Microsoft Word - M.R. Hassan and R.R. Thapa _54-60_.doc M.R. Hassan and R.R. Thapa / BIBECHANA 7 (2011) 54-60 : BMHSS 54 BIBECHANA A Multidisciplinary Journal of Science, Technology and Mathematics ISSN 2091-0762 (online) Journal homepage: http://nepjol.info/index.php/BIBICHANA Applications of KS – variables in celestial mechanics M.R. Hassana , R.R. Thapab ∗∗∗∗ a Dept. of Mathematics, S.M. College, Bhagalpur, India b Dept. of Mathematics, P.G. College, Biratnagar, Morang ,Nepal Article History: Received 5 September 2010; Revised 18 November 2010; Accepted 29 November 2010 Abstract This paper deals with the transformation of KS-Variables and canonically conjugate variables from sideral (inertial) to synodic (rotating) form and their applications in “the circular restricted problem of three bodies in three-dimensional coordinate system” to form generating solutions. Keywords : KS – transformation; KS – Variables; Orbital elements; Generating solutions 1. Introduction Kustannheimo and Steifel [1] presented a very convenient regularization method for the space motion of particles near an attracting centre. This mechanism leads to a transformation, called KS – transformation. The transformation is of the type xk = xk (uj), (k = 1, 2, ……… n), (j = 1, 2, 3 ,…….., n + 1) which increases the number of variables. In this regard Schiefle [2] and Steifel and Schiefle [3] have considered the so – called generalized canonical transformation increasing the number of variables. As in physical space, KS – regularization is the most perfect regularization, so to solve all the problems easily of three dimensional synodic coordinate system by KS – Variables; it is necessary to transform the KS – Variables from sidereal to synodic form. In this paper, we attempted to present the KS – Variables in synodic form and their simple application to the motion of an infinitesimal mass in the restricted problem of the three bodies in three dimensional synodic coordinate system have been shown. In earlier sections we just presented some important tools of KS – Transformations in terms of orbital elements i, σω,,Ω , t etc. in synodic coordinate system. Some related and conjugate variables are also presented here in these sections. In later section, the synodic KS – Variables are used to solve the equations of motion of infinitesimal mass in the circular restricted three – body problem in three dimensional synodic coordinate system in the form of generating solutions. The symbols used are as follows: i = inclination of the orbital plane with the plane of motion of the central body, Ω = longitude of the ascending node, ω = argument of perihelion, φ = true anomaly, e = eccentric anomaly, t = physical time, ∗ Corresponding author: Raju Ram Thapa, Post Graduate Campus, Biratnagar, Tribhuvan University, Nepal, Email:thaparajuram@yahoo.com M.R. Hassan and R.R. Thapa / BIBECHANA 7 (2011) 54-60 : BMHSS 55 s = psedo time, n = mean angular motion , . = , dt d / = , ds d σ = φ + ω 2. Derivation of synodic KS – Variables In 1965 Kustaanheimo and Steifel [1] defined an orthogonal matrix L (u), called KS – matrix, where L(u) =             −− −− −− 1234 2143 3412 4321 uuuu uuuu uuuu uuuu (1) LT (u) L (u) = L(u) LT (u) = (u, u) I. Here LT(u) is the transpose of L(u), I is the 4 × 4 unit matrix and u = (u1, u2, u3, u4). If x = (x1, x2, x3) are three dimensional coordinates of a point, KS – transformation in sidereal frame is given by the single matrix equation: x = L(u) (u) (2) Explicitly, x1 = u1 2 – u2 2 – u3 2 + u4 2 x2 = 2 (u1.u2 – u3.u4) (3) x3 = 2 (u1.u3 – u2.u3) In sidereal frame, the compact parametric representation of KS – Variables in terms of orbital elements i, σ,Ω in any plane curve having its orbital plane in general position, is given by [3] as : u1 = ] 2 )( sin[) 2 sin(] 2 )( cos[) 2 sin( 2 σσ −Ω = −Ω i u i (4) u3 = ] 2 )( cos[) 2 cos(] 2 )( sin[) 2 cos( 4 σσ +Ω −= +Ω i u i The sidereal parameters uj (j = 1, 2, 3, 4) of the particle itself are obtained by multiplying the square root of its distance ‘r’ from the central body. Therefore, u1 = ] 2 )( cos[) 2 sin( σ−Ωi r u2 = ] 2 )( sin[) 2 sin( σ−Ωi r (5) u3 = ] 2 )( sin[) 2 cos( σ+Ωi r u4 = ] 2 )( cos[) 2 cos( σ+Ω − i r Where r = (x1 2 + x2 2 + x3 2)½ = u1 2 + u2 2 + u3 2 + u4 2 = (u, u) (6) It may be noted that i & Ω are fixed orbital elements whereas σ is only the variable orbital element. In terms of orbital elements i, Ω & σ t he sidereal physical coordinates xi s of a point are given by x1 = )cos() 2 (cos)cos() 2 (sin 22 σσ +Ω+−Ω ii x2 = )sin() 2 (cos)sin() 2 (sin 22 σσ +Ω+−Ω ii (7) x3 = sin i. sinσ If x1 x2 plane rotates about x3 – axis with mean angular motion n, then at any time t, the synodic coordinates (y1, y2, y3) of the point, are given by y1 = x1cos nt + x2 sin nt, y2 = - x1 sin nt + x2 cos nt, y3 = x3 (8) Introduction of Eqs. (7) in (8) yields M.R. Hassan and R.R. Thapa / BIBECHANA 7 (2011) 54-60 : BMHSS 56 y1 = )cos() 2 (cos)cos() 2 (sin 22 nt i nt i −+Ω+−−Ω σσ y2 = )sin() 2 (cos)sin() 2 (sin 22 nt i nt i −+Ω+−−Ω σσ (9) y3 = sin i. sinσ If q = (q1, q2, q3, q4) are the synodic KS – Variables corresponding to physical synodic coordinates y = (y1, y2, y3), then Y = L (q) q (10) Explicitly: y1 = q1 2 – q2 2 – q3 2 + q4 2 , y2 = 2(q1q2 – q3q4) , y3 = 2(q1q3 + q2q4) (11) From Eqs. (9) & (11) the parametric representation of a point in synodic KS – Variables are given q1 = ] 2 )( cos[) 2 sin( nti −−Ω σ q2 = ] 2 )( sin[) 2 sin( nti −−Ω σ (12) q3 = ] 2 )( sin[) 2 cos( nti −+Ω σ q4 = ] 2 )( cos[) 2 cos( nti −+Ω − σ Which satisfy all the Eqs.(9) & (11) & properties of KS – Transformations. The synodic parameters qj (j = 1, 2, 3, 4) of the particle itself can be obtained by multiplying by the square root of its distance ‘r’ from the central body. i.e. q1 = ] 2 )( cos[) 2 sin( nti r −−Ω σ q2= ] 2 )( sin[) 2 sin( nti r −−Ω σ (13) q3 = ] 2 )( sin[) 2 cos( nti r −+Ω σ q4 = ] 2 )( cos[) 2 cos( nti r −+Ω σ where r = (y1 2 + y2 2 + y3 2 ) ½ = q1 2 + q2 2 + q3 2 + x4 2 = (q, q) (14) From symbolic representations, we have φωσ += where ω is the constant orbital element. i.e. . . σ φ= 3. Derivation of the generalized momenta corresponding to synodic KS – Variables The set of synodic KS – Variables given in Eq.(13) acquire great importance, while studying the motion of an infinitesimal mass in synodic frame of reference. Moreover, in Hamiltonian dynamics, the generalized momenta corresponding to the generalized coordinates are also of great importance, so here we need to establish the generalized momenta corresponding to the synodic KS – Variables in terms of orbital elements i, σ,Ω etc. If X = (X1, X2, X3) are the momenta corresponding to sidereal physical coordinates x = (x1, x2, x3) and P = (P1, P2, P3, P4) are the generalized momenta corresponding to sidereal parametric coordinates u = (u1 u2 u3 u4), then momenta transformation is given by the single matrix equation: x = (1/2r) L(u) P (15) Explicitly X1 = 2r )uP + uP - uP - u(P 44332211 X2 = 2r )uP - uP - uP u(P 34431221 + (16) X3 = 2r )uP + uP uP u(P 24134231 ++ and T (p, u) = P1u4 – P2u3 + P3u2 - P4u1 = 0 (17) The bilinear relation (16) plays an important key role in the above representations. The inverse transformation of (15) is given as : P = 2H (u) X (18) M.R. Hassan and R.R. Thapa / BIBECHANA 7 (2011) 54-60 : BMHSS 57 . . . . . . . . . . Explicitly from Eqs. (3) and (15) P1 = 2(X1u1 + X2u2 + X3u3) P2 = 2(-X1u2 + X2u1 + X3u4) (19) P3 = 2(-X1u3 + X2u4 + X3u1) P4 = 2(X1u4 - X2u3 + X3u2) Suppose that Y = (Y1, Y2, Y3) are the momenta is synodic system corresponding to y = (y1, y2, y3), then from the definition of momenta Y1 = . 1 2y ny− , Y2 = . 2 1y ny+ + ny1, Y3 = . 3y (20) From Eqs. (7) and (20) we get Y1 = . 2 2[sin ( )sin( ) cos ( )sin( ) 2 2 i i nt ntσ σ σΩ − − − Ω + − Y2 = . σ 2 2[sin ( ) cos( ) cos ( ) cos( ) 2 2 i i nt ntσ σΩ − − + Ω + − (21) Y3 = . σ sin i cos σ If Q = (Q1, Q2, Q3, Q4) are the generalized momenta in synodic system corresponding to q = (q1, q2, q3, q4), then with the help of Eq.(18), the value of Qj s may be represented as : Q = 2H (q) Y (22) Therefore, Q1 = 2(Y1q1 + Y2q2 + Y3q3) = . ( ) 2 sin( )sin[ ] 2 2 i nt r σ σ Ω − − = . 2σ q2 (23) Therefore, Q1 = . 2σ q2, Q2 = - . 2σ q1, Q3 = - . 2σ q4, Q4 = σ2 q3 The set of variables Qj s and qj s satisfy the bilinear relation T(Q, q) = Q1q4 – Q2q3 + Q3q2 – Q4q1 = o (24) Thus in both the systems T (P, u) and T (Q, q) are invariant. Also |Q| 2 = (Q, Q) = 24 . 2 1 4j j Q rσ = =∑ (25) 4. The equations of motion The regularized canonical equations of motion of the infinitesimal mass in synodic coordinate system given by Kurcheeva [4] are : )4,3,2,1(, = ∂ ∂− = ∂ ∂ = j q K ds dQ Q K ds dq j j j j (26) where the Hamiltonian K (regularized at r = 0 only) is given by : K = ½ 4)(2 034431221 2 4 1 2 −+−+−+∑ = CqQqQqQqQQ j j ρ µ4+ [1- 2ρ (q1 2 – q2 2 – q3 2 + q4 2 + 2 11 1 − r (c1+1)] (27) With r = q1 2 + q2 2 + q3 2 + q4 2 = 2ρ (28) = the distance of the infinitesimal mass from the first primary, r1 2 =1 + 2(q1 2 - q2 2 - q3 2 + q4 2) + 4ρ (29) = the square of the distance of the infinitesimal mass from the second pimary. c = ∑ ∞ =0i i ic µ , the jacobi’s constant, (30) the physical time ‘t’ and the pseudo time ‘s’ are connected by the differential equation 244)( ρ== r ds dt (31) M.R. Hassan and R.R. Thapa / BIBECHANA 7 (2011) 54-60 : BMHSS 58 5. Generating solutions For generating solutions, taking 0=µ , the reduced Hamiltonian, K0 = ½ 4)(2 034431221 2 4 1 2 −+−+−+∑ = CqQqQqQqQQ j j ρ (32) The Hamilton canonical equation of motion due to K0 are : )4,3,2,1(, 00 = ∂ ∂ = ∂ ∂ = j q K ds dQ Q K ds dq j j j j (33) Thus the equations of motion of an infinitesimal mass in terms of synodic KS – Variables can easily by derived from (30), (32) & (33) as : q1˝–4( 2ρ +q1 2 +q2 2 )q2΄+4(q1q4–q2q3)q3΄-4(q1q3+q2q4)q4΄=4q1(3 4ρ -c0) (34) q2˝+4( 2ρ +q1 2 +q2 2 )q1΄+4(q1q3+q2q4)q3΄+ 4(q1q4–q2q3)q4΄=4q2(3 4ρ -c0) (35) q3˝-4(q1q4-q2q3)q1΄-4(q1q3+q2q4)q2΄ - 4( 2ρ +q3 2 +q4 2)q4΄ =4q3(3 4ρ -c0) (36) q4˝+4(q1q3+q2q4)q1΄-4(q1q4-q2q3)q2΄ +4( 2ρ +q3 2 +q4 2)q3΄ =4q4(3 4ρ -c0) (37) By multiplying Eqs. (34) to (37) respectively by q1΄, q2΄, q3΄, q4΄ and integrating their sum, one has hcq j j ++−=∑ = 844' 2 0 6 4 1 2 ρρ (38) Here, h is taken as the constant of integration. Again multiplying (34) to (37) respectively by q2, q1, q4 & q3 and subtracting the sum of the second and fourth from the sum of the first and third and integrating we get q1΄q2 – q1 q2΄ + q3΄q4 – q3q4΄ = 2 4ρ + b (39) Here b is the constant of integration to be determined by the initial conditions. It is well known that in the restricted three – body problem, mean angular motion is taken to be unity i.e. n = 1 (as the total mass of the primaries is 1 and separation = 1 ⇒ n = 1) and thus the synodic KS – Variables qj s given in (13) will be of the form : q1 = ] 2 )( cos[) 2 sin( ti −−Ω σ ρ q2 = ] 2 )( sin[) 2 sin( ti −−Ω σ ρ (40) q3 = ] 2 )( sin[) 2 cos( ti −+Ω σ ρ q4 = ] 2 )( cos[) 2 cos( ti −+Ω − σ ρ From Eqs. (27) and (40) one can easily find that )cos816(' '24'22' 4 1 2 iq j j σρρσρρ −++=∑ = (41) And q1΄q2 – q1q2΄ + q3΄q4 – q3q4΄ = 2 icos) 2 1 ( '24 σρρ − (42) The combinations Eqs.(38, 41) and (39, 42) yield hci ++−=−++ 844)cos816() 4 1 ( 2 0 6'242'22' ρρσρρσρρ (43) And - ( bi =cos) 2 1 '2σρ (44) Following [4] and [5] at s = 0, ρ = 0 and which gives b = 0, consequently 'σ =0 i.e. σ = constant = 0σ (say) and the second degree differential equation reduces to M.R. Hassan and R.R. Thapa / BIBECHANA 7 (2011) 54-60 : BMHSS 59 2 0 2' 48 ρρ ch −+= (45) At h = 0, the generating solutions with generalized period s* = k 2 π , k∈N of the system of Eqs. (34) to (37) are given by [6, 7] as : q1 = ] 2 )( cos[) 2 sin( 0 ti −−Ω σ ρ q2 = ] 2 )( sin[) 2 sin( 0 ti −−Ω σ ρ (46) q3 = ] 2 )( sin[) 2 cos( 0 ti −+Ω σ ρ q4 = ] 2 )( cos[) 2 cos( 0 ti −+Ω − σ ρ and Q1 = ] 2 )( cos[) 2 sin( 0' ti −−Ω σ ρ Q2 = ] 2 )( sin[) 2 sin( 0' ti −−Ω σ ρ (47) Q3 = ] 2 )( sin[) 2 cos( 0' ti −+Ω σ ρ Q4 = ] 2 )( cos[) 2 cos( 0' ti −+Ω − σ ρ where 00 ' 2sin(22 ssc −−=ρ ) (48) From (32) t = 10000 0 )]24sin(4)[ 1 ( ssscscc c +−+ (49) where s1 is the constant of integration. Let us introduce two variables θ and ϕ , where θ = ( ) 2 (&) 2 00 tt −+Ω = −−Ω σ ϕ σ (50) Introduction of (31) yield 2'' 2ρϕθ −== (51) q1 = θρ cos) 2 sin( i q2 = θρ sin) 2 sin( i (52) q3 = ϕρ cos) 2 cos( i q4 = ϕρ cos) 2 cos( i − and Q1 = θρ cos) 2 sin(' i Q2 = θρ sin) 2 sin(' i (53) Q3 = ϕρ cos) 2 cos(' i Q4 = ϕρ cos) 2 cos(' i − Now from Eq.(45) 2 0 2' 48 ρρ ch −+= (54) Integrating Eq.(54) with respect to s we get s = ∫ −+ ρ ρ ρ 0 2 048 ch d with the substitution αρ sin 4 )8( 0c h+= one can easily find sc02=α & sc c h 0 0 2sin 4 )8( +=ρ (55) Using Eqs. (58) in (35) and (55) one has t = )]24sin(4[8 )8( 00000 sscsccc h −− + (56) M.R. Hassan and R.R. Thapa / BIBECHANA 7 (2011) 54-60 : BMHSS 60 ), 2 ( 1 t −= ωθ ) 2 ( 2 t −= ωϕ (57) where 1ω and 2ω are the constants of integration. Thus the solutions of the problem in question at 0=µ depend upon three arbitrary constants h, 1ω and 2ω . 6. Discussion A system of differential equations may have infinitely many generating solutions so there are many possibilities of generating solutions of the differential Eqs. (26). Here in our case (50) represents one set of generating solution for the satellite motion. References [1] P. Kustaanheimo and E. Steifel, J. Reine Angew Math. 218 (1965) 204. [2] G. Scheifle, , Cele. Mech., 2 (1970)296. [3] E. Stiefel, and G. Scheifle, Linear and Regular Celestial Mechanics, Berlin, Heidelberg, New York,(1971). [4] I.V. Kurcheeva, Cele. Mech, 15 (1977) 353. [5] A. Ahmad, Cele. Mech & Dynamical Astronomy, 61 (1995)181. [6] G.A. Krashinaski, Bull., I.T.A , 9 (1963)205. [7] M.R. Hassan, Bull. Astr. Soc., India, 2001.