3 - Hamilton's Principle

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Hami ton’s Pr ncip Ie

3.1 Introduction

In the previous chapters the equations of m otion have been presented as differential equa-tions. In this chapter we shall express the equations in the form of stationary values of a timeintegral. The idea of zero variation of a quantity was seen in the method of virtual work andextended to dynamics by means of D’Alembert’s p rinciple. It has long been considered thatnature works so as to m inimize some quantity often ca lled action. One of the first statementswas made by Maupertuis in 1744. The most commonly used form is that devised by SirWilliam Rowan Ham ilton around 1834.

Hamilton’s principle could be considered to be a basic statement of m echanics, especiallyas it has wide applications in other areas of physics, but we shall develop the principledirectly from Newtonian laws. For the case with conservative forces the principle states thatthe time integral of the Lagrangian is stationary with respect to variations in the ‘path’ inconfiguration space. That is, the correct displacement-time relationships give a minimum(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 ac tion integral. There are several different inte-grals which are also known as action integrals.

The calculus of variations has an interesting history with many applications but we shalldevelop only the techniques necessary for the problem in hand.



Derivation of Hamilton Sprinciple 47

3.2 Derivation of Hamilton‘s principle

Consider a single particle acted upon by non-conservative forces F,, F,,Fk nd conservativeforcesf;,J , c which are derivable from a position-dependent potential function. Referring toFig. 3.1 we see that, with p designating mom entum, in the x direction

dF , + f ; =z PI)

with similar expressions for the y and z directions.For a system having N particles D’Alembert’s principle gives

F, +f; - dt (p,) 6xl = 0, 1 5 i S 3N?( d l

1 ?(l +f; ;il

lP 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

summation and integration and then integrating the third term we obtainx1

t 2 I2 d(3.3)

t 2 t2 a v

tl t, axl tl tl

We now impose a restriction on the variation such that it is zero at the extrem e points t , andtz; therefore the third term in the above equation vanishes. Reversing the order o f 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 thenby definition

Fig. 3.1



48 Hamilton 'sprinciple

*

Fig. 3.2

dP ipi=-dt

so the work done over an elemental displacement is

d p .P,&; = -' dr, =xidpidt

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

T = $ x i d p iLet 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)dtti I t , I t , ; t,

1:

(3.5)

where 6 W is the virtual work done by non-conservative forces. This is Ham ilton 'sprinciple.If mom entum is a linear function of velocity then T* = T. It is seen in section 3.4 that thequantity (T* - V) is in fact the Lagrangian.

If all the forces are derivable from potential functions then Hamilton's principle reducesto

6 X d t = O (3.6)

All the comm ents made in the previous chapter regarding generalized co srd ina tes applyequally well here so that Z is independent of the co-ordinate system.



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 ofx 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 kc t ion of time except thatit 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 +d x ) = 6x +d(6x)

Therefore 6 (dr) = d(6x) and dividing by dt gives

d x d

dt dt6 - =- 6x )

m i 2 kx2

(3-7)

For the problem at hand the Lagrangian is

E = - - -2 2

Fig. 3.4



50 Hamilton S rinciple

Thus the integral to be minimized is

The varied integral with x replaced by f =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 for any value oft , that is

m , f + k x = 0 (3 -9)

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

k2

S I t , (;X2 - T ~ 2 )r = 0

Making use of equation (3.7), equation (3.10) becomes

P

Again, integrating by parts,

h x 1 - t:mi 6x dt - kx 6x dt = 04

(3.10)



Lagrange 3 equations deri ve d jk m HamiltonS principle 5 1

or

- t : (m2 + la) 6x dt = 0

and because 6r s arbitrary it follows tha t

& + 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 thegeneralized co -ordinates, the generalized velocities and time, that is P =P qi,qi t).Thuswith

t 2

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 canconsider the case for all q, to be zero excep t 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 i = T*

- V because, with reference to Fig. 3.2, it is the variation of co-kinetic energy which isrelated to the momentum. But, as already stated, when the m omentum is a linear functionof velocity the co-kinetic energy T* = T , he kinetic energy. The use o f co-kinetic energy



52 Hamilton S rinciple

becomes important when particle speeds approach that of light and the non-linearitybecomes apparent.

3.5 Illustrative example

One of the areas in which Hamilton's principle is useful is that of continuous media wherethe number of degrees of freedom is infinite. In particular it is helpful in complex problemsfor which approximate solutions are sought, because approximations in energy terms areoften easier to see than they are in compatibility requirements.As an example we shall lookat wave motion in long strings under tension. The free-body

diagram approach requires assumptions to be made in order that a simple equation of motionis generated; whilst the same is true for this treatment the implications of the assumptionsare clearer.

Figure3.5 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-forcespring at one end and assuming that aulax is small. In practice the elasticity of the string andits 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 thestring 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 ofthe 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



Illustrative example 53

The poten tial energy is -T (-s) = TS giving

(3.14)

I f u is also a function of time then du ldx will be replaced by duldx.

Ifp is

the density and a is the cross-sectional area of the string then the kine tic energy is

The L agrangian is

‘E = J-r =0

According to Ham ilton’s principle we need to find the conditions so that

t2 r = L

6 1 , r = Of [ ( g ) 2 - L ( 2 ) 2 ]axd 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 byParts

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

Integrating by parts gives

(3.19)



54 Hamilton S principle

(3.20)

The first term is zero provided that the e m U re passive, that is no energy is being fed intothe 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 thatmakes 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

- aZu

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 ofbending stiffness of the wire, are to be considered. Extra energy terms can be added to theabove treatment without the need to rework the whole problem. This fact will be exploitedin Chapter 6 which discusses wave motion in more detail.

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