LIE TRANSFORMS FOR ORDINARY DIFFERENTIAL EQUATIONS: TAKING ADVANTAGE OF THE HAMILTONIAN FORM OF...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 40, 2289–2300 (1997)

LIE TRANSFORMS FOR ORDINARY DIFFERENTIALEQUATIONS: TAKING ADVANTAGE OF THE

HAMILTONIAN FORM OF TERMS OF THE PERTURBATION

ROBERTO BARRIO

Grupo de Mec�anica Espacial, Departamento de Matem�atica Aplicada CPS, Universidad de Zaragoza,Mar��a de Luna 3, 50009 Zaragoza, Spain

JES �US PALACI �AN

Grupo de Mec�anica Espacial, Departamento de Matem�atica e Inform�atica, Universidad P�ublica de Navarra,31006 Pamplona, Spain

SUMMARY

In the context of Lie transforms theory applied to ordinary di�erential systems, we deal with the case forwhich part of the perturbing equation admits a Hamiltonian formulation. This fact allows to perform a Lietransform combining a treatment of vector type with another of Hamiltonian character. We show that thisproduces a reduction in the number of operations involved in the algorithms. We illustrate the algorithmswith an example. ? 1997 by John Wiley & Sons, Ltd.

KEY WORDS: di�erential systems; perturbation theories; Lie transforms; canonical variables

1. INTRODUCTION

In this paper we deal with the use of the Lie transforms theory for ordinary di�erential systemswith a small parameter. The interest of applying a perturbation technique in this frame is toperform a transformation of the equations by changing the variables in order to solve or, at least,to simplify the original equations by means of asymptotic developments in power series of thesmall parameter. The method was �rst given by Kamel1 and extended soon after by Henrard.2 Theypresent a generalization of the classical Lie transform theory for Hamiltonian systems introducedby Deprit.3 Lately, Coppola and Miller4 have used this theory for simplifying perturbed non-linearHamiltonians with one degree of freedom.Recently, Meyer5; 6 has generalized these theories presenting some theorems which summarize

and include the works of Deprit, Kamel and Henrard, giving a formal treatment in terms of tensor�eld theory. In his review, Meyer describes some problems that can be solved by Lie transformssuch as the implicit function theorem, catastrophe theory, classical normal forms, Poisson systems,the Floquet exponents and the Liapunov transformations and the computational Darboux theorem.At �rst we present a brief description of the classical Lie transform algorithms for vector and

scalar �elds proposed by Deprit, Kamel and Henrard, together with the main theorem given byMeyer. In this paper we shall follow the notation of Meyer.Thereafter we study the speci�c case in which the right member of the di�erential system appears

as the sum of two functions. Then, we established in Proposition 1 an algorithm to perform the Lie

CCC 0029–5981/97/122289–12$17.50 Received 24 July 1995? 1997 by John Wiley & Sons, Ltd.

2290 R. BARRIO AND J. PALACI �AN

treatment in three steps. In particular, when part of the perturbation is of Hamiltonian form, wepropose in Theorem 1, a mixed Lie algorithm which combines the vector �eld and Hamiltoniansystem formulations of the Lie method. These results are reformulated in Theorem 2 and thecorollary, giving a formal treatment in terms of vector �eld theory. This procedure permits toconserve the form of the di�erential equations, that is

dxdt= I ·∇xH(x) + f(x) −→ dy

dt= I ·∇yH∗(y) + f∗(y)

Also, this alternative reduces signi�cantly the computational complexity of the treatment andbesides, allows to choose some classical transformations designed for Hamiltonian systems. Themethod proposed is illustrated with the Du�ng oscillator in a viscous uid.

2. ON LIE TRANSFORMS FOR DIFFERENTIAL SYSTEMS

Here we start giving a short description of the general algorithms of the Lie transforms, for moredetails see References 2, 3, 5Given an analytic tensor �eld

f(x; ”) =∑i¿0

”i

i!f (0)i (x) (1)

expressed in power series of a small parameter ”, it is transformed into another analytic tensor�eld f∗(y; ”), through a transformation x = x(y; ”) de�ned by a 1-contravariant tensor �eld calledgenerating function or simply generator

W(x; ”) =∑i¿0

”i

i!Wi+1(x) (2)

Then, f∗(y; ”) expressed in terms of ” as a power expansion reads by

f∗(y; ”) =∑i¿0

”i

i!f (i)0 (y) (3)

where the coe�cients f (i)0 are obtained by the successive application of the formula

f (i)j = f (i−1)j+1 +∑

06k6j

(jk

)Lk+1f

(i−1)j−k with i¿1 and j¿0 (4)

This recursive algorithm (4) is easily understood with the aid of the Lie triangle given by Deprit,3

see also Figure 1. In practice there exist two cases; a �rst one occurs when the generating functionW is given, so the Lie algorithm is applied directly to obtain f∗: In the second W is unknownand one wants to �nd it together with f∗ both to be simple. That is, a criterion to obtain f∗ isneeded and afterwards one can obtain W from the Lie equation.As we deal with ordinary di�erential systems we want to transform the system x = f(x; ”) into

y = f∗(y; ”). In that case, the vector �elds are 1-contravariant tensor �elds. This is assumed in thewhole paper. So, for a vector �eld g the Lie operator stands for

Lig =@g@x

·Wi − @Wi

@x· g (5)

Int. J. Numer. Meth. Engng., 40, 2289–2300 (1997) ? 1997 by John Wiley & Sons, Ltd.

LIE TRANSFORMS FOR ORDINARY DIFFERENTIAL EQUATIONS 2291

Figure 1. Lie triangle. The coe�cients f (0)i of the expansion of the old vector �eld are on the left column while the coef-�cients of the transformed vector �eld g, i.e. f (i)0 , are the outputs of the process and are placed on the diagonal

such that @g=@x and @Wi=@x are, respectively, the Jacobians of g and Wi with respect to thevector x.However, for a scalar �eld P = P(x; ”), the transformed �eld is P∗(y; ”) = P(x(y; ”); ”) and

the Lie operator stands for

LiP =@P@x

·Wi

A particular case of a di�erential system occurs when there exits a scalar function H (theHamiltonian) such that

dxdt= J · @

@xH(x; ”) = J ·∇xH(x; ”) (6)

with x = (q; p) where q stands for the co-ordinate vector of a con�guration space while p is themomentum conjugate of q. Besides the symplectic matrix J corresponds to the square matrix

J =

(0 1I

−1I 0

)where 1I is the identity matrix.Accordingly, a change of variables is called a canonical transformation if for every function

H(x; ”) there exists a function H∗(y; ”) such that the transformation of (6) is

dydt= J ·∇yH∗(y; ”) (7)

Now the Lie derivative L acting over a function turns out to be the Poisson bracket of a coupleof scalar functions H and V (the generating function), that is

{H; V} = @H@x

·J · @V@x

(see Reference 7). Hence, we recover the vector formulation with equation (7), as well as for thegenerator

W = J ·∇xV(x; ”) (8)

Meyer5; 6 has given a generalization of the algorithm (4) in the following theorem.

? 1997 by John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng., 40, 2289–2300 (1997)


Meyer’s Theorem. Let {Pi}∞0 , {Qi}∞0 and {Ri}∞0 be sequences of vector spaces of smoothvector �elds de�ned on a manifold M and [· ; ·] a bilinear operator (the so-called Lie bracket).Assume besides that

(i) Qi⊂Pi ; i = 1; 2; : : : ;(ii) f (0)i ∈ Pi ; i = 0; 1; 2; : : : ;(iii) [Pi ; Rj]⊂Pi+j; i; j = 0; 1; 2; : : : ;(iv) for any A ∈ Pi ; i = 1; 2; : : : ; there exist B ∈ Qi and C ∈ Ri ; i = 1; 2; : : : such that

B = A+ [f (0)0 ; C] (9)

Then there exists W with a formal expansion of the form (2) with Wi ∈ Ri, i = 1; 2; : : : ;which transforms the vector �eld f with the formal series expansion given in (1) to the vector�eld f∗ with the formal series expansion given by (3) with f (i)0 ∈ Qi, i = 1; 2; : : :A proof of this theorem is given by Meyer.6 In practice, f (0)0 is given and thus one takes the

subspaces Pi as small as possible. The spaces Qi and Ri come from an analysis of the equationin (9). The Lie bracket [· ; ·] of this theorem corresponds to the Lie derivative given in (5) fordi�erential systems or the Poisson bracket for Hamiltonian functions.One of the most usual applications of Lie transforms to dynamical systems concerns with the

theory of averaging. In this case, Lie transforms are built in order to eliminate one or severalvariables in the original equations. Therefore, the number of degrees of freedom of the prob-lem is reduced. Then the new equations are better suited for performing either numerical in-tegrations or analytical studies concerning the qualitative behavior of the dynamical system (seeReference 8). Actually, the method of averaging can be interpreted as a special case of the classicalnormal form theory for autonomous systems as Meyer points out.

Remark 1. Clearly, the treatment presented in this section is valid for autonomous di�erentialequations. However, in many applications the time is explicitly involved in the equations, i.e.they are of the form x = f(t; x; ”), so one can transform it into the previous case by replacing theoriginal system with the equivalent autonomous one x = f(�; x; ”), � = 1, where � is considered asa new variable. There exists also a version of Meyer’s Theorem5 translated to the non-autonomouscase, based on the method given by Deprit3 for non-autonomous Hamiltonians.

3. SPLITTING THE PERTURBING FUNCTION

In this section we state an alternative to the treatment of Section 2 when the perturbation is dividedinto two parts.

3.1. Algorithms

Proposition 1. Let us consider the following di�erential system:

dxdt= f(x; ”) (10)

such that the analytic vector �eld f is de�ned by two other analytic vector �elds

f(x; ”) = a(x; ”) + b(x; ”) =∑i¿0

”i

i!

(a(0)i (x) + b

(0)i (x)

)(11)



Under these assumptions, system (10) is transformed into the di�erential system

dydt=∑i¿0

”i

i!

(a(i)0 (y) + b

(i)0 (y) + c

(i)0 (y)

)(12)

where the coe�cients a(i)0 ; b(i)0 and c(i)0 are obtained from the recursive relations

a(i)j = a(i−1)j+1 +∑

06k6j

(jk

)LIk+1a

(i−1)j−k

b(i)j = b(i−1)j+1 +∑

06k6j

(jk

)LIIk+1b

(i−1)j−k (13)

c(i)j = c(i−1)j+1 +∑

06k6j

(jk

)[LIk+1b

(i−1)j−k +LII

k+1a(i−1)j−k

+LIIIk+1

(a(i−1)j−k + b(i−1)j−k + c(i−1)j−k

)]for j¿0; i¿1 and c(0)i = 0; ∀i; with the generating function given by

W(x; ”) =WI(x; ”) +WII(x; ”) +WIII(x; ”)

=∑i¿1

”i

i!

(WIi(x) +W

IIi (x) +W

IIIi (x)

)(14)

and the Lie di�erential operator L acting over a vector �eld s de�ned in each case as

Lij s =

@s@x

·Wij −

@Wij

@x· s with i = I; II; III (15)

Proof. The proof of this proposition turns out to be straightforward, by induction on the columns(j¿0) and rows (i¿1) of the Lie triangle and taking into account the linear character of the Lieoperator. Adding and factoring the terms a(i)j , b

(i)j and c(i)j and naming a(i)j + b(i)j + c(i)j by f (i)j

Formula (4) is recovered for j = 0 and ∀i.

An application of the last result is that this decomposition permits to choose di�erent transfor-mation or simpli�cation criteria for each part of the equations. In particular, it is possible to takeadvantage of mixing Hamiltonian and vector Lie treatments once the right member of the systemis adequately split. That is the main result of this paper and it is established in the followingtheorem.

Theorem 1. Given a di�erential system

dxdt= f(x; ”) =

∑i¿0

”i

i!f (0)i (x) (16)

such that x = (q; p) represents a set of canonical variables of coordinates q and momenta p.Besides the functions f (0)i ; i¿0 are decomposed in two di�erent parts

f (0)i (x) = f(0)Hi (x) + f

(0)NHi(x)



such that f (0)Hi come from a Hamiltonian H (0)i , then equation (16) is transformed, through the

generating function

W(x; ”) =∑i¿1

”i

i!

(WIi(x) +W

IIi (x)

)(17)

into another di�erential equation

dydt=∑i¿0

”i

i!

(f (i)H0 (y) + f

(i)NH0(y)

)(18)

in which the terms f (i)H0 ; i¿1 are obtained by calculating f(i)Hj = J ·∇xH (i)

j , and H(i)j by using

the algorithm of Lie transforms for Hamiltonians, that is

H(i)j =H (i−1)

j+1 +∑

06k6j

(jk

){H (i−1)

j−k ; Vk+1} (19)

for j¿0; i¿1 and V =V(q; p; ”) being a scalar generating function.Finally, the terms f (i)NH0 ; i¿1 are calculated through the formula

f (i)NHj = f(i−1)NHj+1 +

∑06k6j

(jk

)[(LIk+1 +L

IIk+1

)f (i−1)NHj−k +L

IIk+1f

(i−1)Hj−k

](20)

also for j¿0, i¿1. Now the Lie operators LI and LII are de�ned in the same manner than inequation (15) of Proposition 1 in accordance with WI and WII. Besides now WI = J ·∇xV isthe generating function built from V.

Proof. The proof of this theorem is simply achieved by substituting in Proposition 1 a by fHand b+ c by fNH. That is, the coupling terms must be included in the vector part.

This theorem can be sketched in Figure 2 with the aid of the Lie staircase.

Figure 2. Lie staircase. This scheme can be understood as a ‘double’ Lie triangle. On the one hand, the canonical part canbe performed totally independent of the rest of the equation while, on the other hand, the intermediate Hamiltonians H(i)

jmust be inserted, order by order of the Lie process, to build the functions f (i+1)NHj

. Naturally, these Hamiltonians must beconverted to vector functions each time, multiplying the symplectic matrix by their corresponding gradients. Finally, the

transformed function is obtained, also order by order, as the sums f (i)H0 + f(i)NH0



The latter theorem can be valid also for non-autonomous equations. First at all the primitivesystem should be converted in an equivalent autonomous one by introducing a new independentvariable � such as it was explained in Remark 1, i.e. x′ = (q; �; p; �′). Then, the canonical part ofthe di�erential equation could be transformed into a non-autonomous Hamiltonian replacing fH byJ ·∇x′H and the generator W I by J ·∇x′V.In practice, we shall consider problems where f (0)NH0 = 0, thus only f

(i)H0 is present. In that case it

is easier to obtain W.

Remark 2. This result gives a new method to build Lie transforms. It is worthy to note thatthe �nal equations are not necessarily the same equations as if one applies directly the classicalLie algorithm; this fact depends on the nature of the transformation chosen.

3.2. Formal theorems

The algorithms stated in Proposition 1 and in Theorem 1 admit to be written in the frame ofthe generalization of Lie transforms theory, i.e. in terms of sequences of vector spaces of smoothvector �elds and bilinear operators:

Theorem 2. Let {P ji }∞i=0, {Q ji }∞i=0 and {R j

i }∞i=0, j= I; II; III; be nine sequences of vector spaceswhose components are smooth vector �elds de�ned on a manifold M . Let [· ; ·] a bilinear operatorde�ned over any two vector spaces Pj

i and Qji ; i = 1; 2; : : : ; j = I; II; III: Let us assume besides

that(i) Q ji ⊂P j

i , i = 1; 2; : : : ; j = I; II; III;(ii) a(0)i ∈ PIi , b(0)i ∈ PIIi , c(0)i = 0, i = 0; 1; 2; : : : ;(iii) [P j

i ;Rjk ]⊂P j

i+k ; i; k = 0; 1; 2; : : : ; j = I; II; III;

[PIIi ; RIk ]⊂PIIIi+k ; [PIi ; R

IIk ]⊂PIIIi+k ;

[PIi ∪PIIi ; RIIIk ]⊂PIIIi+k ; i; k = 0; 1; 2; : : : ;(iv) for any Aj ∈ P j

i , i = 1; 2; : : : ; j = I; II; III; there exist B j ∈ Q ji and Cj ∈ R ji , i =

1; 2; : : : ; j = I; II; III, such that

BI = AI + [a(0)0 ; CI];

BII = AII + [b(0)0 ; CII];

BIII = AIII + [b(0)0 ; CI] + [a(0)0 ; C

II] + [a(0)0 + b(0)0 ; CIII]

Then there exists W with a formal expansion of the form (14) such that

Wi =WIi +W

IIi +W

IIIi

with Wji ∈ R j

i ; i = 1; 2; : : : ; j = I; II; III; that transforms the vector �eld a+ b with the formalseries expansion given in (11) into the vector �eld with the formal series expansion given by (12)with a(i)0 ∈ QIi , b(i)0 ∈ QIIi and c(i)0 ∈ QIIIi , i = 1; 2; : : :

Proof. Using induction on the rows of the Lie triangle. If, by induction hypothesis In let a(i)j ∈

PIi+j, b(i)j ∈ PIIi+j and c(i)j ∈ PIIIi+j for 06i + j6n, Wj

i ∈ R ji , j = I; II; III, a

(i)0 ∈ QIi , b(i)0 ∈ QIIi and

c(i)0 ∈ QIIIi , 16i6n.



I0 is true by assumption and so assume In−1 to be true. By (13)

a(1)n−1 = a(0)n +n−2∑k=0

(n− 1k

)[a(0)n−1−k ; W

Ik+1] + [a

(0)0 ; W

In]

b(1)n−1 = b(0)n +n−2∑k=0

(n− 1k

)[b(0)n−1−k ; W

IIk+1] + [b

(0)0 ; WII

n ]

c(1)n−1 =n−2∑k=0

(n− 1k

){[b(0)n−1−k ; W

Ik+1] + [a

(0)n−1−k ; W

IIk+1]

+ [a(0)n−1−k + b(0)n−1−k ; W

IIIk+1]

}+ [b(0)0 ; WI

n] + [a(0)0 ; WII

n ] + [a(0)0 + b(0)0 ; W

IIIn ]

(21)

The terms that depend on Wn are not covered either by the induction hypothesis or the hypothesisof the theorem. All the other terms in the three equations are, respectively, in PIn, P

IIn and P

IIIn

by In−1 and hypothesis (iii). So,

a(1)n−1 = K11 + [a(0)0 ; W

In]

b(1)n−1 = K21 + [b(0)0 ; W

IIn ]

c(1)n−1 = K31 + [b(0)0 ; W

In] + [a

(0)0 ; W

IIn ] + [a

(0)0 + b(0)0 ; WIII

n ]

(22)

where Kj1 ∈ P jn , j = I; II; III; are known. Now, performing a simple induction over the columns

of the Lie triangle using (13) yields

a(s)n−s = K1s + [a(0)0 ; WI

n]

b(s)n−s = K2s + [b(0)0 ; WII

n ]

c(s)n−s = K3s + [b(0)0 ; WI

n] + [a(0)0 ; WII

n ] + [a(0)0 + b

(0)0 ; WIII

n ]

(23)

where K js ∈ Pjn for s = 1; 2; : : : ; n and j = I; II; III. Then

a(n)0 = K1n + [a(0)0 ; WI

n]

b(n)0 = K2n + [b(0)0 ; WII

n ]

c(n)0 = K3n + [b(0)0 ; WI

n] + [a(0)0 ; WII

n ] + [a(0)0 + b

(0)0 ; WIII

n ]

(24)

By hypothesis (iv) equation (24) is solved for Wjn ∈ R j

n , j = I; II; III, a(n)0 ∈ QIn, b(n)0 ∈ QIIn and

c(n)0 ∈ QIIIn . Thus In is true.

Now Theorem 1 can be deduced easily as a corollary of Theorem 2.

Corollary. Let {PIi}∞0 , {QIi}∞0 , and {RIi}∞0 , be three sequences of vector spaces of scalarfunctions de�ned on a manifold M and {PIIi }∞0 , {QIIi }∞0 , and {RIIi }∞0 be three sequences ofvector spaces whose components are smooth vector �elds on a manifold M ′. Let {· ; ·} a bilinearoperator de�ned over two vector spaces of scalar functions PIi and Q

Ii ; i = 1; 2; : : : ; and [· ; ·]

a bilinear operator de�ned over two vector spaces of vector functions PIIi and QIIi ; i = 1; 2; : : : :

Assume besides that(i) Q ji ⊂P j

i ; i = 1; 2; : : : ; j = I; II;(ii) H(0)

i ∈ PIi , f (0)NHi ∈ PIIi , i = 0; 1; 2; : : : ;



(iii) {PIi ; RIk}⊂PIi+k ; [PIIi ; RIIk ]⊂PIIi+k ; i; k = 0; 1; 2; : : : ;

[J ·∇xPIi ; RIIk ]⊂PIIi+k ; [PIIi ; J ·∇xRIk ]⊂PIIi+k ; i; k = 0; 1; 2; : : : ;

(iv) for any a ∈ PIi , A ∈ PIIi , i = 1; 2; : : : ; there exist b ∈ QIi , B ∈ QIIi and c ∈ RIi , C ∈ RIIi ,i = 1; 2; : : : ; such that

b= a+ {H(0)0 ; c}

B=A+ [f (0)NH0 ; c+ C] + [f(0)H0 ; C];

where c = J ·∇xc, and fH0 (0) = J ·∇xH(0)0 , x being the set of canonical variables in which

the problem is de�ned and J the usual symplectic matrix of an adequate dimension.Then there exists W with a formal expansion of the form (17) with Wi = WI

i +WIIi and

WIi = J ·∇xVi with Vi ∈ RIi ; V given by equation (19) and WII

i ∈ RIIi , i = 1; 2; : : : ; whichtransforms the vector �eld f with the formal series expansion given in (16) to the vector �eldwith the formal series expansion given by (18) with H(i)

0 ∈ QIi and f (i)NH0 ∈ QIIi ; i = 1; 2; : : : :

Proof. It is an immediate consequence of Theorems 1 and 2.

3.3. Shortening the number of basic operations

In Perturbation Theory the number of Lie derivatives computed in the construction of the Lietriangle is given in accordance with the order n of the theory, by the quantity n(n+ 1)(n+ 2)=6.Now we de�ne the number of basic operations as the sum of the number of partial derivatives plusthe number of products and sums of functions needed in the calculation of a Lie derivative. Let uscompare now the usual algorithm for di�erential equations given by (4) with the one presented inTheorem 1. First we assess the same weight for the perturbing functions of both treatments. Thus,if the dimension of an equation verifying the hypotheses of Theorem 1—equation (16)—is 2m,the number of partial derivatives involved in the computation of a Lie is given by 8m2, just thesame than for the products of functions while the number of sums is 8m2−2m. On the other hand,in the canonical case, the number of partial derivatives needed for calculating a Poisson bracketis reduced to 4m while the number of products is 2m; �nally, the number of sums is 2m− 1.If the perturbation of the dynamical system is split into two functions, the number of Lie

derivatives in the whole vector treatment is 2n(n+1)(n+2)=3 (calculation of four di�erent typesof blocks of vector Lie derivatives). However, if use is done of Theorem 1, the number of vectorLie derivatives reduces to n(n + 1)(n + 2)=2(three blocks of Lie derivatives of vector type) plusone block of scalar Lie derivative, which introduces n(n + 1)(n + 2)=6 computations of Poissonbrackets instead of the same number of calculations of vector Lie derivatives.In spite of the �nal equations being di�erent, one may compare the computational e�ort. Thus,

the use of Theorem 1 reduces the number of the operations needed in the calculation, as is shownin Table I by the ‘relative di�erence’ between the common procedure and the approach we propose.This di�erence tends to 25 per cent when the dimension of the problem 2m grows up.

Table I. Relative di�erence in the number of operations when using the normal algorithmof Lie transforms for vector �elds compared versus Theorem 1

Partial derivatives Products Sums

vec− (vec&Ham)vec

2m− 18m

− 38(n+ 1) (n+ 2)m

4m− 116m

8m2 − 4m+ 18m(4m− 1)



4. EXAMPLE

In this section we illustrate the approach presented here by applying the algorithm to a di�erentialsystem with two degrees of freedom: the Du�ng oscillator in a viscous uid. The position q isgiven by the second-order di�erential equation

d2qdt2

= −q− q3 − �dqdt

(25)

If � = 0 the problem is the Du�ng oscillator.This second-order di�erential equation can be written as a system of �rst-order di�erentialequations

dqdt=@H@p

dpdt= −@H

@q− �p

(26)

where the Hamiltonian is

H =12(q2 + p2) +

4q4 (27)

and p stands for the momentum conjugate of q. The parameters and � are assumed to be smalland of similar size ∼ � ∼ ” (” being the small parameter of the theory).Now, system (26) is transformed by using the classical action-angle variables proposed by

Poincar�e (I ,�) into

dIdt=@H@�

− �I(1− cos 2�)

d�dt

= −@H@I

− �2sin 2�

(28)

where now H stands for

H = I + 8I 2(3 + 4 cos 2�+ cos 4�) (29)

so that Theorem 1 can be applied for solving (28). Now we can apply a Lie transform in orderto remove the angle � from the equations (28) and (29). This is precisely the averaging methodtechnique although from the point of view of Lie transforms theory. The idea is to eliminate �from the Hamiltonians and vector functions order by order according to the small parameter ”,while the generating functions (scalar and vector, respectively) are also built up order by order torecover the original expressions.First, we start the procedure for the Hamiltonian (29). Up to second order in ”, H is reduced

to K such that

K = I +3 8I 2 +

17 2

64I 3 + O(”3) (30)

in which I must be understood as the new action variable. The generator associated to this trans-formation up to second order is

V = 4I 2(sin 2�+ 1

8 sin 4�)

+ 2

128I 3(33 sin 2�+ 3 sin 4�− 1

3 sin 6�)+ O(”3) (31)



Hence the functions f (i)Hj are calculated from the Hamiltonians H(i)j , (i; j = 1; 2). Similarly, the

generators WIj are built from Vj; (j = 1; 2). These vectors are thus inserted into the equations of

the Lie staircase algorithm (20) proposed in Theorem 1. In order to complete Formula (18) nowthe functions f (i)NHj , (i; j = 1; 2); must be calculated using the recursive expressions (20) where L

I

now stands for the Lie derivative associated to WI and, order by order, one must calculate f (i)NHj aswell as WII. First f (i)NH0 are taken to be the corresponding averages over � while W

II are chosen toverify the correspondent partial equations of the Lie equation for �rst order di�erential systems.At this point one should realize the di�erence between the usual manner of attacking this prob-

lem, through the Lie transforms, and the modi�cation proposed in this paper. With the treatmentwe are employing one does not need to perform the vector treatment for the whole system (28).Moreover, part of the problem is done with the canonical version and then the amount of terms thatmust be taken into consideration for solving the system has been simpli�ed. Therefore, this reducesthe vector treatment because only the coe�cients f (1)NH0 and f

(2)NH0 have to be obtained following (20).

Lastly, the canonical and the non-canonical terms are joined to give the reduced system, that is

dIdt= −�I + O(”3)

d�dt

= −1− 3 4− 18

(�2 +

51 2

8I 2)+ O(”3)

(32)

in which again (I; �) stand for the new variables. Finally, the generator W = (1W; 2W ) up tosecond order

1W = 2I 2(cos 2�+ 1

4 cos 4�)+�2I sin 2�

+ 2

32I 3(33 cos 2�+ 6 cos 4�− cos 6�) + �

16I 2(17 sin 2�+ sin 4�) + O(”3)

2W = − 2I(2 sin 2�+ 1

8 sin 4�)+�4cos 2�

+ 2

64I 2(−99 sin 2�− 9 sin 4�+ sin 6�) + �

32I(34 cos 2�+ cos 4�) + O(”3)

This completes the total elimination of �. Now, the transformation of the variables may be obtaineddirectly using the generator W and the classical Lie algorithms.The computation has been compared with the direct application of the Lie method to

equation (28), giving the same expressions for either the reduced system and the generator. Thesymbolic processor MATHEMATICA has been used in order to automate the whole set of calculations.This procedure can be extended up to any order, as an example it is shown the number of terms

in the di�erent orders up to order 10 (see Table II).Once the initial system (28) is transformed into (32), di�erent techniques can be followed to

solve de�nitely the problem.

5. CONCLUSIONS

In this paper an adaptation of the general scheme of the Lie transform is proposed. This approachreduces the algorithmic complexity of the transformation of a dynamical system through a Liemethod in the case in which part of the perturbation is Hamiltonian permitting to deal with someproblems intractable with the usual Lie treatment for di�erential systems.



Table II. Number of terms of the reduced equations andthe generator in each order using the normal algorithmof Lie transforms for vector �elds versus the algorithm

presented here

Vector treatment New algorithm

Order Eq. Gen. H NH GenH GenNH

1 2 6 1 1 2 22 2 10 1 1 3 43 2 20 1 1 4 124 3 28 1 2 5 185 3 42 1 2 6 306 4 54 1 3 7 407 4 72 1 3 8 568 5 88 1 4 9 709 5 110 1 4 10 9010 6 130 1 5 11 108

The algorithm of this paper can be employed in some typical cases for which Lie transformsare widely used. We mention here two of them. First, the theory of non-linear oscillators withperturbations both of canonical and non-canonical type. Second, the arti�cial satellite theory, inorder to study the combination between canonical forces of gravitational nature with other type ofperturbations, for which this kind of splitting permits the use of some typical transformations. Astudy on this line is now in progress.

ACKNOWLEDGEMENTS

The authors thank Professors A. Deprit, A. Elipe, S. Ferrer and K. R. Meyer for their fruitfulcomments. This work is partially supported by the Project ESP91–0919 of the Comisi�on Inter-ministerial Cient��ca y T�ecnica (CYCIT) of Spain and by the D�epartement de Math�ematiquesSpatiales at the Centre National d’Etudes Spatiales (Toulouse).

REFERENCES

1. A. A. Kamel, ‘Perturbation method in the theory of nonlinear oscillations’, Celes. Mech., 3, 90–106 (1970).2. J. Henrard, ‘On a perturbation theory using Lie transforms’, Celes. Mech., 3, 107–120 (1970).3. A. Deprit, ‘Canonical transformations depending on a small parameter, Celes. Mech., 1, 12–30 (1969).4. V. T. Coppola and B. R. Miller, ‘Synchronization of perturbed non-linear Hamiltonian oscillators’, Int. J. Non-LinearMech., 30, 69–80 (1995).

5. K. R. Meyer, ‘A Lie transform tutorial II’, in Computer Aided Proofs in Analysis, K. R. Meyer and D. S. Schmidt(eds.), The IMA Volumes in Mathematics and Its Applications, Vol. 28, Springer, New York, 1991, pp. 190–210.

6. K. R. Meyer, ‘Perturbation analysis of nonlinear systems’, in E. Tournier (ed.), Computer Algebra and Di�erentialEquations, Cambridge University Press, Cambridge, 1994, pp. 103–139.

7. K. R. Meyer and G. R. Hall, ‘Introduction to Hamiltonian dynamical systems and the N–body problem’, AppliedMathematical Sciences, Vol. 90, Springer, New York, 1992.

8. J. A. Sanders and F. Verhults, ‘Averaging methods in nonlinear dynamical systems’, Applied Mathematical Sciences,Vol. 59, Springer, New York, 1985.


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