s principle is useful is that of continuous media where

the number of degrees of freedom is infinite.

In

particular it is helpful in complex problems

for which approximate solutions are sought, because approximations in energy terms are

often easier to see than they are in compatibility requirements.

As

an

example we shall

look

at wave motion in long strings under tension. The free-body

diagram approach requires assumptions to be made in order that a simple equation of motion

is generated; whilst the same is true for this treatment the implications of the assumptions

are

clearer.

Figure 3.5 shows a string of finite length. We assume that the stretching of the string is neg-

ligible and that no energy is stored owing to bending. We further assume that the tension T in

the string remains constant.

This

can be arranged by having a pre-tensioned constant-force

spring at one end and assuming that aulax is small. In practice the elasticity of the string and

its

supports is such that for small deviations the tension remains sensibly constant.

We need an expression for the potential energy of the string in a deformed state.

If

the

string is deflected from the straight line then point B will move to the left. Thus the neg-

ative of the work done by the tensile force at B will be the change in potential energy of

the system.

The length of the deformed string is

If we assume that the slope dddx is small then

1

= O

For small deflectionss Q

L so the upper limit can be taken

as

L. Thus

r

Fig. 3.5

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Illustrative example 53

The po tential energy is

-T -s)

=

TS

giving

(3.14)

I f

u is also a function of time then duldx will be replaced by

duldx.

If

p

is

the density and

a

is the cross-sectional area

of

the string then the kinetic energy is

The Lagrangian is

E = J-

r

= 0

According to Hamiltons principle we need to find the conditions so that

t2

r = L

6 1 ,

r = Of [ g ) 2 - L 2 ) 2 ]ax

d x d t = O

Carrying out the variation

t 2 = L

s . L o [

( & ) 6 ( ) - T ( $ ) 6 ( $ ) p d 2 = O

(3.15)

(3.16)

(3.17)

(3.18)

To keep the process as clear as possible we will consider the two terms separately. For the

first

term the order

of

integration is reversed and then the time integral will be integrated by

Parts

because 6u = 0 at t, and t2. The second term in equation (3.18) is

Integrating by parts gives

(3.19)

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54 Hamilton

S

principle

3.20)

The first term is zero provided that the em U re passive, that is no energy is being fed into

the string after motion has been initiated. This means that either

6u = 0

or du/dx

= 0

at

each end. The specification of the problem indicated that 6u

=

0

but any condition that

makes energy transfer zero at the extremes excludes the first term.

Combining equations

3.19)

and 3.20) and substituting into equation 3.18) yields

and because 6u is arbitrary the integrand must s u m o zero so that finally

-

a u

p a , , ,

- T s

3.21)

This is the well-known wave equation for strings. It is readily obtained from free-body dia-

gram

methods but this approach is much easier to modify if other effects, such as that of

bending stiffness of the wire, are to be considered. Extra energy terms can be added to the

above treatment without the need to rework the w hole problem. This fact will be exploited

in Chapter

6

which d iscusses wave motion in more detail.