E X P A N S I O N F O R M U L A E I N C A N O N I C A L T R A N S F O R M A T I O N S

D E P E N D I N G O N A S M A L L P A R A M E T E R

AHMED ALY KAMEL Stanford University, Stanford, Calif., U.S.A.

(Received 9 April, 1969)

Abstract. The theory of perturbation based on Lie transforms is considered. Deprit's equation is reduced to a form which enables us to generate simplified general recursion formulae. These expansions are then modified to speed up the implementation of such perturbation theory in the computerized symbolic manipulation.

1. Introduction

I f a system is described by a Hamiltonian depending on a small parameter, then canonical transformation can sometimes be obtained using a yon Zeipel generating function (see for instance Brouwer and Clemence, 1961). In such a case, the transfor- mation is implicit because the generating function is in mixed variables (the old coordinates and the new momenta).

The shortcomings of von Zeipel's method, when the generating function itself depends on a small parameter, were felt by Breakwell and Pringle (1966), and Deprit (1966), when they used a von Zeipel generating function to remove the short period terms from the Hamiltonian of a particle in the neighborhood of the triangular points in the restricted problem of three bodies. Breakwell (see Schechter, 1968) recognized, after comparing with Deprit et al. (1967), that the long period part of the second order Hamiltonian derived in mixed variables was misleading, and that it was possible to obtain a different representation in terms of new variables only. Using this suggestion, Schechter (1968) obtained a more valid second order expression. Deprit (1968) at- tacked the problem using Lie transforms and extended the expansion to include higher orders. In this paper Deprit 's recursive algorithm is reduced to a form which enables us to generate simplified and modified general formulae (Section 3 and Section 4).

2. Background

A Lie transform may be defined by the differential equations

dx d~ = Wx(x , X , t; e) ( la)

dX d~ = - Wx(x' X , t; e) ( lb)

dt - 0 ( l c )

de

Celestial Mechanics 1 (1969) 190-199; �9 D. Reidel Publishing Company, Dordrecht-Holland

EXPANSION FORMULAE IN C A N O N I C A L TRANSFORMATIONS 1 9 1

dF d~ = Wt(x, X, t; e) ( ld)

with the initial conditions x =y, X = Y, t = t, and F = 0 at e = 0. The foregoing equations define a canonical t ransformation. This can be shown as follows: for any e, the

differentials dx, dX, 5x, 5X, and bF produced by the initial changes dy, d Y, dt, 6y,

and 6 Y satisfy the equat ion*

d de [dx. 6 X - d X . 6x - dt 6F] = 0. (2)

Hence, dx" b X - d X " 3 x - d t b F is independent of e and equals its value at ~ = 0, so that

for F = H ( x , X, t; e ) - K ( y , Y, t; e)

2" 6 X - X . 6x -- OH = p. b Y - ~ . 6y - f K . (3)

Therefore, if x and X satisfy the canonical equations

2 = Hx, 2 = - H~, (4)

then, also y and Y satisfy the canonical equations

.9 = Ky, Y = - K, . (5)

Now, take any indefinitely differentiable function f ( x , X, t; e) that can be expressed in terms of x, X, t and e as a power series in z, in the form

2 f ( x , X , t ; e ) = ~ ~ T J ( x , X , t ; ~ ) = ~ f . ( x , X , t ) (6) e = 0

n = O n = O

then, in terms of y, Y, t, and c as a power in e, it takes the form

s" d" s" f ( , ) t) (7) f ( x , X , t; ~) = ~ d~ . f ( x , X , t; e) = ~ (y, Y,

n = O n = O

where

and

j . ( x , x , t) = x , t; ~) , n > / 0 ; (8) e = 0

df (x, X, t; e) = ~f dx d X (9a) de " ~ + f~'de~ + I x " d~ '

[d~ ] f ( " ) ( y , Y , t ) = d ~ , f ( x , X , t ; e ) , n>~O.

/ ~ = 0

* Notice that

d -- dx -- dWx = Wxx.dx -k Wxx .dX -k Wxt dt, de

d -- bx -- 6Wx = Wxx.6x + Wxx.bX, etc. de

(9b)

1 9 2 AHMED ALY KAMEL

Notice that

fo (x ,X , t )= f (x ,X, t ;O) , and f ( O ) ( y , Y , t ) = f ( y , Y,t;O).

Using Equations (la) and (Ib), Equation (9a) can be written as

df Of - + Lwf (10)

de Oe

where L w is a linear operator called Lie derivative generated by W, and is defined by

Lwf = ( f ; W) = fx 'W x - fx'Wx. (11)

Taking f = x, X, and F in (7), and using (la), (lb), and (ld), one obtains the following ct)

x = y + ~. y(")(y, Y, t)

n = l

at)

x = r + ~ r ( " ~ ( y , r , 0

n = l

u = ~: - ~ R ~~ r , t),

n = l

where, for n/> 1 we have

{d "-1 y~"~ = \ ~ Wx)~= o

[d ~-1 y~.~ = _ \d~.~_, Wx)~=o

//d .-1 R(") = - \ ~ Wt)~= o "

In particular, for a generating function W of the form

~/" (X, X , t; e) = ,T~ Wn+ l ( x , X , l ) ,

n = 0

and f(x, X, t ; e) of the form given by Equation (6), Equation (10) yields

df ~ e " de = n.1 f~a)(x, X, t)

n~>O

with f (1)(x ,X, t )=f~+l+ ~ C~Lm+af~-m

O<<.m<~n

(12a)

(12b)

(12c)

(13a)

(13b)

(13c)

(14)

(15)

(16)

EXPANSION FORMULAE IN CANONICAL TRANSFORMATIONS 193

where

and

n~ C~ - (17a) (n - m)! m!'

Lpf = ( f ; Wp), p ~> 1. (17b)

In general, for k/> 1, n ~> O, one obtains

with

n dk ~t f,(k)(x, X, t) (lS) de k f = .

n~O

f 2 k ) ( x , X , t) = f ~ 1) + 2 ~m~m+lJn-mr"t r . 0 9 ) O~m~n

Now, letting e = 0 in the above equation we get the following. (For the remainder of this paper, this equation will be referred to as Deprit's equation.)

where

f~(k)(Y' Y ' t ) = f,Ck+-11) + Z ~m~m('nI + 1J~e(k- 1),_,, (20) O<~m<~n

Lpf -- ( f ; Wp) = f r" Wpr - f r" Wp, , p >~ l . (21)

Notice that

f(O)(y, y, t )= f~(y, Y, t), and f(ok)(y, Y, t )= f(k)(y, Y, t).

f3

fo = f(o)

It f(l)

) / i (a) f(~)

Fig. 1. Recursive transformation of an analytic function under a Lie transform.

Deprit's equation, together with the functions H ("), R ("), y("), and Y(") can be best visualized from the triangles of Figure 1 and Figure 2.

194 AHMED ALY KAMEL

/ ~a) H(z) / ~ R ( 2 ) 1/HIa)/H(3) /-W3t/R' 2) R (3)

/u(o) /u(2) /uCa) w(4) RC4)

H- TRIANGLE R-TRIANGLE

Fig. 2.

Wly= y(I) /-Wly=Y(I)

y(a) �9 / ~ y ( Z )

,2) y{3) /_W~y /y,2)y(,)

y-TRIANGLE Y-TRIANGLE

Triangles for the Hamiltonian H, the coordinates y, the momenta Y, and the remainder R.

Finally, the inverse t ransformation can be written as

oo

y = x + - - x (") (x, X, t) n!

r / = l

(22a)

oo

Y = X + n! X(") (x, X, t).

n=l (22b)

To find the relation between the x(")'s and y(")'s, X(")'s and Y(")'s, one may eliminate x - y and X - Y between Equations (12a), (12b), and (22), and define the functions q(x, X, t ; e) and Q(x, X, t ; 5) as follows:

oo oo

q(x, X , t; e) = ~). x(")(x, X , t) = - ~ y(")(y, Y, t)

n = l n = l

(23a)

8n ~ 8n Q(x, x , t; 5) = n~). X(")(x, X , t) = - ~ Y("l(y, Y, t)

n = l n = l

(23b)

EXPANSION FORMULAE IN CANONICAL TRANSFORMATIONS 195

Therefore, for n ~> I we have

q. = x ("~ (x, X, t),

Q. = x (")(x, x , t),

q(") = -- y(")(y, Y, t),

Q(")= - Y("~ (y, Y, t).

(24a)

(24b)

3. Simplified General Expansions

Given the functions f , , f , _ a ..... andfo, Deprit (1968) constructed the required functions f ( , ) , f ( , - a ) ..... a n d f (~ by introducing the auxiliary functions f(k) and by moving recursively from the left diagonal of Figure 1 towards the right diagonal. One might as well construct the function f (") only in terms o f f , a n d f ("-*) , f("-2) . . . , f (o) o r f , in terms o f f ("), f ( " -2 ) ..... and f(0) (which will be useful in the construction of the inverse transformation) by introducing a suitable linear operator. To show how this can be done, let us write Deprit's equation as

n - 1 Z , - . - s T r . - ~ , . ~ m + l a . - r . - - 1 , n > 1, k > O. (25)

m = 0

By successive elimination of the functions on the right hand side of the above equation one would eventually obtain f , ~k) in terms o f f (k+"), f (k+,-1) , .. . ,f(k). Thus, one may assume the following form for f(k)

f(k) = f(k+,) __ ~ C~Gjf(k+,-j~; n > 1, k >>, O, (26) j = l

where Gj is a linear operator and is a function of Lj, L j_, , ..., and L~. Substitution of Equation (26) into Equation (25) yields the following recursion relation for the linear operator Gj

G j = L j -

For example

G~ = L 1

j - 1 . (27) Cm Lm+IGj-, . -1, l ~ j ~ n O<~m<~j--2

(28a)

Ge = Lz - L1L1 (28b)

G 3 = L 3 - L 1 ( L 2 - - L 1 L 1 ) - 2L2L1. (28c)

Using Equation (26) with f = y and Y, and taking k = 1, we obtain the following general recursive relations for y("3 and Y(") of Equations (12a) and (12b)

y(") = W,y + ~ C~-*Gjy ("-j), n ~> 1, (29a) l <<.j<~n-1

Y(") = - W.y + ~ C~.-1GjY ("-j), n ~> 1. (29b) l ~ j ~ n - 1

196 A H M E D A L Y K A M E L

Using Equation (26) with f = q and Q of Equations (23a) and (23b), and taking k=0 , we obtain the following general formulae for x (') and X (").

x (~) = - y(") + ~ C~Gy ("-j), n >/1 (30a) l < ~ j ~ n - 1

X (") = - Y(") + ~ C~GjY ("-j), n ~> 1. (30b) l <~j<~n-1

Now, x(")(x, X, t) and X(")(x, X, t) of Equations (22a) and (22b) are simply given by

x ("~ (x, X, t) = [x(")]y=:, , (31a) Y = X

X (") (x, X, t) = [X(")]y=x. (31b) Y = X

Next, consider an indefinitely differentiable function v(x, X, t) not explicitly de- pendent on s. Using Equations (6), (7), and (26) withfo=v(x , X, t), f . = 0 for n>~ 1, and k = 0 , we obtain the following general formula

v (x, X, t) = n! v(") (y' Y' t) (32a)

n = 0

v(")(y, Y, t) = ~ C~Gjv("-J)(y, Y, t), n ~> 1 (32b) l<<.j<~n

where r , t) = r , t). (32c)

Also the inverse relation can be written as

v (y, Y, t) = v (x, X, t) + nt V(") (x, X, t), (33a)

n = l

elimination of v(y, Y, t ) - v ( x , X, t) between Equations (32a) and (33a), and using Equations (6), (7), (26) with k = 0 , and (32b) leads to

V (") (x, X, t) = - [G.v (y, Y, t)],=x �9 (33b) Y = X

Lastly, given the Hamiltonian H (x, X, t ; e) in the form

H (x, X, t; e) = n! H. (x, X, t) (34)

n = 0

one can construct the transformed Hamiltonian K(y, Y, t; e) in the form

K(y, Y, t; e) = ~ K.(y, Y, t) (35)

n = 0

E X P A N S I O N F O R M U L A E IN C A N O N I C A L TRANSFORMATIONS 197

and this can be done as follows. Using Equation (7), H can be written as oo

t t (x, X , t; ~) = n~ H(') (y' Y ' t). (36)

n = O

Combination of Equation (36) with Equation (12c) and Equation (35) yields

Ko = Ho (37a)

K n = H (') + R (n), n>11. (37b)

Setting k = 1 and f = H + R in Equation (26) leads to

1/(1) -'1- R(1) = Kn+l - i CjGjKn-j+I, // • 1. (38) j = l

But from the R and H triangles of Figure 2, we have

R~I)_ ~W.+I , n >i 0 (39) Ot

H ( I ) = Hn+ 1 -.}- ~,, CnmLm+lHn_m, n ~ O. (40) m=0

Therefore, the simplified general recursive relation of the transformed Hamiltonian is given by

Ko =/4o (41a)

K, 14,+ ~ , -2 " - 'GjK, ,_ j ) DW, = ( C j _ I L j H , _ j + Cj. , n >~ 1 (41b) l~j<~n-1 Dt

where DW, OWn

- L . H o , n >~ 1. (42) Dt Ot

For example

Ka = It1 - DW1/Dt (43a)

K2 = H2 + LIH1 + GIK1 - DW2/Dt (43b)

K 3 = H 3 + L1H 2 + 2 L z H 1 + 2GIK 2 + GzK I - DW3/Dt (43c)

1s = H4 + LIH3 + 3L2H2 + 3L3H~ + 3G~K3

+ 3G2K 2 + G3K I - DW4/Dt (44)

the operators GA, G2, and G 3 being as defined for Equations (28a) to (28c).

4. Modified General Expansions

In the simplified formulae obtained in Section 3, the rate of increase of the number of the Poisson brackets with respect to the order of perturbation can be reduced if

198 A H M E D A L Y K A M E L

one uses intermediate functions like f y . . = Gi f . or G f f (") to be saved for later use in computa t ion . Thus, this leads to the following recursive relationships:

a) For y("), x ("), Y("), and X (") of Equat ions (29) and (30)

Y(")= W.r + ~. C ~ - l y j , . _ j (45a) l<~j<~n-1

x (") = - y(") + ~ C ~ y j , . _ j (45b) l< . j~n-1

Y(")= - W.y + Z C~ i - IYJ , . - J (45c) l<~j<~n--1

X ( " ) = - Y(") + ~ C ~ Y j , . _ j (45d) l~j<~n-1

where Yi, i L f f (0 ~ j - 1 (46a) = -- Cm L r . + l Y j - m - l , i

O<~m<<.j-2

y j i Ljy(1) ~ j - 1 , = - - C m L m + l Y j _ m _ l , i , (46b) O<<.m<<.j-2

(b) For v (") and V (") o f Equat ions (32b) and (33b)

v(")= ~ C~vi , ._ ~ (47a) l<<.j<~n

V ("~ (x, X, t) = - Ivy, o]y = ~ (47b) Y=X

v j, i L1 v(1) ~. ~- 1 . (48) -- C m L m + l l ) j - m - l , i , O<~m<~j--2

(c) For K. of Equat ion (41b)

where

n- n -1 DW. 14.. = 14. + Z ( C i - I L j H . - J + Cj Kj , . _ j ) , n >t 1 (49a)

l <<j<~n-1 D t

K] i L j K I 2 j - I 1Kj_m _ . ( 4 9 b ) , = - C , . L , . + 1,1 O~m<~j-2

Notice that for Ki = 0, Kj, i = 0 for all j ' s . For example,

K2 = H2 + L1H1 + K1, l - DW2/Dt

K3 = H3 + L1H2 + 2L2Ha + 2K1, z + K2, 1 - DW3/Dt

(50a)

(SOb)

where

K 4 = H 4 + L 1 H 3 + 3 L 2 H z + 3L3H t + 3K1, 3 -'1- 3K2, 2 "t- K3, 1 - DW4/Dt

(5Oc)

K1, 1 = L1K1 (51a)

K1, 2 = L~Kz , K2, 1 = L2K1 - LxK1, a (5~b)

K1, 3 = L 1 K 3 , K2, 2 = L 2 K 2 - L1K1, 2,

K3, 1 = L3K1 - L1K2, t - 2L2K1, 1 �9 (51c)

EXPANSION FORMULAE IN CANONICAL TRANSFORMATIONS 199

The construction of the transformed Hamiltonian using the scheme presented by Equations (49a), (49b), and (42) is simpler and requires less computer time and storage than the scheme presented by Deprit (1968). A considerable amount of this reduction is due to the fact that the sums H ( " ) + R (") as well as H(,1)+R(~ 1) in Equations (37b) and (38) were considered as single quantities as if the transformed Hamiltonian was constructed f rom a single triangle whose end products are H (") + R (") and whose starting elements are H,(1)+ R(, 1). Further reduction in the computer requirements can be achieved if some of the Ki's vanish, in which case Ki, i also vanishes for all possible values of j .

Equations (41) or (49) and (42) are directly applicable to nonlinear resonant problems in which H o is a function of only the action variables X, while H,(n>~ 1) depend trigonometrically on the angle variables x and possibly the time t. I t is desira- ble to transform to new variables so that the resulting Hamiltonian contains, together with the new action variables Y, only certain slowly-varying ' long-period' combi- nations of the new angle variables y and the time t. Equation (41) or (49) may be used to define the W,'s successively so as to remove all 'short-period' terms from the K, 's; such a W, is unique up to an arbitrary additive long-period term.

The equations obtained are now being used in a fourth-order analysis of the motion (stability and periodic orbits) of a particle in the neighborhood of L4 of the earth- moon system in the presence of the sun. In this problem, the following parameters are treated as first order small quantities: distance from LJea r th -moon distance, eccentricity of the moon 's orbit around the earth, moon mass/earth mass, mean motion of the sun/mean motion of the moon. The additional parameter (earth-moon distance/earth-sun distance) is treated as a second order small quantity.

Acknowledgements

The author gratefully thanks his advisor, Professor John V. Breakwell, for his helpful suggestions and assistance. The suggestions and comments of Dr. Andr6 Deprit, Dr. Ali Nayfeh, and Dr. Jacques Henrard are appreciated.

This research was supported by the National Aeronautics and Space Adminis- tration, under Contract No. NsG 133-61.

References

Breakwell, J. V. and Pringle, Jr., R. : 1966, Progress in Astronautics, Vol. 17, Academic Press, New York, pp. 55-73.

Brouwer, D. and Clemence, G. M. : 1961, Methods of Celestial Mechanics, Academic Press, New York. Deprit, A. : 1966, Proceedings ofLA. U., Symposium No. 25, Academic Press, New York, pp. 170-175. Deprit, A., Henrard, J., and Rom, A. R. M.: 1967, Icarus 6, 381-406. Deprit, A.: 1968, 'Canonical Transformations Depending on a Small Parameter', Boeing Scientific

Research Labs., The Boeing Company, Seattle, Washington, document No. D1-82-0755. Schechter, H. B.: 1968, AIAA Y. 6, No. 7.