Teodorescu P.P. Mechanical Systems, Volume III: Analytical Mechanics

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MECHANICAL SYSTEMS, CLASSICAL MODELS

662

The LEM equivalent (24.1.59') was obtained by differentiating (24.1.59) with

respect to

t and by replacing the derivatives

x from the non-linear system (24.1.58''').

The usual notation

(;D)

stands for the differential polynomial associated to

x(; )

ft . The second LEM equivalent (the system (24.1.59'')) is obtained from the first

one, by searching the unknown function

v in the class of analytic with respect to

functions

∞

∑

(; ) 1 ()

vt v t

ξ .

(24.1.60)

The LEM system (24.1.59'') may be also written in the matric form

∈

≡− = = =

VAVVVV

() , ( ) , ( )



S .

(24.1.60')

The LEM matrix

A has a special form, being always row-finite and, in the case of

polynomial operators, also column-finite

11 12 13 1, 1 1

22 23 2, 1 2

33 3, 1 3

() () () () ()

() () () ()

() () ()

()

ttt t t

tt t t

ttt

−

⎡

⎤

⎢

⎥

⎢

⎥

⎢

⎥

⎢

⎥

≡

⎢

⎥

⎢

⎥

⎢

⎥

⎢

⎥

⎢

⎥

⎣

⎦

000 0



 



 

AAA A A

AA A A

AAA

(24.1.60'')

The cells

A ()

t on the main diagonal are squares, of + 1s rows and columns, and are

generated by the coefficients of the linear part of the operator – namely, those

()

for which

=||1μ . The other cells

()

kk s

t contain only those ()

for which

=+|| 1sμ . More precisely, the diagonal cells contain the coefficients of the linear

part, on the next upper diagonal we find the coefficients of the second degree in

x etc.

In case of polynomial operators of degree

m , the associated LEM matrix is

band-diagonal the band being made up of

m lines.

It should be mentioned that this particular form of the LEM matrix allows the

calculus by block partitioning, which represents a considerable simplification.

Consider now for (24.1.58) or – equivalently – (24.1.58'''), the initial conditions

=∈xx

000

() ,ttI.

(24.1.61)

By LEM, they are transferred to

=∈

(;) e ,

vt 

ξξ,

(24.1.61')

Dynamical Systems. Catastrophes and Chaos

663

a condition that must be associated to the partial differential equation, and

=∈x

() ,vt

γ  ,

(24.1.61'')

indicating an initial condition for the system (24.1.59'') or – equivalently – (24.1.60').

For the matric form, the initial conditions (24.1.61'') become

()

∈

=Vx

()t

γ 

(24.1.61''')

Let us note that, in order to get back to the solutions of the polynomial Cauchy

problem (24.1.58'''), (24.1.61), the partial differential equation should be conveniently

defined on some space of analytic with respect to

functions, uniformly for ∈tI. To

this aim, we introduce

{

}

() : ;(;) () , , ,

IvI vt vt v KM

γγ

≥

≡×→ = ≤ ∈

∑

 A

(24.1.62)

where

i is the “sup” norm in

()CI and

{

}

()

max , 0,1,2,...,

ffjm is

the norm in

()

CI.

Another space may be similarly introduced

()

IB – the isomorphic with ()

space of infinite vectors

V , of components satisfying the same inequalities as in

(24.1.62). The isomorphism is emphasized by the application

() ():

tt→AB that

associates to

v the infinite vector of the coefficients in the development, i. e.

= V((; )) ()vt tτ

The relationships among the above introduced operators are suggestively explained

in the following diagram

:( ()) ( ())

:() ()

CI CI

→

↓

→

↓

→

LA A

SB B

We note that the above diagram is not closed; yet, it may be used to turn back to the

solutions of the polynomial system. In this respect, it was proved

Theorem 24.1.8 (I. Toma). Suppose that

∞

∈ ()

fCI

. Then the solution of the initial

problem (24.1.61''') formally allows the representation

=−VV

() ( ) ( )ttttΠ ,

(24.1.63)

MECHANICAL SYSTEMS, CLASSICAL MODELS

664

where the infinite matrix

Π is given by

≥

−

−=

∑

()

() ()

tt t

(24.1.63')

The matrices

()k

are determined by the recurrence

−

=+ =

AAAAE

(1)

() ( 1) 0

d()

() () (), ()

tttt

(24.1.63'')

where

E is the infinite unit matrix. The components v

of V are consistent on

intervals

, I

γ ∈  , centred at

, whose length depends on

, on γ and on

In particular, the first

n components of V coincide with the Taylor series

expansion of the solution of the Cauchy problem (24.1.58'''), (24.1.61) around

t .

The first

n rows of Π represent in fact the inverse of the non-linear operator F in

matrix form. Thus, the representation separates the contribution of the operator from

that of the initial data.

Let us mention that, as the series form cannot be completely computed, if we wish to

stop at some level

k , all the involved computation up to this level is finite.

In the case of constant coefficients, one may state

Theorem 24.1.9 (I. Toma). If the coefficients , 1,2,...,

fj n

= , are constant, then the

solution of the non-linear initial problem (24.1.58'''), (24.1.61)

(i) coincides with the first

n components of the infinite vector

(

() e

−

(24.1.64)

where the exponential matrix

() ()

−

−− −

=+ + ++ +

EA A A

00 0

() 2

e ... ...

1! 2! !

tt k

tt tt tt

(24.1.64')

can be computed by block partitioning, each step involving finite sums;

(ii) coincides with the series

∞

=+ =

∑∑

() () , 1,2,...,

jj j

xt x u tx j n

(24.1.64'')

where

()

are solutions of the finite linear ordinary differential system

TT T

... , 1,2,..., , ( ( )) ,

kk kkk

klut

=+++ = =

AU AU AU U

(24.1.64''')

Dynamical Systems. Catastrophes and Chaos

665

that satisfy the Cauchy conditions

( ) , ( ) , 2,3,...,

ttsl===0UeU

(24.1.64

)

The representation is very much alike a solution of a linear ordinary differential

system with constant coefficients. There is more: the computation is even easier due to

the fact that the eigenvalues of the diagonal cells are always known. The representation

(24.1.64'') is called the normal LEM representation and was used in many applications

requiring the qualitative behaviour of the solution.

Suppose now that the coefficients

of the non-linear operator are also

polynomials, of maximum degree

q , written in the form

=−= ∈

∑

( ) ( ) , 1,2,.., ,

ft ftt j n

μμ

μ  .

(24.1.65)

Then, the linear equivalent system becomes

∈

=+− +− ++−

⎡⎤

⎣⎦

AA A AV

VV V

00 0 0

()() ...() ,

() , () .

tt tt tt



(24.1.66)

Let us mention that in (24.1.66) the matrices

are all of them constant and,

obviously, of LEM construction. Each of the LEM matrices

is set up using only the

coefficients

. One can formally write (24.1.66) in integral form

]

000 0 0

() ()() ...() ()d

tutututuu= + +− +− ++−

⎡

⎣

∫

VV A A A AV

(24.1.66')

and apply to this linear integral equation the successive approximations

(0)

() ( 1)

() ,

() ( ) ( )d .

ljl

tutuu

−

⎡⎤

=+ −

⎢⎥

⎣⎦

∑

∫

VV AV

(24.1.66'')

With these preparations, using the same techniques as in Theorem 24.1.7, one can

obtain LEM representations in the case of polynomial coefficients.

The representation (24.1.63) is more suitable for numerical applications, while the

normal LEM representation suits better to study the qualitative behaviour of the solution.

24.1.2.5 One-Pass Numerical Methods of Calculation

The first one-pass method used has been the method of expansions into Taylor

series; in case of only one differential equation, for the Cauchy problem (24.1.37'),

(24.1.37''), we can write the formula (24.1.52) in the form

MECHANICAL SYSTEMS, CLASSICAL MODELS

666

=+ + ++

()

00 0 0

() () () () ... ()

1! 2! !

hh h

xt xt xt xt x t

 ,

(24.1.67)

where

=−

htt, while ∈ ()

xCI.

Retaining only the first two terms in (24.1.67), we can write

000

(; )xxhftx;

(24.1.68)

thus, from a geometric point of view, the integral curve

= ()xxt has been replaced by

its tangent

(; )tx . After obtaining

x , one deduces =+

21 11

(; )xxhftx a.s.o., as in

Fig. 24.10; the algorithm of this method, called Euler’s method, is

( ; ), , 0,1,2,..., 1

iii i

xxhftxhtti n

=+ = − = −.

(24.1.68')

Euler’s method is an explicit one-pass method. The error which is made contains the

truncation error (due to the truncation of the Taylor series) and the rounding error;

these two types of error appear at each step and, finally, they accumulate. The choice of

the step must be made carefully. Indeed, at a great number of steps, the truncation error

is decreasing, but the rounding error is increasing; the computation depends much on

the skill of the user.

Fig. 24.10 Euler’s numerical method

One cannot take many terms in a Taylor expansion because of the difficulties of

calculation. The Runge–Kutta method tries to pass over these difficulties, introducing a

function

h , which can be easily handled and calculated. We take thus

x of the form

∑

rj j

xx pk,

(24.1.69)

where

rp∈  are constants which must be determined, while =,1,2,...,

kj r, are

certain values of

(; )ftx , multiplied by h ; more precisely,

−

==+=+=

∑

( , ), , , 1,2,...,

jjjj jj jii

khf t h x kj rτξ τ α ξ β ,

(24.1.69')

Dynamical Systems. Catastrophes and Chaos

667

, , 1,2,..., 1, 1,2,...,

jji

ijjrαβ =−=, are constants and where we assume that

1111 1

0, 0, ,txαβ τ ξ====. We are thus led to

00 0 0

122211

33311322

4 4 41 1 42 2 43 3

( ; ), ( ; ),

(; ),

( ; ), ...

khftxkhft hx k

khft hx k k

khft hx k k k

αβ

αββ

αβββ

==++

=+ ++

=+ +++

(24.1.69'')

Hence, following constants

kli

k αβ can be calculated starting from the previous

ones; to do this, we expand

x and +

()xt h into Taylor series after h , and the

coefficients of the powers of

h must coincide, till a certain power, arbitrarily chosen.

The algebraic equations thus obtained, which must specify the constants to be

determined, are – in general – in a greater number that the number of these constants;

one can thus obtain several formulae of Runge–Kutta type for a given

r . We give now

the formulae which are obtained by this method, for various values of

r .

For

= 1r one obtains again the formula (24.1.68) of Euler’s method.

For

= 2r we can write

=+ + + +

000 0000

(; ) ( ; (; ))

x x ft x ft hx hft x

(24.1.70)

and, in general,

=+ + + + =

( ; ) ( ; ( ; )), 0,1,2,...,

iii iiii

xxftx fthxhftxi n

;

(24.1.70')

this algorithm is called the Euler’s modified method.

For

= 3r , it results

=+ + +

1123

(4 )

xx kkk

(24.1.71)

where

()

⎛⎞

==++

⎜⎟

⎝⎠

=+−+

(; ), ; ,

ii i i

khftxkhft x

khfthxk

(24.1.71')

For

= 4r one obtains two variants much used. In the Kutta–Simpson first variant

we can write

[]

=+ + + +

11234

2( )

xx kkkk,

(24.1.72)

with

MECHANICAL SYSTEMS, CLASSICAL MODELS

668

⎛⎞

==++

⎜⎟

⎝⎠

⎛⎞

=++ =++

⎜⎟

⎝⎠

343

(; ), ; ,

;,(;),

ii i i

ii ii

khftxkhft x

khft x khfthxk

(24.1.72')

while in the Kutta–Simpson second variant we have

[]

=+ + + +

11234

3( )

xx kkkk,

(24.1.73)

where

⎛⎞

==++

⎜⎟

⎝⎠

⎛⎞

=++− =++−+

⎜⎟

⎝⎠

324321

(; ), ; ,

;,(; ).

ii i i

ii ii

khftxkhft x

k hft xk k hfthxkkk

(24.1.73')

One can make also a study of the propagation of the error in the Runge–Kutta type

methods.

We notice that Euler’s method and Euler’s modified method can be used together,

being led to an implicit predictor–corrector algorithm. Thus, Euler’s method determines

the predictor value

(; )

iii

xxhftx,

(24.1.74)

the corrector value, which is an implicit one, being obtained by means of Euler’s

modified method

=+ + +

(; ) ( ; )

iii i

xxftx fthx

(24.1.74')

The method is iterative, so that one can obtain corrector values starting from a

previous one (which replaces the predictor value). We continue the calculation till the

difference between two successive corrector values is smaller, in absolute value, than

given

ε (hence, till we get a wanted approximation).

The methods presented above can be extended to the systems of non-linear

differential equations.

24.1.2.6 Perturbations Method

The fundamental problem of celestial mechanics is the problem of

n particles, for

which one obtains an exact solution only if

= 2n (see Sect. 11.1.2.8), putting thus in

evidence Kepler’s laws; but if a third particle appears (

= 3n ), e. g., the Moon besides

the Sun and the Earth or an interplanetary vehicle besides the Earth and the Moon etc.,

then one can no more obtain an exact solution. In this case, an important method of

calculation is the perturbations method.

Dynamical Systems. Catastrophes and Chaos

669

Let thus be the autonomous differential equation

=+ ∈(,) (,),xfxx gxxxε    ,

(24.1.75)

where

ε is a small parameter.

The linear particular equation

=− +xxε

(24.1.76)

leads to (with the initial condition

(0)xx)

()

−−

=+−

() e 1 e

xt x ε ,

(24.1.76')

while if

= 0ε , then we get

−

() e

xt x



, so that

()

−

−=−≤() () 1 e

xt xt εε



(24.1.76'')

the solution

()xt remaining in the neighbourhood of the solution ()xt



As well, the particular linear equation

=+xxε , (24.1.77)

leads, with the same initial condition, to

()

=+−

() e e 1

xt x ε

(24.1.77')

while if

= 0ε , then we obtain =

() e

xt x



, wherefrom

()

−=−() () e 1

xt xt ε



;

(24.1.77'')

the solution

()xt can differ very much from ()xt



if > ln2t , the difference tending to

infinite for

→∞t . Hence, in general, one cannot obtain the solution of the equation

(24.1.75) making

= 0ε

, neither even if the equation is linear.

Let us consider, in general, the system of non-linear autonomous differential

equations

=∈xfx x(,),





ε ,

(24.1.78)

where

ε is a small parameter; assuming that both the solution x()t and the function f

are analytical with respect to the parameter

ε , we use the expansions into power series

=+ + +xx x x

() () () () ...,tt t tεε

(24.1.79)

=+ + +fx f x f x f x

( , ) ( ) ( ) ( ) ...,εεε ,

(24.1.79')

writing the system (24.1.78) in the form

MECHANICAL SYSTEMS, CLASSICAL MODELS

670

−+ −+ −+xf xf x

11 22

( ) ( ) ...tεε

 

(24.1.78')

Equating to zero the coefficients of the powers of the small parameters, we obtain the

system of differential equations

===xfxxfxxfx

11 22

(), (), ()



,...;

(24.1.78'')

integrating this system, the searched solution is given by the expansion (24.1.79).

The problem is put to use a finite number of terms in the above expansion into series

so as to can obtain a solution with a sufficiently small error; indeed, one cannot always

truncate these expansions, because there can appear – in the solution – some terms

which tend to infinite for

→∞t . The terms are called secular terms in problems of

celestial mechanics, and one tries to eliminate them; the solution can be used, in this

case, only in a certain finite interval of time. Lindstedt succeeded to eliminate them in

various stages of calculation, searching periodic solutions and remaining, essentially, in

the frame of the perturbations method (see Sect. 24.3.1.3).

An evaluation of the difference between the exact solution and a truncated one,

obtained by the perturbations method, is given by the

Theorem 24.1.10 Let be the differential equation

(; ) (; ) (; ) ... (; ) (; , ), ,

mm n

tt t t tεε ε ε

=+ + ++ + ∈





xf x f x f x f x Rx xε

(24.1.80)

with the initial conditions

00 0

() , | |ttth=−≤xx , for which:

(i)

( ; ), 0,1,2,...,

ti m=fx , are functions continuous in t and x , −+1mi times

differentiable and Lipschitzian with respect to

x ;

(ii)

Rx(; , )t ε is a continuous and bounded function in ,t x and ε .

In this case, the solution

=+ + ++xx x x x

() () () () ... ()

tt t t tεε ε,

(24.1.80')

obtained by integrating the differential equations

=== =xfxxfxxfx x f x

11 22

( ; ), ( ; ), ( ; ),..., ( ; )

ttt t

 

(24.1.80'')

with the initial conditions

00 0 0

( ) , ( ) , 1,2,...,

ttim===0xxx , represents an

approximation with an error given by

()

⎡⎤

−++ ++ =

⎣⎦

xx x x x

() () () () ... ()

ttt t tεε ε εO .

(24.1.80''')

24.1.2.7 The Averaging Method. The Van der Pol Plane

Let be the system of non-autonomous non-linear differential equations

() () (; ), , (0)

tt tε=+ ∈ =





xAxfxx x x,

(24.1.81)

Dynamical Systems. Catastrophes and Chaos

671

where

A()t is a square matrix, x()t and fx(; )t are column vectors, smooth functions

t and x , while ε is a small parameter; to study this system, we use the averaging

method, rigorously introduced by J.-L. Lagrange, in 1788, in the study of the motion of

three particles (Sun and two particles).

The solution of the linear equation, corresponding to

= 0ε , is

=xx

() ()ttΦ ,

(24.1.82)

where

()tΦ is a square matrix, which – putting the initial conditions – has the property

= I()tΦ , where

is the unit matrix. By the change of variable

() () ()tttΦ ,

(24.1.83)

where

()t is a column vector, and introducing in (24.1.81), we obtain

()

() () () () ; ()tttt ttε+= +



ΦΦ Φ Φ;

taking into account that (24.1.82) is the solution of the linear equation, it results that

= A() () ()ttt



ΦΦ

, so that

()

() ; ()tttε=



ΦΦ,

wherefrom

()

() ; () (; )tt t tεε

−



ΦΦ ,

(24.1.83')

because the matrix

()tΦ is non-singular.

If we succeed to get the solution of this non-autonomous system, then we obtain

subsequently the solution

x()t too; otherwise, assuming that

()t varies a little in a

certain interval of time

(which can be the period of motion in case of a periodic

motion), we can write, by averaging,

∫

yFyFy

(; )d ( )





.

(24.1.84)

The system of differential equations in

y()t thus obtained is autonomous. The relation

between

y()t and y()t (the mode in which the solution y()t approximates the

solution

y()t ) is put in evidence by

Theorem 24.1.11 If the function Fy(; )t and the functional determinant

Fy|D (; )/D |ty are definite and bounded by a constant M , independent of the small

parameter

ε , in the domain D of the variables yy,  , if Fy(; )t is periodic in t , of

period

T , independent of the small parameter ε , and if y is always in a set interior to

the domain

D , then