Iran J Sci Technol Trans Sci (2017) 41:169–178 DOI 10.1007/s40995-017-0195-z

RESEARCH PAPER

Extension of the Lie Transform Theory Depending on a Small Parameter for Multi-parametric Dynamical Systems F. A. Abd El-Salam1,2

Received: 12 November 2014 / Accepted: 24 May 2015 / Published online: 14 March 2017  Shiraz University 2017

Abstract Lie transform method derived by Deprit and Hori in the 1960s allowed researchers to solve the perturbation problems depending on a small parameter. But actually the real systems are very complicated. In the astrodynamics, one usually encounters problems involving several perturbations which in turn yields dynamical system with several small parameters, e.g., oblateness of the massive objects, radiation pressure, mass loss, relativistic effects, drag perturbations, etc. To involve as many perturbations as the system requires, the theory of canonical Lie transform depending on a small parameter is extended to N-small parameters. Lie transform based on one small parameter is briefly surveyed. Some useful lemmas are proved. Then, the generalized Lie transformation method is developed. Keywords Lie transform  N-small parameters  Generating function  Canonical perturbations  Multiparametric dynamical systems

1 Introduction The canonical perturbation theory, especially Lie series and Lie transformation, have found a wide range of applications, particularly in celestial mechanics, especially in artificial satellite theory and in the elliptic restricted & F. A. Abd El-Salam [email protected] 1

Department of Mathematics, Faculty of Science, Taibah University, Medina, Kingdom of Saudi Arabia

2

Department of Astronomy, Faculty of Science, Cairo University, Cairo 12613, Egypt

three-body problem, magnetic optics, light optics, neutron transport, plasma physics, and other areas. The relevant methods were used in perturbation theory with a focus on the Lie transform method derived by Deprit and Hori in the 1960s. The single-parameter Deprit–Hori method was described in the original papers of Deprit and Hori as well as the more recent works of Meyer et al. (2009), Boccaletti and Pucacco (1996, 1999), Hori (1966) and Deprit (1969). Following a description of the single-parameter method, an original derivation is provided detailing an extension of the Deprit–Hori method to non-autonomous, two-parameter systems that may be compared to similar methods derived in Varadi (1985), Ahmed (1993) and Andrade (2008); validation of the two-parameter Deprit–Hori method is achieved through the normalization of the single-degree of freedom damped oscillator. Finally, a preliminary investigation into incorporating control terms in the normalization is conducted with the goal of deriving control laws in the transformed phase space. Extensive references to the applications literature are provided. The applications are of two types: expanding solutions of Hamilton’s equations and simplifying, i.e., reducing Hamiltonians to normal form. The expansions are not power series, but rather factored product. These expansions have the advantage of the approximating systems also being Hamiltonian. If the solutions of Hamilton’s equations are thought of as a mapping from the initial data to the solution at some fixed time, then the factored product expansion writes this mapping as a composition of simpler maps. Naturally, the approximating maps become more complicated as the order of the expansion increases. It is shown that there is a one-to-one relation between vector fields on a manifold and families of transformations of the manifold onto itself. This relation is essential in the study of various symmetries.

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Definition 1 Let M be a differentiable manifold. A oneparameter group of transformations, U, on M, is a differentiable map from M  R onto M such that Uðx; 0Þ ¼ x and Uððx; tÞ; sÞ ¼ Uðx; t þ sÞ, 8 x 2 M; t; s 2 R. Defining Ut ð xÞ ¼ Uðx; tÞ, then, for each t 2 R. Ut is a differentiable map from M to M and thus Utþs ð xÞ ¼ Uðx; t þ sÞ ¼ UðUðx; tÞ þ sÞ ¼ UðUt ð xÞ þ sÞ ¼ Us ðUt ð xÞÞ ¼ Us  Ut ð xÞ that is, Utþs ð xÞ ¼ Us  Ut ð xÞ ¼ Ut  Us ð xÞ (since t ? s = s ? t). U0 is the identity map of M since U0 ð xÞ ¼ U0 ðx; 0Þ ¼ x for all x 2 M. We have then Ut  Ut ð xÞ ¼ Ut  Ut ð xÞ ¼ U0 , which means that each map Ut has an inverse, Ut , which is also differentiable. Therefore, each Ut is a diffeomorphism, fUt jt 2 Rg is an Abelian group of diffeomorphisms of M onto M, and the map t ! Ut is a homomorphism from the additive group of the real numbers into the group of diffeomorphisms of M.

Lie derivative of f with respect to X and is denoted by LX f . From the expression, L X f = X f . Definition 3 Let H and W be assumed real and analytic in a bounded domain of the phase space. This insures a convergent, real-valued series expansion for the resultant canonical transformation in a neighborhood of the unperturbed system (that is, for small parameter, say e). The Lie derivative of H generated by W is defined by  N  X oH oW oH oW D LW H ½H; W  ¼  ð1Þ oqi opi opi oqi i¼1

Further, the nth Lie derivative is defined as LnW H ¼   LW H Ln1 W H and the zero-order derivative is the identity operator L0W H ¼ H. 2.1 Lie Transform Based on One Small Parameter Let the Hamiltonian function be represented by the series expansion

2 Lie Derivative of Functions Definition 2 Let U be a one-parameter group of transformations or a flow on M. As pointed out in Definition 1, the map Ut : M ! M, defined by Ut ðxÞ ¼ Uðx; tÞ, is a differentiable mapping. For f 2 C 1 ðMÞ, Ut f ¼ f  Ut also belongs to C 1 ðMÞ; the limit Ut f  f t!0 t

1 X en ð0Þ H ðq; p; mÞ n! n n¼0

This Hamiltonian system is transformed under the nearidentity state transformation ðq; pÞ ! ðq0 ; p0 Þ given explicitly by q ! Qðq0 ; p0 ; e; mÞ;

lim

represents the rate of change of the function f under the family of transformations Ut . If X is the infinitesimal generator of U, the curve Ux given by Ux ð xÞ ¼ Ux ðx; tÞ is the integral curve of X that starts at x; therefore t exists and depends on U only through its infinitesimal generator. This limit is called the Lie derivative of f with respect to X and is denoted by L X f . From the expression      Ut f  f f ðUt ð xÞÞ  f ð xÞ lim ð xÞ ¼ lim t!0 t!0 t t   f ðUðx; tÞÞ  f ð xÞ ¼ lim t!0 t   f ðUx ðtÞÞ  f ðUx ð0ÞÞ ¼ lim t!0 t 0

¼ Ux ð0Þ½ f  ¼ X x ½ f  ¼ Xf ð xÞ which shows that, for any differentiable function, the limit   limt!0 ð1=tÞ Ut f  f ð xÞ exists and depends on U only through its infinitesimal generator. This limit is called the

123

Hðq; p; e; mÞ ¼

p ! Pðq0 ; p0 ; e; mÞ

and defined implicitly through a generating function W ðq0 ; p0 ; mÞ, which is also represented by its series expansion W ðq0 ; p0 ; e; mÞ ¼

1 X en W nþ1 ðq0 ; p0 ; mÞ n! n¼0

this generating function is expressed in terms of the transformed state variables. The transformed Hamiltonian function is constructed term by term in the series form by H ðq0 ; p0 ; e; mÞ ¼ ¼

1 n X e

n! n¼0

Hn ðq0 ; p0 ; mÞ

1 n X e ðnÞ ðnÞ H 0 ð q0 ; p0 ; m Þ þ R 0 ð q0 ; p0 ; m Þ n! n¼0

~ ðq0 ; p0 ; e; mÞ þ Rðq0 ; p0 ; e; mÞ ¼H ~ ðq0 ; p0 ; e; mÞ ¼ HðQðq0 ; p0 ; e; mÞ; Pðq0 ; p0 ; e; mÞ; e; mÞ where H and R is a remainder function in the manner of ðdW=dmÞ.

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Theorem 4 Consider a non-autonomous Hamiltonian function expanded about a small parameter e as represented by the series 1 X en ð0Þ H ðq; p; mÞ Hðq; p; e; mÞ ¼ n! n n¼0

~ ¼ W

1 X en ~ ðnÞ W n! 0 n¼0 ð0Þ

~ ¼ W n and constructed using the recursive with W n equation ~ ðrþ1Þ ¼ W ~ ðrÞ þ W n nþ1

A canonical transformation is generated from the function 1 X en W nþ1 ðq0 ; p0 ; mÞ W ðq0 ; p0 ; e; mÞ ¼ n! n¼0 such that the transformed Hamiltonian function may be constructed recurrsively as 1 X en  ðnÞ 0 0 ðnÞ H ðq0 ; p0 ; e; mÞ ¼ H 0 ð q ; p ; m Þ þ R 0 ð q0 ; p0 ; m Þ n! n¼0 where the recursive equations are given by n   X n ðr Þ ðr Þ ðnÞ ðn1Þ Hðnrþ1Þ ¼ Hnþ1 þ LW nþ1 Hni ; R0 ¼ S 0 i i¼0 and Rðn0Þ ¼ 

o W nþ1 ; om ðrÞ

S ðnrþ1Þ ¼ S nþ1 þ

n   X n i¼0

i

ðrÞ

LW nþ1 S ni

The expansion of the explicit state transformation equations q ¼ Q ðq0 ; p0 ; e; mÞ and p ¼ P ðq0 ; p0 ; e; mÞ is represented by the series 1 n X e ðnÞ 0 0 q ðq ; p ; mÞ; q ¼ Qðq0 ; p0 ; e; mÞ ¼ n! 0 n¼0 p ¼ Pðq0 ; p0 ; e; mÞ ¼

1 n X e

n! n¼0

ðnÞ

p 0 ð q0 ; p0 ; m Þ

ð0Þ

where q0 ¼ q0 , p0 ¼ p0 and qðn0Þ ¼ q0 , pðn0Þ ¼ 0 for n [ 0. Moreover, the inverse transformation q0 ¼ Q0 ðq; p; e; mÞ ¼

1 n X e

n! n¼0

with the recursive equations ¼

0ðrÞ qnþ1

p0nðrþ1Þ ¼

0ðrÞ pnþ1

q0nðrþ1Þ

þ

1   X n n¼0

þ

1   X n n¼0

0ð0Þ q0

i i

0ðrÞ

LW~ iþ1 qni 0ðrÞ

LW~ iþ1 pni

0ð0Þ p0

where ¼ q; ¼ p and q0nð0Þ ¼ p0nð0Þ ¼ 0; for n [ 0. For more details, please refer to the proof included in the original works on the topic of Lie transform by pioneers Deprit (1969) and Hori (1966) and the previously mentioned works of Meyer et al. (2009), and Boccaletti and Pucacco (1996, 1999). In the following sections, we aim to generalize the theorem to a non-autonomous Hamiltonian function expanded about N-small parameters e1, e2,…,en. In order to achieve this goal, we are going to prove some basic lemmas introduced by the author.

3 Basic Lemmas (Abd El-Salam) The following equation holds   o o LW k2 ð:Þ  LW k2 ð2Þ ¼ LoW k2 ð:Þ oe1 oe1 oe1

Lemma 5

Proof

and may be constructed recursively as n   X n ðr Þ ðr Þ qðnrþ1Þ ¼ qnþ1 þ LW nþ1 qni i i¼0 n   X n ðr Þ ðr Þ pðnrþ1Þ ¼ pnþ1 þ LW nþ1 qni i i¼0 ð0Þ

1   X n ~ ðr Þ LW iþ1 W ni i n¼0

0ðnÞ

q0 ðq; p; mÞ

p0 ¼ P0 ðq0 ; p0 ; e; mÞ ¼

h

1 n X e n¼0

From the definition of the Lie derivative, we have     o o o oð:Þ oW k2 ¼ LW k2 ð:Þ  LW k2 op oe1 oe1 oe1 oq       o oð:Þ oW k2 oð:Þ o oW k2 oð:Þ o oW k2 þ   oq op oq oe1 op oq oe1 op oe1       o o oW k2 o o oW k2   op oq oe1 op oe1 oq     oð:Þ o oW k2 oð:Þ o oW k2  ¼ LoW k2 ð:Þ ¼ oq op oe1 op oq oe1 oe1

n!

0ðnÞ

p0 ðq; p; mÞ

Lemma 6 (Abd El-Salam) The following equation generalizes the previous lemma

is obtained by defining the inverse generating function

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o o o o o o  LW k1  LW k1  oes oes1 oe2 oe2 oe3 oes ¼ LQ s  o i¼2



Lemma 9 (Abd El-Salam) The following equation generalizes the previous lemmas to some extent ð3Þ

W k1

oei

Proof     o o o o o o ð:Þ  LW k1 ð:Þ  LW k1  oes oes1 oe2 oe2 oe3 oes   os ð : Þ os ð:Þ ¼ LW k 1 ð: Þ  LW k 1 oe2    oes1 oes oe2    oes1 oes   sþ1 o ð: Þ oW k1 oð:Þ o o s W k1 þ ¼ oe2    oes1 oes oq op oq op oe2    oes1 oes   sþ1 o ð: Þ oW k1 oð:Þ o o s W k1 þ ¼ oe2    oes1 oes oq op oq op oe2    oes1 oes   s o o ð: Þ oW k1 o os ð : Þ oW k1  þ oq oe2    oes1 oes op oe2    oes1 oes oq op     oð : Þ o o s W k1 oð : Þ o o s W k1 ¼  oq op oe2    oes1 oes op oq oe2    oes1 oes  ¼ L os ð:Þ ¼ L Q ð:Þ oe2 ...oes1 oes W k1

s

o i¼2 oei

W k1

h Lemma 7 (Abd El-Salam) It is easy also to prove the following lemma based on the way it was introduced in previous lemma   o o o o o o  LW s  LW s  ð: Þ oe1 oe2 oes1 oes1 oe2 oe1 ð4Þ ¼ LQs1 ð:Þ o

  o o o o o o o o o  LW k   LW k  oes oes1 oekþ1 oek1 oe2 oe1 oe1 oe2 oek1   o o  ð:Þ oekþ1 oes   ! k1  s  Y Y o o ð:Þ  LQk1 ð:Þ ¼ LQs o o W k i¼1 oei W k i¼kþ1 oei i¼kþ1 oe i¼1 oe i

Lemma 8 (Abd El-Salam) The following equation holds   o o o o o o o  LW k2  LW k2  ð:Þ oes oes1 oe3 oe1 oe1 oe3 oes    ! s  Y o ð : Þ o   L ð: Þ ¼ L Q s o oe1 W k2 i¼3 oei o oe W k2

oei

1

ð5Þ Proof      o o o o o o o  LW k2 LW k 2   ð: Þ oes oes1 oe3 oe1 oe1 oe3 oes        o oð:Þ o os W k 2 o oð:Þ o  ¼ oq oe1 op oe3 . . .oes1 oes op oe1 oq   o s W k2 oe3 . . .oes1 oes    ! s  Y oð : Þ o   ¼ L Qs L ð:Þ o o W k2 oe1 W k2 i¼3 oei oe i¼3 oe i

1

h

123

ð6Þ

i

Proof    o o o o o o o o  LW k   oes oes1 oekþ1 oek1 oe2 oe1 oe1 oe2   o o o  LW k  ð:Þ oek1 oekþ1 oes  !  k1  s  o Y o o Y oW k ð:Þ ¼ oq i¼1 oei op i¼kþ1 oei !    k1 s  o Y o o Y o  ð: Þ Wk op i¼1 oei oq i¼kþ1 oei "  ! Y   s  Y o o o k1 o  ð:Þ Wk oq i¼kþ1 oei op i¼1 oei  ! Y   # s  Y o o o k1 o  ð: Þ Wk op i¼kþ1 oei oq i¼1 oei  k1  Y o  ¼ L Qs ð:Þ  LQk1 o o W k i¼1 oei Wk oe i¼kþ1 i i¼1 oei   s Y o ð: Þ oe i i¼kþ1

Ws

i¼1 oei

i¼3



h

4 Special Cases 4.1 When we have the case as is proved in Lemma 1, we can obtain the same result, by just setting k = k1 we have   o o o o o o  LW k1  LW k1  oes oes1 oe2 oe2 oe3 oes ¼ L Qs ð7Þ o i¼2 oei

W k1

4.2 When we have the case as is proved in Lemma 2, we can obtain the same result, by just setting k = s, we have   o o o o o o  LW s  LW s  ð: Þ oe1 oe2 oes1 oes1 oe2 oe1 ð8Þ ¼ LQs1 ð:Þ o i¼1 oei

Ws

4.3 When we have the case as is proved in Lemma 3, we can obtain the same result, by just setting k = 2 we have

Iran J Sci Technol Trans Sci (2017) 41:169–178



173

 o o o o o o o  LW k2  LW k2  ð:Þ oes oes1 oe3 oe1 oe1 oe3 oes    ! s  Y oð : Þ o   L ð:Þ ¼ L Qs o o W k2 oe1 W k2 i¼3 oei oe i¼3 oei 1

ð9Þ (Abd El-Salam) The following equation holds

Lemma 10

LW 1 ðLW 2 ð:ÞÞ  LW 2 ðLW 1 ð:ÞÞ ¼ LLW 1 W 2 ð:Þ

ð10Þ

Proof LW 1 ðLW 2 ð:ÞÞ  LW 2 ðLW 1 ð:ÞÞ   o oð:Þ oW 2 oð:Þ oW 2 oW 1  ¼ oq oq op op oq op   o oð:Þ oW 2 oð:Þ oW 2 oW 1   op oq op op oq oq   o oð:Þ oW 1 oð:Þ oW 1 oW 2   oq oq op op oq op   o oð:Þ oW 1 oð:Þ oW 1 oW 2  þ op oq op op oq oq    oð:Þ o oW 2 oW 1 oW 2 oW 1  ¼ oq op oq op op oq    oð:Þ o oW 2 oW 1 oW 2 oW 1   op oq oq op op oq

¼ LL L

W 3 ðW 2 Þ

ðW 1 Þ ð:Þ

h Corollary 13 LW 1 ðLW 2 ðLW 3 ð:ÞÞÞ  LW 3 ðLW 2 ðLW 1 ð:ÞÞÞ ¼ Lð:Þ ðLW 1 ðW 2 ÞÞ

¼ LLW 1 W 2 ð:Þ h Corollary 11 LW 1 ðLW 2 ð:ÞÞ  LW 2 ðLW 1 ð:ÞÞ ¼ Lð:Þ ðLW 1 ðW 2 ÞÞ

  o oð:Þ oW 3 oð:Þ oW 3 oW 2 oW 1    op oq op op oq oq oq    o o oð:Þ oW 1 oð:Þ oW 1 oW 2   oq oq oq op op oq op    o oð:Þ oW 1 oð:Þ oW 1 oW 2 oW 3   op oq op op oq oq op    o o oð:Þ oW 1 oð:Þ oW 1 oW 2  þ op oq oq op op oq op    o oð:Þ oW 1 oð:Þ oW 1 oW 2 oW 3   op oq op op oq oq oq

   oð:Þ o oW 3 o oW 2 oW 1 oW 2 oW 1  ¼ oq op oq op oq op oq op   oW 3 o oW 2 oW 1 oW 2 oW 1   op oq oq op op oq

   oð:Þ o oW 3 o oW 2 oW 1 oW 2 oW 1   op oq oq op oq op op oq   oW 3 o oW 2 oW 1 oW 2 oW 1   op oq oq op op oq

ð11Þ

Proof

ð13Þ

Proof LW 1 ðLW 2 ð:ÞÞ  LW 2 ðLW 1 ð:ÞÞ ¼ LLL

W 3 ðW 2 Þ

ðW 1 Þ ð:Þ

¼ Lð:Þ ðLW 1 ðLW 2 ðW 2 ÞÞÞ ¼ Lð:Þ ðLW 1 ðLW 3 ðW 2 ÞÞÞ

LW 1 ðLW 2 ð:ÞÞ  LW 2 ðLW 1 ð:ÞÞ ¼ LLW 1 W 2 ð:Þ ¼ Lð:Þ ðLW 1 ðW 2 ÞÞ

5 Generalized Lie Transformation Method

Lemma 12 (Abd El-Salam) The following equation generalizes the previous lemma LW 1 ðLW 2 ðLW 3 ð:ÞÞÞ  LW 3 ðLW 2 ðLW 1 ð:ÞÞÞ ¼ LLL ðW Þ ðW 1 Þ ð:Þ W3

2

Proof LW 1 ðLW 2 ðLW 3 ð:ÞÞÞ  LW 3 ðLW 2 ðLW 1 ð:ÞÞÞ    o o oð:Þ oW 3 oð:Þ oW 3 oW 2  ¼ oq oq oq op op oq op    o oð:Þ oW 3 oð:Þ oW 3 oW 2 oW 1   op oq op op oq oq op    o o oð:Þ oW 3 oð:Þ oW 3 oW 2   op oq oq op op oq op

ð12Þ

The Deprit–Hori method provides a robust methodology for normalizing a perturbed Hamiltonian system about its unperturbed form as parameterized by the small parameter. Following the same way obtained by Duffy (2012), we can extend the procedure to n-parameters as follows with the help of the previous lemma obtained by the author of this manuscript. Consider an n-expanded Hamiltonian system 1 X 1 X Hðq; p; e1 ; e2 ; . . .; es ; mÞ ¼ n1 ¼0 n2 ¼0 1 ns ns1 X es es1 en 1 Þ    1 Hðn0;0;...;0  ðq; p; mÞ s ;ns1 ;...;n1 n ! n ! n ! s s1 1 n ¼0

ð14Þ

S

whose higher-order terms are parameterized by s-quantities: e1, e2, … and, es. This s-parameter case was treated for

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autonomous system by Varadi (1985), (Deprit 1969), whose analysis was subsequently extended to three parameters by Ahmed (1993) and to N parameters by Andrade (2008). While it is possible to apply Varadi’s method to non-autonomous systems, it requires an expansion of the phase space to incorporate the independent variable and Hamiltonian function as extra state variables. This renders the system autonomous and allows for the application of Varadi’s formulation. To circumvent the need to expand the phase space, an original theorem is presented in the sequel in which the two-parameter Deprit– Hori method is formulated directly in terms of the original non-autonomous system by applying a remainder function in the tradition of the single-parameter method presented in Theorem 4. As before, the Lie transform operator defined in (1) is applied to formulate the canonical transformation, but now through a number of s generating functions W k ; k ¼ 1; 2; . . .; s W k ðq0 ; p0 ; e1 ; e2 ; . . .; es ; mÞ ¼

1 X 1 X 1 ns ns1 X es es1 n !n ! n ¼0 n ¼0 n ¼0 s s1 1

2

S

en1 ð0;0;...;0Þ    1 W k;ns þ1;ns1 þ1;...;n1 þ1 ðq0 ; p0 ; mÞ n1 !

ð15Þ

2

1 X 1 X



n1 ¼0 n2 ¼0

1 ns ns1 X es es1 n !n ! n ¼0 s s1 S

en 1 Þ    1 Hðn0;0;...;0 ðq; p; mÞ n1 ! s ;ns1 ;...;n1

ð17Þ

A canonical transformation is generated from the pair of functions 1 X 1 1 s1 X X ens s ens1 W k ðq0 ; p0 ; e1 ; e2 ; . . .; es ; mÞ ¼  n !n ! n ¼0 n ¼0 n ¼0 s s1 1

2

S

en1 ð0;0;...;0Þ    1 W k;ns þ1;ns1 þ1;...;n1 þ1 ðq0 ; p0 ; mÞ n1 !

ð18Þ

such that the transformed Hamiltonian function may be constructed term by term in the series 1 X 1 X

1 s1 X ens s ens1 en1  1 n ! n ! n1 ! n1 ¼0 n2 ¼0 nS ¼0 s s1  ðns ;ns1 ;...;n1 Þ 0 0 ðn ;ns1 ;...;n1 Þ 0 0  H0;0;...;0 ðq ; p ; mÞ þ Rk; s 0;0;...;0 ð q ; p ; mÞ

H ðq0 ; p0 ; e1 ; e2 ; . . .; es ; mÞ ¼



ð19Þ by the extended recursive equations

The transformed Hamiltonian function is constructed term by term in the series 1 X 1 1 s1 X X ens s ens1  H ðq0 ; p0 ; e1 ; e2 ; . . .; es ; mÞ ¼ n !n ! n ¼0 n ¼0 n ¼0 s s1 1

Hðq; p; e1 ; e2 ; . . .; es ; mÞ ¼

S

en11 ð0;0;...;0Þ 0 0 H ðq ; p ; mÞ n1 ! ns ;ns1 ;...;n1 1 X 1 1  ns ns1 X X es es1 en1 ðns ;ns1 ;...;n1 Þ    1 H0;0;...;0 ... ¼ ns ! ns1 ! n1 ! n1 ¼0 n2 ¼0 nS ¼0 ðns ;ns1 ;...;n1 Þ 0 0 ðq0 ; p0 ; mÞ þ R0;0;...;0 ð q ; p ; mÞ 

ðr ;r

;...;r Þ

s1 1 s1 ;...;rk þ1;...;r1 Þ Hðnrss;n;rs1 ¼ Hnss;ns1 ;...;n1 ;...;nk þ1;...;n1     ns n1 X n2 X X n2 n1 þ   i1 i2 i1 ¼0 i2 ¼0 iS ¼0   ns ðr ;r ;...;r1 Þ  LW k;is þ1;is1 þ1;...;i1 þ1; Hnssis1 s ;ns1 is1 ;...;n1 i1 is

and ( ðns ;ns1 ;...;n1 Þ R0;0;...;0

¼

ð0;0;...;n 1;...;0Þ

k S k; 0;0;...;0

;

ðn ;ns1 ;...;nk 1;...;n1 Þ S k; s0;0;...;0 ;

nk 6¼ 0 nk 6¼ 0; k ¼ 1; 2; . . .; s

~ 0 ; p0 ; e1 ; e2 ; . . .; es ; mÞ þ Rðq0 ; p0 ; e1 ; e2 ; . . .; es ; mÞ ¼ Hðq ð16Þ ~ represent the old Hamilwhereas before, the terms in H tonian written explicitly in terms of the transformed state variables 0

0

0

0

0

0

~ ðq ; p ; e; c; mÞ ¼ HðQðq ; p ; e; c; mÞ þ Pðq ; p ; e; c; mÞ; e; c; mÞÞ H and the terms in R comprise the remainder function and are generated from a pair of intermediary functions S and T . Theorem 14 (Abd El-Salam) Consider a non-autonomous Hamiltonian function expanded about s-small parameters, namely e1, e2, …, es as represented by the series

123

ð20Þ

ð21Þ where ð0;0;...;0Þ

S k;ns ;ns1 ;...;n1 ¼  ðr ;r

;...;r þ1;...;r Þ

o W k;ns þ1;ns1 þ1;...;n1 þ1 om ðr ;r

ð22Þ

;...;r Þ

1 1 ¼ S k;snss1 S k;ns s ;ns1s1 ;...;nk 1 ;ns1 ;...;nk þ1;...;n1   nS  n1 X n2 X X n2 n1 þ   i i2 1 i1 ¼0 i2 ¼0 iS ¼0   ns ðr ;r ;...;r1 Þ  LSk;is þ1;is1 þ1;...;i1 þ1; S k;s ns1 s is ;ns1 is1 ;...;n1 i1 is

ð23Þ

Moreover, the terms appearing in the expanded transformation containing ‘‘mixed parameters’’ (i.e., of the form s1 ens s ens1 . . .en11 ; 8ns 6¼ 0) may be obtained equivalently using

Iran J Sci Technol Trans Sci (2017) 41:169–178

175

any of generating functions W k such that these generating functions satisfy the Deprit commutation conditions, o W k1 o W k2   LW k2 W k1 ¼ 0; oek1 oek2

k1 6¼ k2

and incorporating the additional terms from (29) for the expanded phase space, the extended Deprit operators are defined as

ð24Þ D

EWk ¼

which term by term implies

o oW k o oW k o ¼ DW k þ þ LW k þ oek om oH om oH

Note that the original Deprit operators satisfy the following conditions: the zeroth-order Deprit operators act as identity operations, D0W k H ¼ H and successive applications of the Deprit operators are given by  s1  s2 o o DsW1 k DsW2 k ¼ þ LW k1 þ LW k2 ; ð32Þ 1 2 oek1 oek2

W k1 ;ns þ1;ns1 þ2;...;n1 þs ¼ W k2 ; ns þ1;ns1 þ2;...;n1 þs   nS  n1 X n2 X X n2 n1 þ  i1 i2 i1 ¼0 i2 ¼0 iS ¼0   ns ðr ;r ;...;r1 Þ  LW k;is þ1;is1 þ1;...;i1 þ1; W k1s; ns1 s is ;ns1 is1 ;...;n1 i1 is ð25Þ Proof The canonical transformation from coordinates (q, p) to ðq0 ; p0 Þ may be formulated in terms of the generating functions W k through the pair of non-autonomous systems 0 1 o 0 0   W ð q ; p ; e ; e ; . . .; e ; m Þ k 1 2 s B op C o q C ¼B ð26Þ @ A o oek p 0 0  W k ðq ; p ; e1 ; e2 ; . . .; es ; mÞ oq with initial conditions qðe1 ; e2 ; . . .; es ¼ 0Þ ¼ q0 and pðe1 ; e2 ; . . .; es ¼ 0Þ ¼ p0 . Notice that both these systems are in Hamiltonian form where W k take the place of the Hamiltonian function and and serve as the independent variables. For well-behaved functions, the general solution for this coupled system is a pair of analytic functions defining the near-identity state transformation equations,  q ¼ Qðq0 ; p0 ; e1 ; e2 ; . . .; es ; mÞ; ð27Þ p ¼ Pðq0 ; p0 ; e1 ; e2 ; . . .; es ; mÞ

The question of commutativity will be treated in the sequel. When applied to a function F ðq; p; e1 ; e2 ; . . .; es ; mÞ with no explicit dependence on the Hamiltonian, the right-most terms in (32) are zero and the extended Deprit operators reduce to the original Deprit operators, E W k F ¼ DW k F. Furthermore, under the transformation q ¼ Qðq0 ; p0 ; e1 ; e2 ; . . .; es ; mÞ and p ¼ Pðq0 ; p0 ; e1 ; e2 ; . . .; es ; mÞ, the partial derivative of F with respect to ek is o F ðq; p; e1 ; e2 ; . . .; es ; mÞjq¼Qðq0 ;p0 ;;e1 ;e2 ;...;es ;mÞ oek p¼Pðq0 ;p0 ;e1 ;e2 ;...;es ;mÞ   oF oF oq oF op  þ þ j 0 ;p0 ;e ;e ;...;e ;mÞ s 1 2 oek oq oek op oek q¼Qðq p¼Pðq0 ;p0 ;e1 ;e2 ;...;es ;mÞ   oF oF oW k oF oW k  ¼ þ j 0 ;p0 ;e ;e ;...;e ;mÞ s 1 2 oek oq op op oq q¼Qðq p¼Pðq0 ;p0 ;e1 ;e2 ;...;es ;mÞ   oF  þ LW k F jq¼Qðq0 ;p0 ;e ;e ;...;e ;mÞ s 1 2 oek p¼Pðq0 ;p0 ;e ;e ;...;e ;mÞ 1 2

where Qðe1 ; e2 ; . . .; es ¼ 0Þ ¼ q0 es ¼ 0Þ ¼ p0 .

and

Pðe1 ; e2 ; . . .;

In the spirit of Deprit’s original proof, one may define an extended phase space by appending the independent variable and the Hamiltonian function to the original state variables q ! fq; mg;

p ! fp; Hg

ð28Þ

To apply the Lie operator within the extended phase space but still maintain its definition within the original (q, p) phase space, one must incorporate the additional terms LW k

oW k o oW k o oW k o þ ¼ LW k þ  oH om om oH om oH

ð29Þ

wherein the generating functions will vary with v, but will not be explicitly dependent on the Hamiltonian itself such k that oW oH ¼ 0. The Deprit operators are defined as D

DW k ¼

o þ LW k oek

ð31Þ

ð30Þ

¼ DW k Fj

ð33Þ

s

q¼Qðq0 ;p0 ;e1 ;e2 ;...;es ;mÞ p¼Pðq0 ;p0 ;e1 ;e2 ;...;es ;mÞ

By extension, one may also express the mixed e1 e2 . . .es derivatives to any order by on1 on2 on s    F ðq; p; e1 ; e2 ; . . .; es ; mÞjq¼Qðq0 ;p0 ;e ;e ;...;e ;mÞ s 1 2 oen11 oen22 oens s p¼Pðq0 ;p0 ;e1 ;e2 ;...;es ;mÞ ¼ DnW1 1 DnW2 2 . . .DnWs s F q¼Qðq0 ;p0 ;e1 ;e2 ;...;es ;mÞ p¼Pðq0 ;p0 ;e1 ;e2 ;...;es ;mÞ

ð34Þ Upon examining (34), it is evident that the Deprit operators may be applied within an expansion series in e1 ; e2 ; . . .; es . This approach is subsequently applied to the Hamiltonian function under the transformation ðq; pÞ ! ðq0 ; p0 Þ. Consider the canonical transformation of a non-autonomous Hamiltonian function resulting in a transformed Hamiltonian function in the form

123

176

Iran J Sci Technol Trans Sci (2017) 41:169–178

H ¼ HðQðq0 ; p0 ; e1 ; e2 ; . . .; es ; mÞ; 0

Pðq ; p ; e1 ; e2 ; . . .; es ; mÞ; e1 ; e2 ; . . .; es ; mÞ þ Rðq0 ; p0 ; e1 ; e2 ; . . .; es ; mÞ ~ ðq0 ; p0 ; e1 ; e2 ; . . .; es ; mÞ þ Rðq0 ; p0 ; e1 ; e2 ; . . .; es ; mÞ ¼H

1

2

S

 ðHðq; p; e1 ; e2 ; . . .; es ; mÞÞj q¼Qðq0 ;p0 ;e1 ;e2 ;...;es ;mÞ p¼Pðq0 ;p0 ;e1 ;e2 ;...;es ;mÞ



ð36Þ For the time being, the Hamiltonian function is treated as in function F used previously, that is, with no explicit dependence on the Hamiltonian as a state variable. Equation (36) can be represented in terms of the extended Deprit operators by substituting (34) for the partial derivatives in (36) such that (with H ¼ Hðq; p; e1 ; e2 ; . . .; es ; mÞ, ~ ¼H ~ ðq0 ; p0 ; e1 ; e2 ; . . .; es ; mÞ H 1 X 1 1 ns ns1 X X es es1 ~¼  H n !n ! n1 ¼0 n2 ¼0 nS ¼0 s s1 n1  e    1 DnW1 1 DnW2 2    DnWs s H n1 !

q¼Qðq0 ;p0 ;e1 ;e2 ;...;es ;mÞ 0 0

ð37Þ Further introducing the subscripted and superscripted formulation H¼

1 X



n1 ¼0 n2 ¼0

1 ns X es

s1 ens1

n !n ! n ¼0 s s1



S

DnW1 1 DnW2 2    DnWs s H ¼

1 X

1 X

n1 ¼0 n2 ¼0



en11 n1 !

s1 ;...;r1 Þ Hðnrss;n;rs1 ;...;n1

en11 n1 !



Þ Hðn0;0;...;0 s ;ns1 ;...;n1

1 ns X es



n1 ¼0 n2 ¼0

1 X

ens s

s1 ens1

n !n ! n ¼0 s s1

n !n ! nS ¼0 s s1 ð38Þ



S

en11 n1 !

ðm;nÞ

s

ð39Þ e1 ;e2 ;...;es ¼0

ðn ;n

s s1 H0;0 H0;0;...;0

;...;n1 Þ

ðq0 ; p0 ; mÞ ð40Þ

Within the subscripted and superscripted formulation, the Lie derivatives satisfy     ns  n1 X n2 X X n1 n2 ns ;...;r1 LW k Hrnss;r;ns1 ¼     s1 ;...;n1 i1 i2 is i1 ¼0 i2 ¼0 is ¼0 1 LW k ;is þ1;is1 þ1;...;i1 þ1 Hrnss;ris1s ;n;...;r s1 is1 ;...;n1 i1

ð41Þ such that the terms included in the expansion series under the Deprit operator satisfy

n1 ¼0 n2 ¼0



1 s1 X ens s ens1 en1 ðr ;r ;...;r þ1;...;r1 Þ    1 Hk;ns s ;ns1s1 ;...;nk 1 n ! n ! n1 ! nS ¼0 s s1

! 1 s1 X ens s ens1 en11 ðrs ;rs1 ;...;rk ;...;r1 Þ  H ¼ DW k  n ! n ! n1 ! k;ns ;ns1 ;...;n1 n1 ¼0 n2 ¼0 nS ¼0 s s1 ! 1 X 1 1 s1 X o X ens s ens1 en11 ðrs ;rs1 ;...;rk ;...;r1 Þ ¼  H  oek n1 ¼0 n2 ¼0 nS ¼0 ns ! ns1 ! n1 ! k;ns ;ns1 ;...;n1 ! 1 X 1 1 s1 X X ens s ens1 en11 ðrs ;rs1 ;...;rk ;...;r1 Þ þ LW k  H  n ! n ! n1 ! k;ns ;ns1 ;...;n1 n1 ¼0 n2 ¼0 nS ¼0 s s1  1 X 1 1 s1 X X ens s ens1 en1 ðr ;rs1 ;...;r1 Þ    1 Hnss;ns1  ¼ ;...;nk þ1;...;n1 n ! n ! n ! s s1 1 n1 ¼0 n2 ¼0 nS ¼0    ! n1 X n2 nS  X X ns n1 n2    þ i i is 1 2 i1 ¼0 i2 ¼0 iS ¼0 1 X 1 X

;...;r Þ

1 LW k;is þ1;is1 þ1;...;i1 þ1; Hnssis1 s ;ns1 is1 ;...;n1 i1

ð42Þ Thus, comparing (41) and (42) to (38), all the unknown ðr ;r

;...;r þ1;...;r Þ

1 functions Hk;ns s ;ns1s1 ;...;nk 1 may be constructed term by term using recursive equations

ðr ;r

s1 ens1

the expansion of the original Hamiltonian function is repÞ resented by the series of functions Hðn0;0;...;0 and the s ;ns1 ;...;n1 ~ expansion of H is represented by

123

1 X

ðr ;r

p¼Pðq ;p ;e1 ;e2 ;...;es ;mÞ e ;e ;...;e ¼0 s 1 2

1 X

¼

1 X

1 X 1 X

e1 ;e2 ;...;es ¼0



1 2

ð35Þ ~ represents the original Hamiltonian function where H written explicitly in terms of the state transformation equations and R represents a remainder function. The ~ ¼H ~ ðq0 ; p0 ; e1 ; e2 ; . . .; es ; mÞ in a Taylor expansion of H series about ek ¼ 0 is   n1 n2 1 X 1 1  ns ns1 X X es es1 en 1 o o on s ~¼  1     H ns ! ns1 ! n1 ! oen11 oen22 oens s n1 ¼0 n2 ¼0 nS ¼0 ~ 0 ; p0 ; e1 ; e2 ; . . .; es ; mÞ Hðq e1 ;e2 ;...;es ¼0   n1 n2 1 X 1 1  ns ns1 X X es es1 en11 o o on s  ¼     ns ns ! ns1 ! n1 ! oen11 oen22 oes n ¼0 n ¼0 n ¼0

1 X 1 X

1 s1 X ens s ens1 n !n ! n1 ¼0 n2 ¼0 nS ¼0 s s1 n1 e    1 DnW1 1 DnW2 2    DnWs s H q¼Qðq0 ;p0 ;e ;e ;...;e ;mÞ s 1 2 n1 ! p¼Pðq0 ;p0 ;e ;e ;...;e ;mÞ

~¼ H

0

;...;r Þ

s1 1 s1 ;...;rk þ1;...;r1 Þ Hðnrss;n;rs1 ¼ Hnss;ns1 ;...;n1 ;...;nk þ1;...;n1   nS  n1 X n2 X X n2 n1 þ   i1 i2 i1 ¼0 i2 ¼0 iS ¼0   ns ðr ;r ;...;r1 Þ  LW k;is þ1;is1 þ1;...;i1 þ1; Hnssis1 s ;ns1 is1 ;...;n1 i1 is

ð43Þ

referred to as the extended recursive equations. The formulae in (18) are recursive in the sense that each successive term is dependent only on terms preceding it, starting

Iran J Sci Technol Trans Sci (2017) 41:169–178

177

Þ with the original Hamiltonian Hðn0;0;...;0 . Thus far, the s ;ns1 ;...;n1 method has provided the means to construct the explicit substitution of the state transformation equations into the original Hamiltonian through the Deprit operators DW k . ~ in (35), one must still While this provides the function H account for the remainder function. Applying the extended Deprit operators to the Hamiltonian state variable yields

oW k E W k Hk ¼ DW k Hk  om E nW1 1 E nW2 2    E nWs s H ¼ DnW1 1 DnW2 2    DnWs s H s s Y X oW k  DnWi i Dk1 Wk om k¼1 i¼1;i6¼k

2

S

ð49Þ DW k2 DW k1 H ¼

1 X

1 X



n1 ¼0 n2 ¼0

ð45Þ

o W k;ns þ1;ns1 þ1;...;n1 þ1 om



en11 n1 !

Þ Hðn1;1;...;1 s ;ns1 ;...;n1

DW k1 DW k2 H ¼ DW k2 DW k1 H

ð51Þ

The Lie derivatives are not themselves commutative, but instead satisfy the condition LW k 2 LW k 1 ¼ LW k 1 LW k 2  LL W k

1

ð52Þ

W k2

wherein the last term represents the Lie derivative generated by LW k1 W k2 . Expanding the Deprit commutative condition in (52) based on the definition of the Deprit operator in Eq. (31) yields the constraint DW k1 DW k2    DW ks

  s s Y X o o ¼ LW i þ LW k oek oei i¼1 k¼1;k6¼i s s X X o LW k þ þ oei i¼1 i¼1 ð53Þ

and DW ks DW ks1    DW k1 ¼

s s Y X i¼1 k¼1;k6¼i

 LW siþ1 þ

and the recursive formulae ðr ;r

nS ¼0

ns ! ns1 !

s1 ens1

ð50Þ

ð47Þ

;...;r þ1;...;r Þ

ens s

ð46Þ

such that in (45), the remainder function R is constructed term by term according to ( ð0;0;...;n 1;...;0Þ ; nk 6¼ 0 S k;0;0;...;0k ðns ;ns1 ;...;n1 Þ ¼ R0;0;...;0 ðns ;ns1 ;...;nk 1;...;n1 Þ S k;0;0;...;0 ; nk 6¼ 0; k ¼ 1; 2; . . .; s

ðr ;r

1 X

For consistency sake, the terms on the right-hand side should be the same in either case, which implies that the Deprit operators are commutative, that is,

The first term appearing in (45) is equivalent to the explicit substitution of the state transformation equations into the original Hamiltonian function while treating it as being independent of the Hamiltonian state variable. This effectively generates the autonomous part of the transfor~ in (35). The other two mation represented by the term H Ps Qs ni k1 oW k terms appearing in (45), k¼1 i¼1;i6¼k DW i DW k om comprise the remainder function and may be constructed term by term in the same manner as the Hamiltonian k function, but as acting on the functions oW om . To do so in a constructive manner, define the intermediary functions S k by ð0;0;...;0Þ

1

ð44Þ

and for higher-order and mixed terms, k ¼ 1; 2; . . .; s

S k;ns ;ns1 ;...;n1 ¼ 

Based on the preceding discussion, one may derive either of the following representations 1 X 1 1 s1 X X ens s ens1 en 1 Þ    1 Hðn1;1;...;1  DW k1 DW k2 H ¼ s ;ns1 ;...;n1 n ! n ! n ! s s1 1 n ¼0 n ¼0 n ¼0

s Y

o oeskþ1

þ

LW skþ1 þ

i¼1

oesiþ1 s Y o i¼1

 LW skþ1

oesiþ1 ð54Þ

;...;r Þ

1 1 ¼ S k;snss1 S k;ns s ;ns1s1 ;...;nk 1 ;ns1 ;...;nk þ1;...;n1     nS n1 X n2 X X n2 n1 þ   i1 i2 i1 ¼0 i2 ¼0 iS ¼0   ns ðr ;rs1 ;...;r1 Þ  LW k;is þ1;is1 þ1;...;i1 þ1 S k;ns s i s ;ns1 is1 ;...;n1 i1 is

o

wherein the last two terms that satisfy ð48Þ

Together with (18), (21) and (22), define the complete transformation of the Hamiltonian function into H as formulated in (16). To complete the proof, one must address the issue of commutativity in the Deprit operators.

s s Y Y o o ¼ oei i¼1 oesiþ1 i¼1

ð55Þ

are equivalent operations according to the symmetry of nth-mixed-order partial derivatives (as a generalization of Clairaut’s theorem). Substituting the Lie commutation condition in (52) into (53) and re-arranging terms yields

123

178

0¼

Iran J Sci Technol Trans Sci (2017) 41:169–178

  s s Y X o o LW i þ LW k oek oei i¼1 k¼1;k6¼i   s s Y X o o LW siþ1 þ LW skþ1  oeskþ1 oesiþ1 i¼1 k¼1;k6¼i

which implies the condition 0¼

þsi¼1 LW k si¼1 LW skþ1 ð56Þ Moreover, the partial derivatives and Lie operators satisfy the conditions s s s s X Y X X o o 0¼ LW k  LW siþ1 oei oeskþ1 i¼1 k¼1;k6¼i i¼1 k¼1;k6¼i ! s s s s Y Y X X o o  LW skþ1  LW i oesiþ1 oek i¼1 k¼1;k6¼i i¼1 k¼1;k6¼i þ

s Y

LW k 

i¼1

s Y i¼1

  s s X Y o o LW k  LW siþ1 oei oeskþ1 i¼1 k¼1;k6¼i

¼ LQs1

o Ws i¼1 oei



þ LQs

 s s Y X i¼1 k¼1;k6¼i

þ

o W k1 i¼2 oei

s Y

LW k 

i¼1

o i¼3 oei

 o oei

W1

ð58Þ

o o LW  LW ¼ LoW oc oc oc

ð59Þ

oesiþ1 s Y

þ LQs



o

LW skþ1  LW i

o oek

LW skþ1

i¼1

o o LV  LV ¼ LoV ; oe oe oe

where the last terms are the Lie derivatives generated by oV oe and oW , respectively. Substituting (59) into (56) yields oe 0 ¼ LoV  LoW þ LLW V ¼ LðoVoW þLL V Þ oe oc oe oc W

123

ð61Þ

This condition is herein referred to as the Deprit commutation condition in reference to its derivation from (51). By applying the recursive algorithms used previously, the Deprit commutation condition can also be expressed term by term as 0 ¼ V mþ1;nþ2  W mþ2;nþ1  n X m   X n m LW jþ1;iþ1 V mjþ1;niþ1 þ i j i¼0 j¼0

ð62Þ

References

LW skþ1 ð57Þ

0¼

oV oW  þ LW V oe oc

ð60Þ

Ahmed M (1993) Multiple-parameter Lie transform. Earth Moon Planets 61:21–28 Andrade M (2008) N-parametric canonical perturbation method based on Lie transforms. Astron J 136:1030–1038 Boccaletti D, Pucacco G (1996) Theory of orbits, vol 1: integrable systems and non-perturbative methods. Springer, Berlin Boccaletti D, Pucacco G (1999) Theory of orbits, vol 2: perturbative and geometrical methods. Springer, Berlin Deprit A (1969) Canonical transformation depending on a small parameter. Celest Mech 1:12–30 Duffy B (2012) Analytical methods and perturbation theory for the elliptic restricted three-body problem of astrodynamics. A Dissertation submitted to the Faculty of the School of Engineering and Applied Science of the George Washington University Hori G (1966) Theory of general perturbations with unspecified canonical variables. Publ Astron Soc Jpn 18(4):287–296 Meyer KR, Hall GR, Offin D (2009) Introduction to Hamiltonian dynamical systems and the N-body problem, 2nd edn., Applied mathematical sciencesSpringer, New York Varadi F (1985) Two-parameter Lie transforms. Celest Mech 36:133–142