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

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MECHANICAL SYSTEMS, CLASSICAL MODELS

652

==∈∈

112212 12

() ,() , , , ,xt a xt a t t Ia a  ,

(24.1.42)

which, associated to the equation (24.1.40'), form the bilocal (two-point) problem. We

encountered such a situation, e.g., in the case of the fundamental problem of ballistics

(see Sect. 7.1.2.1).

The first difficulty in talking this problem is to get appropriate hypotheses ensuring

the existence and uniqueness of the solution, as in this case we have no more the benefit

of such a powerful tool as the Cauchy–Picard theorem.

We shall suppose that

∈

([ , ])fCtt

for any

∈,xx

. Obviously, there are

infinitely many integral curves passing through the point

(,)ta. But it is possible, even

in simple cases, that none of these integral curves does reach the point

(, )ta . In other

words, it is possible that the solution of the two-point problem not even exist.

Hereafter, we give some of the most common conditions, each of them ensuring the

existence and uniqueness of the solution of the two-point problem (24.1.40), (24.1.42):

(i) (; , )ftxx is bounded;

(ii)

|| ||, 3 /( )fCxC ttπ<<−, for sufficiently great values of ||x ;

(iii)

is Lipschitzian with respect to

,xx

on any finite interval and

+(;,)/(||||)ftxx x x tends to zero, uniformly, on

[, ]tt if +→∞||||xx .

(iv)

f is Lipschitzian with respect to ,xx on any finite interval and has the form

≡+(; , ) (; ) (; , )ftxx tx txxϕψ,

where

+(; , )/(| | | |)txx x xψ  tends to zero uniformly on

[, ]tt if

+→∞||||xx ;

(v)

f allows continuous partial derivatives with respect to ,xx and

∂∂

<<+<

∂∂

,, 1

αβαβ



, with ∂∂≥/0fx .

A particular case of interest is that of the two-point problem

===

(;),(0)0,()0xftxx xt .

One can prove the existence and uniqueness of its solution provided that

∈×

([0, ] )fC t  and that there exist two numbers

0, 0cc≥> such that

(; )d , 0,

ft cx c t

ττ

⎛⎞

≥− − ∈

⎜⎟

⎝⎠

∫

In some particular cases, the solution of the non-linear differential equations of

second order can be simplified by reducing their order with a unity. Thus, if the

differential equation is of the form

Dynamical Systems. Catastrophes and Chaos

653

(; , )ftxx,

(24.1.43)

hence if it does depend explicitly on the unknown function

x , then, by a change of

function

=xp

, one obtains an equation of the form

=(; , ) 0Ftpp .

(24.1.43')

If the equation (24.1.40) does not depend explicitly on

t , hence if

=(,,) 0Fxxx ,

(24.1.44)

we make the same change of function

=xp , resulting the equation

()

;, 0

Fxp

(24.1.44')

As well, if the function

(; , , )Ftxxx is homogeneous of degree m in the variables

,,xxx, i. e.

=(; , , ) (; , , )

F t sx sx sx s F t x x x ,

(24.1.45)

then one can make the substitution

= /uxx , obtaining a differential equation of the

form

(;1, , ) 0Ft uu u .

(24.1.45')

A system of

n non-linear differential equations of first order can be written in the

implicit form

12 12

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

Ftx x x x x x i n  ,

(24.1.46)

where

F are considered definite on the same +(2 1)n -dimensional domain and

sufficiently regular. If the hypotheses of the theorem of implicit functions – extended to

systems of functions – hold, then we can obtain the canonical form of this system of

differential equations

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

xftxx xi n .

(24.1.46')

We notice that any differential equation of order

n , written in the normal form

(explicitly with respect to

()n

)

()

−

() ( 1)

,,,,...,

x f txxx x ,

(24.1.47)

MECHANICAL SYSTEMS, CLASSICAL MODELS

654

may be written also in the form of the canonical system of differential equations

−

== ==

1223 1 12

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

nn n

xxxx x xx ftxx x   ,

(24.1.47')

using the notations

(1)

123

, , ,...,

xxxxxxx x

−

=== = . Conversely, one can state

that a canonical system (in the normal form) of

n differential equations of first order is,

in general, reducible to only one differential equation of order

n .

As in case of a non-linear differential equation, we wish to determine that solution of

the system (24.2.46') that satisfies the initial conditions

( ) , 1,2,...,

xt x i n,

(24.1.46'')

of Cauchy type, where the point

10 20

( ; , ,..., )

tx x x belongs to the

+(1)n -dimensional domain on which the system (24.1.46') has sense. We can thus

state

Theorem 24.1.6 (Cauchy–Picard–Lipschitz). If the functions

, 1,2,...,

fi n=

, are

continuous with respect to all the variable on the

+(1)n -dimensional interval

{

}

=−<−<=

( ; , ,..., ) | , , 1,2,...,

tx x x t t x x bi nαD ,

and satisfy Lipschitz’s conditions

12 12

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

, 1,2,..., ,

ftXX X f tXX X

KXXi n

−

<−=

∑

for any pair of points

12 12

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

tXXX tXXX in D , where

Lipschitz’s constant, then the solution of Cauchy’s problem (24.1.46'), (24.1.46''), exists

and is unique on the real interval

−+

[, ]tTtT, where = min{ , / }TabM, while

(, , ,..., )

max sup ( ; , ,..., )

tx x x

Mftxxx

≤≤

∈

⎧

⎫

⎨

⎬

⎩⎭

The general solution of the non-linear system of differential equations (24.1.46')

depends on

n arbitrary constants and can be written in the form

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

xtCCCi nϕ ,

(24.1.46''')

wherefrom we can write

===

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

tx x x C i nϕ .

(24.1.46

)

These constants are determined by initial conditions in the case of Cauchy’s problem.

Dynamical Systems. Catastrophes and Chaos

655

The Theorem 24.1.6 has been applied for the equations of motion of the discrete

mechanical systems (see Sect. 11.1.1.4), for the equation of notion of the rigid solid

(see Sect. 14.1.1.8), for Lagrange’s equations (see Sect. 18.2.2.4) and for Hamilton’s

canonical equations (see Sect. 19.1.1.4), which are systems of differential equations of

first order. In connection with the last two mentioned systems of equations, various

methods of integration (first integrals, brackets, multipliers etc.) have been studied.

We notice that the limit problem which we wish to solve must be well put in the sense

of Hadamard, in other words: (i) it admits a unique solution; (ii) the solution has a

continuous dependence on the data of the problem (e.g., initial conditions, bilocal

conditions etc.).

In some particular cases, the solutions of the differential equations can be obtained in

an analytical way and can be expressed by means of elementary and special functions

etc. But, frequently, the solutions cannot be obtained in this way and we are led to very

intricate calculations. We are obliged, in this case, to use approximate methods of

calculation, which can be analytical methods, leading to solutions expressed in an

approximate analytical form (e.g., the method of successive approximations, the method

of power series expansions, the linear equivalence method etc.) or numerical methods,

where the solution is obtained in the form of a sequence of numerical values, e. g.,

starting from the initial conditions, in case of a Cauchy type problem, approximating

thus the integral curves.

The numerical methods of calculation can be methods with separated steps (one-step

methods) or methods with linked steps (multi-step methods).

Let be the equation (24.1.37'), with ∈⊂tI  . We assume that the interval I is

divided by a net

, ,...,

tt t, of step

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

ht ti n

=− = −, which can be

constant or variable. The solution at the point

t is = ()

xxt. In a one-step method,

if one knows

(, )

tx , then one can calculate

++11

(, )

tx ; we mention thus the method

of expansion into a Taylor series, Euler’s method and the Runge–Kutta method.

In a multi-step method, to determine

++11

(, )

tx it is necessary to know

11 11

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

tx t x tx

−−

; we mention thus the Adams–Moulton method and

Milne’s method.

An interesting semi-analytical method is the method of spline functions.

In both methods of calculation (one-step and multi-step) one can use explicit or

implicit algorithms. In the case of a one-step method, e. g., an explicit algorithm is of

the form

=+ = −

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

iii

xxhtxhi nϕ ,

(24.1.48)

while an implicit algorithm is given by

=+ = −

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

iii

xxhtxxi nψ ,

(24.1.48')

assuming that the step is constant.

A method to improve the results thus obtained is the predictor–corrector method;

thus the upper index

p corresponds to the predictor value, while the upper index c

corresponds to the corrector value.

MECHANICAL SYSTEMS, CLASSICAL MODELS

656

In the case of bilocal problems, especially the “shooting” method, the finite

difference method or the method of collocation by cubic alpine functions are used.

The considerations made above can be extended to the case of systems of non-linear

differential equations.

The study of the stability of the systems of non-linear differential equations plays a

very important rôle, as it has been seen in the previous considerations too. In this order

of ideas, we may state

Theorem 24.1.7 (Poincaré, Lyapunov). If

=++xAxBxfx() (; )tt



(24.1.49)

is a matric differential equation, where

A()t is a square matrix of order n ,

continuous with respect to time and which verifies the condition

→∞

=Alim ( ) 0

(24.1.49')

B is a constant square matrix of order n , the eigenvalues of which have a negative

real part, while

fx(; )t is a column vector, Lipschitzian in x , which verifies the

condition

→∞

lim ( ; )

(24.1.49'')

uniform in

t and for which =f0 0(; )t , then =x0 is a stable asymptotic solution.

General criteria of stability have been given in Sect. 23.1.

24.1.2.2 Method of Successive Approximations

Let us consider the non-linear differential equation of first order (24.1.37'). Let be

the rectangle

{( ; ) | | | ,| | }tx t t a x x b=−≤−≤D . For (; )ftx continuous and

Lipschitzian on

D a recurrent sequence formed by the functions

()

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

xt x f x j nττ

−

=+ =

∫

(24.1.50)

is set up. One can show that the sequence

{(), }

xtj∈  converges uniformly to the

unique solution of the Cauchy problem (24.1.37'), (24.1.37''), on an interval centred at

t . More precisely, the inequality

∞

−≤ − −<

∑

() () ,

xt xt tt tt h

(24.1.50')

takes place, where

K is Lipschitz’s constant corresponding to f , while h is defined

Dynamical Systems. Catastrophes and Chaos

657

{

}

∈

(,)

min , , sup ( , )

haM ftx

(24.1.50'')

The inequality (24.1.50') allows a sufficiently fine evaluation of the distance between

the approximate solution and the exact one.

In the case of Cauchy’s problem (24.1.46'), (24.1.46''), let us suppose, as well, that

the conditions of existence and uniqueness of the solution hold. One can prove that the

sequence of approximate solutions

()

11 1

( ) , , ,..., d ,

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

jjjj

xt x f x x x

xxi nj m

ττ

−− −

== =

∫

(24.1.50''')

is uniform convergent on the interval

−+

[, ]thth, where h is determined by the

theorem of existence and uniqueness. The construction of the

n recurrent sequences

which tend to the unique solution of the Cauchy problem is a direct generalization of

what has been said previously for the case

= 1n . More precisely, the functions ()

xt,

definite by recurrence by the relation (24.1.50'''), satisfy the inequality

()

∞

−

−< =

∑

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

nK t t

xt xt i n

nK j

(24.1.50

)

where

−<

||tt h, while K is Lipschitz’s constant.

The method of successive approximations is called the Picard–Lindelöff method too.

24.1.2.3 Method of Expansion in a Power Series

If the function

(; )ftx is sufficiently regular on the definition domain D , then ()xt

will admit an expansion into a Taylor series around

()

00 0

()

() () () () ...

1! 2!

() (,),

tt tt

xt xt xt xt

xt Rtt

−−

=+ + +

−



(24.1.51)

where

(, )

Rtt is the remainder of the expansion, which can be estimated in

Lagrange’s form

()

−

=∈

(1)

(, ) ( ), ( , )

(1)!

Rtt x tt

ττ ,

(24.1.51')

so that

−

−≤

()

(1)!

Rt t M

for

∈

()

(,)

sup ( ; )

ftx M

(24.1.51'')

MECHANICAL SYSTEMS, CLASSICAL MODELS

658

The solution of Cauchy’s problem (24.1.37'), (24.1.37'') can be approximated by the

polynomial in the right member of the expansion (24.1.51), neglecting the rest, i. e.

00 0

()

() () () ()

1! 2!

() ()

() ... (),

3! !

xt xt xt xt

tt tt

xt x t

−

=+ +

−−

+++



(24.1.52)

where the coefficients

()

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

xtk n= , are successively calculated in the

form

()

== =

∂

∂∂∂∂

⎡⎤

=+ + + +

⎢⎥

∂∂∂

∂

⎣⎦

000 00 00

000

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

() (; ),

() 2 (; ),

....................................................................................

xt x xt ft xt ft x

xt f t x

ffff

xt f f f f t x

xxx







 



(24.1.52')

In the particular case in which

(; )ftx is expanded into a double series in the

rectangle

{( ; ) | | | ,| | }tx t t a x x b=−≤−≤D , i. e.

∞∞

=−−

∑∑

(; ) ( )( )

ftx a t t x x ;

(24.1.53)

representing

x also in a power series in the form

∞

=+ −

∑

() ( )

xt x c t t ,

(24.1.53')

convergent for

−<

||tt h, with h determined by (24.1.50''), we obtain

∞∞ ∞ ∞

−

== = =

⎡⎤

−−=−

⎢⎥

⎣⎦

∑∑ ∑ ∑

00 0

00 1 1

() () ()

jn m

jk n m

att ctt ctt .

(24.1.53'')

Starting from this relation, we deduce the coefficients

c , by identification

=+ + +

210011

320111021012

...................................................

ca ac

ca acacac

(24.1.53''')

A differential equation of first order being given, before the application of any

method to determine its solutions, we try to be sure if the hypotheses of the theorem of

Dynamical Systems. Catastrophes and Chaos

659

existence and uniqueness are verified, to can solve a Cauchy problem; to do this, we see

if the inequality (24.1.38) is satisfied or not. As we have seen before, there exist large

classes of Lipschitzian functions. Thus, the analytical functions with respect to

x are

Lipschitz functions; in particular, the linear equations, as well as the polynomial ones

have this property. More general, the functions differentiable with respect to

x are

Lipschitzian too.

In the case of a system of non-linear differential equations of the form (24.1.46'), let

us suppose that the functions

( ; , ,..., )

ftxx x have continuous partial derivatives of

any order in a neighbourhood of

10 20

( ; , ,..., )

tx x x ; the solution can be then

searched in the form of Taylor series, i. e.

00 0

()

() () () ...

1! 2!

()

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

ii i

xt x xt xt

xt Rtti n

−

=+ + +

−

++=



(24.1.54)

where

(, )

Rtt are the remainders which can be estimated in Lagrange’s form

−

=∈

(1)

()

(, ) ( ), ( ,)

(1)!

Rtt x tt

ττ ,

(24.1.54')

hence

()

{}

≤≤

∈

−

−<

=−<−<=

()

( ; , ,..., )

() ,

(1)!

max sup ( ; , ,..., ) ,

; , ,..., | , , 1,2,..., .

tx x x

Rtt M

Mftxxx

tx x x t t a x x bi n

(24.1.54'')

As in the one-dimensional case, the solution of Cauchy’s problem (24.1.46'),

(24.1.46'') can be approximated by the polynomials in the right member of the

expansions (24.1.54), neglecting the remainders, so that

00 0

()

() () () ...

1! 2!

()

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

ii i

xt x xt xt

xti n

−

=+ + +

−



(24.1.55)

The corresponding coefficients are successively calculated in the form:

000 0 0

10 20

000

10 20

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

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

........................................................................

iiii n

xt x xt ftx x x

xt f f t x x x

∂

⎛⎞

⎜⎟

∂

⎝⎠

∑





(24.1.55')

MECHANICAL SYSTEMS, CLASSICAL MODELS

660

Here also, one can consider the case in which

f admits an expansion into a series

analogue to (24.1.53), hence of the form

≥

∑

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

! !.... !

ftxx x f txx x

μμ

μμ μ

;

(24.1.56)

ρ is a constant introduced for further identifications,

( , , ,..., )

μμμμ μ= ,

=++++

| | ...

μμ μμ μ, is a +(1)n -dimensional multi-index, while the

coefficients

()

are continuous on the interval [0, ]a , satisfying on it the inequality

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

! !.... !

...

ft i n

rr r

μμ μ

<=∈

(24.1.56')

A and =, 1,2,...,

rk n, being determinate constants. Then, the solution of the system

(24.1.46'), which satisfies the initial conditions (24.1.46''), can be expanded in a power

series

≥

∑

10 20

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

xt t xx x i n

μμ

ϕρ ,

(24.1.57)

which is absolute convergent in the domain determined by the inequality

10 20 0

(1)

... e .

kaAn

rr r r k

−

∞

−+

++++ <

∑

(24.1.57')

In the particular case in which

()

ft at

μμ

we take =

()

tct

μμ

ϕ ; the

coefficients

are obtained by identification.

But this procedure is particularly difficult to apply in practice; an efficient mode to

obtain the coefficients

iμ

ϕ is given in the following subsection.

24.1.2.4 Linear Equivalence Method (LEM)

The linear equivalence method or, briefly, LEM, has been introduced to find

convenient, both quantitative and qualitative representations of the solutions of

non-linear ordinary differential equations and of systems of non-linear ordinary

differential equations via the methods in use for the linear ones. The method was

elaborated and applied by Ileana Toma beginning with the eighth decade of twentieth

century. The method, initially introduced for first order polynomial differential systems,

was extended to first order ordinary differential systems with right side analytic with

respect to the unknown functions. The case of polynomial operators involves some

simplified formulae for the LEM representations and even more simplifications are

emphasized in the case of constant coefficients.

There, let us consider the system (Soare, M., et al., 2007; Toma, I., 1995)

Dynamical Systems. Catastrophes and Chaos

661

[

]

()

[]

1,2,...,

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

ttft CIIab

=≡ ∈=⊆





xfxfx x x

(24.1.58)

where

x(; )

ft are analytic functions, uniformly with respect to ∈tI, i. e.

{}

()

∞

==∈∪

∑

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

ft f t j n

μ  ,

(24.1.58')

where

μ and ||μ have the significations given in the preceding subsection and where

≡x

...

xx x

μμ

(24.1.58'')

The coefficients

→:

 are supposedly at least of class

()CI

. We deal here

only with ordinary differential equations with null free terms, this is why the sums in

(24.1.58') are starting from

1 on.

The system may be also written putting into evidence the differential operator

≡− =xxfx() (;)t 0



F .

(24.1.58''')

LEM considers an exponential mapping depending on

n parameters ∈



( , ,..., )

ξξ ξ

, namely

≡=

∑

(; ) e , ,

vt xξ

ξξ ,

(24.1.59)

that associates to the above non-linear ordinary differential equation two linear

equivalents:

(i) a linear partial differential equation, always of first order with respect to

∂

≡− >

∂

f(; ) , (;D) 0

vt t

ξL ξ ;

(24.1.59')

(ii) a linear, while infinite, first order ordinary differential system

1,2,...,

() , ,

jjj j

ftv

μμ

γγ

∞

+−

==δ∈

⎡⎤

⎣⎦

∑∑



e .

(24.1.59'')

In (24.1.59'), the formal operators

∞

∂

≡≡

∂∂ ∂

∑∑

, ( ;D) ( ;D ), ( ;D ) ( )

...

jj j j

tftftft

ξξ

μμ

ξξ ξ

(24.1.59''')

make sense on Exp-type spaces.