Families of Canonical Transformations by Hamilton-Jacobi-Poincare equation. Application to Rotational and Orbital Motion

F amilies of Canonical T ransformation s b y Hamilton-Jacobi-P oincar ´ e equation. Application to Rotational and Orb i ta l Motion S. F errer ∗ and M. Lara † Octob er 17, 201 8 Abstract The Hamilton-Jaco bi equation in the sense of P oinca r ´ e, i.e. formulate d in the extended phase sp a ce and in c luding regularization, is revisited b uilding canonical transformations with the pur pose of Hamilto nian r e duction. W e illustrate our ap- proac h dealing with orbital and attitude dynamics. Based on the use of Whittak er and An do y er sym plec tic c harts, for whic h all bu t one co ordinates are cyclic in the Hamilton-Jaco bi equation, w e provide whole families of canonical transform ations, among whic h one recognizes the familiar ones used in orb i tal and attitude dyn amics. In addition, new canonica l transf orm a tions are demonstrated. 1 In tro d uction T r a ns formation of v ariables, including or not the c hange of t ime used as ev olution pa- rameter, is a basic to ol in almost an y problem of ph ysics. A r ecen t illustration of the usefulness of this tec hnique, whic h original application go es back to Laplace and Euler, ma y b e found in [19]. Changes of v ariables b elong to Geometry and, therefore, do not dep en d on the ph ysical system to whic h they ma y b e applied. Ho wev er, t rans formations are commonly deriv ed for sp ecific purp oses and, hence, they are tig h tly related to the particular problem they are applied to. A renew ed in terest on transformation of v ariables tailored to specific problems has recen tly emerged in three areas. On one hand there is a contin uous searc h for more efficien t n umerical integrators demanded in many applied fields, ranging from researc h gr o up s endo we d with p o w erful computing facilities, t o researc h inv olv ed in v ery long p erio ds of time (see [5, 16] and references therein). On the ot her hand the Hamilton-Jacobi (H- J) equation is receiving new attention in extended phase space, connecting this tec hnique with the v ariational principles of La grangian and Hamiltonian Mec hanics (see [31] and references therein). Finally a third area, whic h has its ro ots in Poincar ´ e ’s [30] fundamen ta l ∗ Departamento de Ma tem´ atica Aplicada , Universidad de Murcia, 3 0071 E s pinardo, Spa in † Real Observ ato r io de la Armada , 1 1110 San F ernando, Spain 1 problem of dynamics , deals with Hamilto nian systems of the form H = H 0 + H 1 where H 0 defines a n in tegrable system, and H 1 is considered a p erturbation. The H-J equation is built from H 0 b y searc hing for a generating function suc h that t h e new canonical v ariables satisfy some requiremen ts of perturba tion t heories (see, f or instance, [37, 17]). The conten t of this pap er fo cus es in this last area. Finally note that the H-J approac h has b een used by man y authors, along the history of this equation, lo oking for v aria bles leading to separabilit y . W e do not consider that problem as suc h, alt ho u gh it migh t b e included. There is a long traditio n in Classical and Celestial Mec hanics —no w adays a common ground in man y o the r fields— of using b oth canonical transformations and c hanges of indep e nden t v arible dubb ed as regula r iz ation. The H-J equation is the usual approa c h, in whic h the canonical transformation is obtained from either a princip al or a char acteristic function [18]. The characteristic function approac h do es not rely on the v anishing of the new Hamil- tonian. It only requires a non-zero transformed Hamiltonian that is cyclic in a ll v ariables but dep ends on one or sev eral momen ta. The a dequate selection o f the new Hamiltonian will ma ke the transformation succ essful for specific purp oses. Th us, f or instance, if o ne w ants the transformed Hamiltonian to r e tain the top ology of the original problem, prob- ably sev eral momen ta m ust b e retained in the new Hamiltonian. On the con tra r y , useful transformations f o r p erturbation theory ma y lo ok for tr a ns formed Hamiltonians that de- p end only on o ne momenta, thus simplifying the subsequen t application of p erturbation metho ds (whic h, in turn, hav e limited application to sp ecific solutions as, for instance, quasi-p eriodic motion). Sen tences lik e “af t er sev eral trials”, whic h are frequen t in the literature, migh t pro duce the impression that succeed ing in the c hoice o f the reduced Hamiltonian is a matter of educated guess or alquimia . Ho we v er, t he new Hamiltonian do es not need to b e sp ecified since the b eginning and, quite o n the contrary , it ma y b e handled formally throughout the pro cess o f computing the transformation [35]. Therefore, Hamiltonian problems solv ed by the H- J equation giv e rise to whole families of canonical transformations from whic h the selection of a particular one requires only the materialization of the new Hamiltonia n 1 . W e prop ose to obtain canonical transformations from a general f orm o f a H-J equation that is form ulated in the extended phase space. In principle, the only requiremen t imp osed to the Hamiltonian function (6) is that H 0 has to b e in tegrable. More precisely , the t wo families studied are presen ted using symplectic c harts where the unp erturbed part defines a 1 - DOF. In other w ords, w e only deal with canonical transformations derive d from in tegrable Hamiltonians that are cyclic in all but one v ariable, and p erform tw o sp e cific applications. In attitude dynamics , w e find a family o f tra nsfor ma t ions t ha t 1 T o make more clear our p oin t from [35] (p. 412 ), where E denotes the Hamiltonia n in the new v ariables, we quote “W e are still free to c ho ose the functional form of E . A conv enient (and conv entional) choice is E ( p 0 , p 1 , p 2 ) = − mµ 2 / (2 p 2 0 ). With this choice the momentum p 0 has dimensions of angula r momentum , and the conjugate co ordinate is a n angle.” As w e will show this is not the only wa y of reasoning . Indeed, we arr iv e to the same choice when we imp ose the function called Kepler equation as one of the express ions defining the trans f ormation. In other words, when the mean anomaly is chosen a s one of the new co ordinates. As Delaunay showed, in doing so he av oided the prese nc e of mix e d sec ula r terms in this theory of the Mo on. F or a r e cen t study on these v ariables fro m a geo m etric p oin t of view the re ader sho uld co ns ult [7] 2 affords for complete reduction of the Euler-P oinsot pro blem. In this case, w e c hec k that Sado v’s [32, 33], Kinoshita’s [2 4], Deprit’s [12 ], and the recen t F ukushim a’s [17] reduced Hamiltonians are just particular cases of o ur fa mily . Notably , our approac h pro vides explicit transformations from Sadov ’s and Kinoshita’s elemen ts t o Ando y er v ariables [3], th us o v ercoming the ma jo r ob jection to b oth sets of elemen ts of b eing constructed through implicit relat io ns . Besides, w e show that other selections o f t he new Hamiltonian provid e new canonical transformations that may help in simplifying p erturbation a lgorithms in attitude dynamics. In orbital dynamics w e obtain three differen t families of iso chr onal canonical trans- formations —derive d fro m H´ enon’s iso c hronal cen tral p oten tial [21, 6]— each one link ed with a differen t form of the regularizing function. F rom our families w e reco v er a v ariet y of canonical transformatio ns in the literature, ranging from historic ones lik e Delaunay [8], Levi-Civita [25, 2 6 ], or Hill [22], to the recen t transformation of Y ang ua s [37]. In addition, w e sho w that the set of Delaunay elemen ts yields the simplest transformation of the iso c hro nal family that reduces the Ha m iltonian o f the t w o b o dy problem to one momen tum and, b esides, preserv es the original form of the Kepler equation. The tw o charts used enjo y similar characteristics . Both express the unp erturbed part of the Hamiltonian families a s g e n uine 1-D OF systems due to the cen tr a l and a xial sym- metries that b oth p ossess . They belong to the category of sup erin tegrable systems, but not maximally superintegrable (see F ass` o [14] and Ortega and Ra t iu [29]); only f o r par- ticular c ho ic es of the para me ters b oth systems f a ll in to this last category . Moreov er, from the computational p oin t o f view, these c harts in t r oduce great simplification in in termedi- ary expressions whic h, in turn, lead to a deep insight ab out eac h family . Man y authors still rely on Eulerian v ariables when dealing with rot a tional dynamics (see for instance Marsden and Ratiu [28 ]) , and spherical v ariables for orbital type motions (see Goldstein et al. [18 ], Jos´ e and Saleta n [23 ] and Sussman and Wisdom [35]). In spite of the fact that t he tw o families considered are indep enden t and consequen tly , they are studied in a tota lly selfcon t a ine d manner, we make b oth part of the pa p er b ecause this a llows to o bta in a b etter insigh t in the w ay w e approach the H-J formalism. A final commen t is due. Although b oth families ha ve parameters, w e ha ve made the analysis for t he generic case o nly . W ith resp ec t to the orbita l case, as an illustration, w e ha ve added details when w e restrict to Keplerian systems, showing the connection of the new v ariables with t he anomalies . W e w ould lik e to make clear tha t we a re no t doing a review of regularizations. In other w ords, it is left for the reader to try other canonical transformations within t h e general sche me presen ted here. 2 A General F orm of the Hamilton-Jacobi Equation W e o nly deal with Hamiltonians of the type K ( x 0 , x, X 0 , X ; µ ) ≡ ( X 0 + H ) χ, (1) where x = ( x 1 , . . . , x n ) are co ordinates and X = ( X 1 , . . . , X n ) conjugate momen ta ; x 0 is the independen t v ariable and X 0 its conjugate momenta in the extended phase space form ulation, in which w e restrict to the manifold K = 0; µ is a ve ctor of parameters, and 3 the Hamiltonian H as w ell as the “regularizing factor” χ may dep end on a ll or some of the pa rame ters defining µ : χ = χ ( x 0 , x, X 0 , X ; µ ) , H = H ( x 0 , x, − , X ; µ ) . (2) A dash in the place of a v ariable is used to remark that the corresp onding v ariable is not presen t. Hamilton equations are d x 0 d τ = ∂ K ∂ X 0 , d X 0 d τ = − ∂ K ∂ x 0 , d x i d τ = ∂ K ∂ X i , d X i d τ = − ∂ K ∂ x i , ( i = 1 , . . . , n ) (3) where τ is the ev o lu tion parameter of the flo w. Note that the first of the previous equations reads d x 0 d τ = χ (4) whic h tells the f unction χ ough t to v erify that χ > 0 in its domain. Moreo ver, in t he case of conserv ativ e systems H = H ( − , x, − , X ), X 0 is a n in tegral, and the manifold K = 0 ma y b e also seen as X 0 = −H = constant. W e a re in terested in canonical transformations ( x 0 , x, X 0 , X ) T Φ − → ( y 0 , y , Y 0 , Y ) (5) in the sense of Poincar ´ e . More precisely , w e lo ok for transformations suc h that they simplify Hamiltonian systems defined by functions H whic h can b e written as H = H 0 + H 1 (6) where H 0 = H 0 ( − , x, − , X ) defines an in tegr a ble system, and H 1 = H 1 ( x 0 , x, X ) is a p er- turbation. Sp ecifically , w e fo cus on canonical t r ans formations suc h that the new Hamil- tonian K = K 0 + K 1 satisfies K 0 = ( X 0 + H 0 ) χ = Φ( − , − , Y 0 , Y ) , (7) i.e. t he full r eduction of the unp erturbed part is carried out. This Eq. (7) is what we refers as the v arian t o f Poincar ´ e to the H- J equation; the classical case c ho oses Φ( − , − , Y 0 , Y ) ≡ 0. The transformations are defined b y X i = ∂ W ∂ x i , y i = ∂ W ∂ Y i , i = 0 , . . . , n (8) deriv ed from a generator W = W ( x 0 , x, Y 0 , Y , µ ) t ha t is a complete solution o f the gener- alized H- J equation " ∂ W ∂ x 0 + H 0 x, ∂ W ∂ x !# χ x, x 0 , ∂ W ∂ x 0 , ∂ W ∂ x ! = Φ( Y 0 , Y ) . (9) 4 Th us, the Hamiltonian K in the new v ariables will take the f orm K = K 0 + K 1 = Φ + H 1 χ (10) where H 1 and χ are expresse d in the new v ariables. Note that in what follow s w e tak e H 1 = 0 . In t his pap er w e limit to generators of the form W = X 0 ≤ i < n − 1 x i Y i + R ( x n , Y 0 , Y ) , (11) and regularizing function χ = χ ( x n , X 0 , X 1 , . . . , X n − 1 ). Hence, fro m Eq. (9) w e ma y write H 0 x, Y 1 , . . . , Y n − 1 , ∂ R ∂ x n ! = Φ( Y 0 , Y 1 , . . . , Y n ) χ ( x n , Y 0 , Y 1 , . . . , Y n − 1 ) − Y 0 . (12) Dep ending o n the form of H 0 , Eq. (12) may b e solv ed for ∂ R /∂ x n and, therefore, R is computed from a quadra t ur e, whic h solution will dep end on the choice s made for Φ and χ . Other p ossibilities a r e under inv estigation [15]. Note that, in fact, there is no reason wh y w e should imp ose on H 0 to b e cyclic in x 0 . What we ha ve presen ted ab o v e, prop erly adapted, remains v alid if w e lift that constrain t. This is referred in the literature as nonautonom uos systems; the drive n oscillator, the relativistic particle, etc are just simple examples within that categor y (for more recen t systems of in terest see Struc kmeier [31]). In the families w e will study b elo w the p ossible presence of x 0 o ccurs in the p erturbing part. Apart from the general case of Eq. (1 2 ), transformat ions non-based on the homoge- neous fo r ma lism ( χ ≡ 1, Φ = Y 0 + Ψ), adopt the simpler formulation H 0 x n , Y 1 , . . . , Y n − 1 , ∂ R ∂ x n ! = Ψ( Y 1 , . . . , Y n ) . (13) In t he t w o ty p es of Hamiltonian systems w e study b elo w, instead o f ( x 0 , x, X 0 , X ) w e will start eac h Section considering briefly the tw o symplectic c harts on which our study relies. In doing so we will use the standard notations, altho ug h there are more than one: Ando yer v ariables [3] for rotat io nal motion a nd Whittak er (p olar-no dal) v ariables [36 ] for orbital dynamics. With resp ect to the new v ariables ( y 0 , y , Y 0 , Y ), w e will use ( d, γ , v , u , D , Γ , Υ , U ) fo r the r o tational fa milies a nd ( f , g , h, u, F , G, H , U ) for the orbital ones. F or the differen t sets within eac h family , w e do not find necessary to distinguish among them, lik e using subindices, etc. Eac h one distinguishes itself b y the expression tak en b y K 0 . 3 Rigid b o dy transformations Within the frame set up b y Eqs (7) and (8) w e will consider first canonical transformations deriv ed from free rigid b o dy Hamilto nia n function. As it is w ell kno wn, mean while Euler of v ariables con tinues to b e the ones used in classical and recen t b o o ks (see [18], [28]) at the same time in researc h pap ers is already customary t o consider those systems formulated 5 in Andoy er v ariables. More precisely , the Hamiltonian of the f r ee rigid b o dy tak es the form H = H ( − , − , ν , − , M , N ; a i ) = 1 2 ( a 1 sin 2 ν + a 2 cos 2 ν ) ( M 2 − N 2 ) + 1 2 a 3 N 2 . (14) where 0 < a 3 ≤ a 2 ≤ a 1 are parameters. Then, according to our nota tion Eq. (1), w e deal with the Hamiltonian K = h T + 1 2 ( a 1 sin 2 ν + a 2 cos 2 ν ) ( M 2 − N 2 ) + 1 2 a 3 N 2 i χ ( ν, Λ , M , T ) , (15) where ( λ, µ, ν, Λ , M , N ) ar e Andoy er v ariables [3], and M 2 ≥ N 2 making the Hamiltonian, (14), strictly p ositiv e and 1 2 χ a 3 M 2 ≤ H ≤ 1 2 χ a 1 M 2 . As w e said in what follow s we consider only t he g eneric case a 3 < a 2 < a 1 . W e do not need to giv e more details on those v ariables here. The interes ted reader may consult the classical note of Deprit [9] o r in more geometric terms F a ss` o [1 3 ]. 3.1 The general structure of the transformation W e lo ok for canonical transformatio ns ( λ, µ, ν, t, Λ , M , N , T ) T Φ − → ( d, γ , υ , u, D , Γ , Υ , U ) (16) that con v ert (1 5) in a certain function Φ( D , Γ , Υ , U ) dep ending only on momen ta. The transformation will b e defined through a function S = S ( λ, µ , ν, t, D , Γ , Υ , U ) in mixed v ariables suc h that d = S D , γ = S Γ , υ = S Υ , u = S U , Λ = S λ , M = S µ , N = S ν , T = S t . (17) where w e use the no tation S x = ∂ S /∂ x . Th us, fro m (15) w e set the H-J equation h 1 2 ( a 1 sin 2 ν + a 2 cos 2 ν )  S 2 µ − S 2 ν  + 1 2 a 3 S 2 ν + S t i χ ( ν, S λ , S µ , S t ) , = Φ( D , Γ , Υ , U ) (18 ) and S is c hosen in separate v ariables as S = U t + Υ λ + Γ µ + W ( ν, D , Γ , Υ , U ) (19) b ecause t , λ , and ν are cyclic in (15). Then, W = Γ Z ν ν 0 q Q ( ν, D , Γ , Υ , U ) d ν, (20) where Q = a 1 sin 2 ν + a 2 cos 2 ν − A a 1 sin 2 ν + a 2 cos 2 ν − a 3 , A = 2 Γ 2 Φ χ − U ! , (21) and √ Q m ust b e real f or all ν ; therefore, Φ < χ ( 1 2 a 2 Γ 2 + U ). 6 If we assume that the regularizing factor dep ends only on the co ordinate ν , the trans- formation (17) a r e d = 1 Γ Φ D I 3 , (22) γ = µ + I 1 + 2 U Γ 2 I 2 −  2Φ Γ 2 − 1 Γ Φ Γ  I 3 , (23) υ = λ + 1 Γ Φ Υ I 3 , (24) u = t − 1 Γ I 2 + 1 Γ Φ U I 3 (25) N = Γ q Q, (26) T = U, Λ = Υ , M = Γ , (27) where I 1 = Z ν ν 0 q Q d ν, I 2 = Z ν ν 0 Q A √ Q d ν, I 3 = Z ν ν 0 Q A χ √ Q d ν. (28) 3.2 The case χ = 1 W e o nly deal with transformations with χ = 1. Therefore, I 2 ≡ I 3 . Note that a restriction of Ando ye r v ariables is | N | ≤ M , whic h, b ecause of (26), further constrains the v alues of the new Hamiltonian to b e in to the p o sitiv e in t erv al Φ ∈ ( 1 2 a 3 Γ 2 , 1 2 a 2 Γ 2 ). In addition, one should b e a ware that the sign of √ Q m ust b e tak en in accordance with the sign o f N . Again, the transformat io n dep ends on the in tegration of tw o quadratures, whic h closed form solution requires well kno wn c ha ng es o f v ariables. Th us, in t ro ducing the parameters 2 ǫ = a 1 − a 2 a 1 − a 3 , f = a 1 − a 2 a 2 − a 3 , (29) and the functions δ = δ ( D , Γ , Υ , U ; a 1 , a 2 , a 3 ), m = m ( δ ), defined by δ = a 1 − a 2 Γ 2 a 1 − 2(Φ − U ) Γ 2 , (30) m = δ − ǫ 1 − ǫ = a 1 − a 2 Γ 2 a 1 − 2(Φ − U ) 2(Φ − U ) − Γ 2 a 3 a 2 − a 3 , (31) the quadra t ur es in (28) are solv ed to giv e I 1 = m F ( ψ | m ) − ( m + f ) Π( ψ , − f | m ) √ δ f , (32) I 2 = Γ F ( ψ | m ) √ a 2 − a 3 q Γ 2 a 1 − 2(Φ − U ) . (33) 2 F or the sake of linking with recent literature w e used F ukushima’s no tation [17]. Deprit [1 2] calls δ ≡ α 3 , ǫ ≡ α 0 3 , and m ≡ k 2 3 , and Sadov [3 2] names m ≡ λ a nd f = κ 2 . 7 where F ( ψ | m ) is the elliptic in tegral of t he first kind of mo dulus m which amplitude ψ is unam biguo usly defined through cos ν = sin ψ √ 1 − ǫ cos 2 ψ , sin ν = √ 1 − ǫ cos ψ √ 1 − ǫ cos 2 ψ , d ν = − √ 1 − ǫ 1 − ǫ cos 2 ψ d ψ . (34) On the other hand, Π( ψ , − f ; m ) is the elliptic in tegral of the third kind of mo dulus m , c hara cteristic − f , and amplitude ψ . Alternative ly , one can write I 1 = s f δ (1 − δ ) Π( φ, δ | m ) , (35) where the new amplitude φ is unambiguously defined throug h cos ν = cos φ q 1 − δ sin 2 φ , sin ν = √ 1 − δ sin φ q 1 − δ sin 2 φ , d ν = √ 1 − δ 1 − δ sin 2 φ d φ. (36) If w e further in tr o duce σ = Γ q Γ 2 a 1 − 2(Φ − U ) √ a 2 − a 3 = √ δ f a 1 − a 2 = 1 a 2 − a 3 s δ f , (37) then, the transformat io n equations ab o v e, (2 2)–(26), a dopt the compact form d = σ Γ Φ D F ( ψ , m ) (38) γ = µ − σ Γ h (Γ a 3 − Φ Γ ) F ( ψ , m ) + Γ ( a 1 − a 3 ) Π( ψ , − f | m ) i (39) = µ − σ Γ   2 Φ − U Γ − Φ Γ  F ( ψ , m ) −  2 Φ − U Γ − Γ a 2  Π( φ, δ | m )  (40) υ = λ + σ Γ Φ Υ F ( ψ , m ) = λ + Φ Υ Φ D d (41) u = t + σ Γ (Φ U − 1) F ( ψ | m ) = t + Φ U − 1 Φ D d, (42) N = Γ v u u t ǫ (1 − δ cos 2 ν ) δ (1 − ǫ cos 2 ν ) = Γ r ǫ δ (1 − m sin 2 ψ ) , (43) where we main tain tw o expressions for γ to ease comparisons with transformat io ns in the literature. R emark that Φ remains arbitrary . Therefore, (37)–(43) define a family of canonical transformations that pr ovide complete reduction of the Euler-P oinsot problem. Note, b esides, from (42) that reduced Hamiltonians o f the form Φ = U + Ψ( D , Γ , Υ) are required for tra nsfor ma t io ns preserving the time scale. Despite the family of transfor ma t ions ab o v e has b een obtained f r om a sp ecific dynam- ical syste m, the rigid-b o dy problem, once it has b een deriv ed it ma y b e applied to an y problem w e wish. How ev er, successful tr a nsformations for differen t problems are closely dep enden t o n the selection of Φ. Th us, for instance, if w e are to apply one of the tra ns- formations of the family (3 8)–(4 3) a b ov e to reduce the rigid b o dy dynamics or to study 8 a p erturb ed rigid b o dy , w e should consider sp ecific facts. Because neither Λ nor T are altered b y the transformation it seems natural to c ho ose Φ = U + Ψ( D , Γ). Then, u is the t ime, υ , Υ, U , Γ and D , remain constan t, and γ = γ 0 + Φ Γ u, d = d 0 + Φ D u. (44) Because the sup er-inte grability of the Euler-Poins ot problem the closure of a g eneric tra jectory is a t w o-t o rus [4], Therefore, Ψ should dep end b oth on D and Γ if we w an t to retain the top ology of the problem. Sp ecific selections of Ψ dep ending only on D —whose app earance is mandator y f o r equation (38) to b e defined— will pro duce just p erio dic solutions. Ev en particular selections Ψ = Ψ( D , Γ) may reduce the solutions to just p erio dic orbits. F or instance, if we imp ose i Ψ Γ = j Ψ D , (45) i , j , integer, any computed Φ = U + Ψ( i D + j Γ) with arbitr a ry Ψ will provide only p erio dic solutions. W e discuss here some other p o ssible c hoices of Φ. Without any application in mind, one criteria might b e ‘to simplify’ the co efficien ts of the v ariables in the transfor ma t io n. Th us, for instance, σ = D in (37) implies Φ = U + Γ 2 2 a 1 − D 2 a 2 − a 3 ! (46) where Γ has dimensions of momen tum and D of in v erse of momen tum of inertia. This Hamiltonian simplifies the transformation equations, (3 8)–(4 3), to d = − Γ D 2 a 2 − a 3 F ( ψ , m ) (47) γ = µ + D a 1 − a 2 − D 2 a 2 − a 3 ! Π( φ, δ | m ) (48) υ = λ (49) u = t, (50) N = Γ D v u u t 1 − m sin 2 ψ ( a 1 − a 3 ) ( a 2 − a 3 ) , (51) T o discuss other simplifications, w e find it con v enien t to in tro duce s 1 = Γ σ Φ D , s 2 =  2 Φ − U Γ − Φ Γ  1 Φ D , s 3 = Φ Υ Φ D , (52) and consider the p ossible choices of Φ that mak e s i ( i = 1 , 2 , 3) indep enden t of the momen ta. 9 3.2.1 Case s 1 = s 1 , 0 and s 3 = s 3 , 0 constan t. Then, from the first and third of (52) and from (37), we get Φ Υ = s 3 , 0 Φ D = s 3 , 0 s 1 , 0 √ a 2 − a 3 q a 1 Γ 2 − 2 ( Φ − U ) , (53) and, therefore Φ − U = a 1 2 Γ 2 − a 2 − a 3 2 s 2 1 , 0 [ s 3 , 0 Υ + Ψ(Γ , D , U )] 2 . (54) with Ψ arbitrary . Because the top olog y o f the pro blem, we can a dd the condition Φ Υ = 0, and t herefore s 3 , 0 ≡ 0. In addition, if we w an t to preserv e the time scale (42 ), we restrict to the case Φ − U = a 1 2 Γ 2 − a 2 − a 3 2 s 2 1 , 0 Ψ(Γ , D ) 2 . (55) Sado v’s transformation. The c hoice Ψ ≡ s 1 , 0 s a 1 − a 3 a 2 − a 3 D giv es Φ − U = a 1 Γ 2 2 " 1 − a 1 − a 3 a 1  D Γ  2 # = a 1 Γ 2 2 " 1 − a 1 − a 3 a 1 κ 2 κ 2 + λ 2 # , (56) that is Sadov ’s [32] reduced Hamilto nian in which κ 2 ≡ f and λ 2 = κ 2 Γ 2 D 2 − 1 ! . (57) Kinoshita’s case. If w e instead c ho ose Ψ ≡ s 1 , 0 v u u t a 1 ( a 1 + a 2 − 2 a 3 ) ( a 2 − a 3 ) ( a 1 + a 2 ) D , (58) w e find Φ − U = 2 a 1 a 1 + a 2 ˜ H , ˜ H = 1 2  a 1 + a 2 2 Γ 2 + 1 b D 2  , 1 b = a 3 − 1 2 ( a 1 + a 2 ) < 0 , (59) where ˜ H is Kinoshita’s reduced Hamiltonian [24, see Eq. (10’) on p. 43 3 ]. Alternativ ely , the scaling ma y b e av oided in our general form ulation by choo sing a regula r izing factor χ = 2 a 1 / ( a 1 + a 2 ) indep enden t o f the v ariables. Th us, the ma jor ob j ection to Sadov’s and Kinoshita’s elemen ts of b eing related to Ando yer canonical v ariables through implicit transformations is easily circum v en ted using the general tra nsformation (37)–(43) ab ov e particularized f o r either Sadov ’s or Kinoshita’s Hamiltonians, Eq. (56) and (59) resp ectiv ely . 10 F ukushima prop osal. If w e choo se now Ψ ≡ s 1 , 0 s a 1 a 2 − a 3 √ Γ 2 − D 2 , (60) w e get F ukushima’s prop osal [17] Φ = U + 1 2 a 1 D 2 , (61) that, dep ending on less than the required momen ta, constrain the top ology of the trans- formed Hamiltonian t o p erio dic solutions only . Nev ertheless, this wa y o f pro ceeding ma y b e p erfectly adequate for a p erturbation theory , as sho wn in [37]. TR-t yp e mapping. In a similar w ay to F ukushima, w e prop ose to c ho ose Ψ ≡ s 1 , 0 s a 1 a 2 − a 3 (Γ − D ) , (62) leading to Φ − U = a 1 D  Γ − 1 2 D  (63) that is formally equal to the reduced Hamiltonian of the TR-mapping [34, 11] giv en in Eq. (125) b elo w. It is quadratic, dep ends on tw o momenta — as it should b e for the more general case of a sup er- in tegra ble three degrees of freedom problem with f o ur indep enden t in tegra ls [4]— and the momen ta k eep the dimensions of the or iginal pro blem b ecause the Hamiltonian is multiplied by the in v erse of the momen tum of inertia a 1 . Then, the reduced system has four elemen t s ( υ , D , Γ , Υ) and tw o v ariables that ev olv e linearly with time γ = γ 0 + a 1 D t, d = d 0 + a 1 (Γ − D ) t, the conditio n Γ /D ratio nal providing t he subset of p erio dic solutions. 3.2.2 Case s 2 = s 2 , 0 and s 3 = s 3 , 0 constan t F rom the second and third of (52) w e get s 2 , 0 Φ D = s 2 , 0 s 3 , 0 Φ Υ = 2 Φ − U Γ − Φ Γ ⇒ Φ = U + Γ 2 Ψ( D − s 2 , 0 Γ + s 3 , 0 Υ , U ) (64) where Ψ is arbitrary . A simple choice satisfying the condition in (64 ) could b e Φ = U + Γ 2 ( D − s 2 , 0 Γ), for instance. P articular cases in the literature mak e s 2 , 0 = 0. Thus , Deprit and Elip e [12] c ho ose Φ − U = 1 2 Γ 2 D , (65) Other solutions could b e, for instance, Φ − U = Γ 2 D 2 , Φ − U = Γ 2 D . (66) 11 4 Orbital C anonical transformation s What w e hav e said at the b eginning of the previous Section for canonical v ariables in rotational dynamics, it a pp ears again in orbita l v ariables. Th us, mean while Hamiltonian particle dynamics still happ ens to b e in tro duced in spherical cano nical v ariables (see for instance [6], [18], [23], [35]), at the same time in space researc h ano ther cano nical set of v ariables is widely in use; w e refer to no dal- p olar v ariables, already considered for planetary theories almost a cen tury ago [22]. The main difference of b oth set of v ariables is that no dal- p olar v ariables carries out a double reduction (axial and cen tral symmetries ) of Kepler t yp e systems, mean while spherical v ariables only deals with the axial symmetry . As in t he searc h of orbital canonical t ransformations we build up our H- J equation based on no dal-p ola r v ariables, w e briefly giv e a description ab out them. T aking a ref- erence f r ame ( O , e 1 , e 2 , e 3 ), they are intro duced in a natural w ay a sso ciated with the “instan ta neous pla ne of motion” whose c hara cteristic v ector is x × X . The no dal-p olar v ariables ( r , ν, θ ) are defined as follow s: the v a riable r is k x k ; the ang le ν called the as- cending no de , is defined by cos ν = e 1 · ℓ and sin ν = e 2 · ℓ , where ℓ is the unit v ector defined b y e 3 × ( x × X ); the v ariable θ is a p olar angle in the plane of motion giving the p osition of x rec k oned from the vec tor ℓ . The v ariables ( R , N , Θ) are the corresp ond- ing conjuga te momenta. F or more details and explicit expressions of the transfor ma t io n ( x, y , z , X , Y , Z ) → ( r , ν, θ , R , N , Θ), called b y some authors Whittak er transformation, see for instance [10]. 4.1 The general structure of the transformations Giv en the function H = " 1 2 R 2 + Θ 2 r 2 ! + V ( r ) + T # χ ( r , Θ , N , T ) (67) where ( r , θ , ν, t, R , Θ , N , T ) are Hill o r p olar-no dal v ariables in t he extended phase space and V only dep ends on distance, w e lo ok f or a canonical transformation ( r , θ , ν, t, R, Θ , N , T ) T Φ − → ( f , g , h, u, F , G, H , U ) that conv erts (67) in a certain function Φ( F , G, H , U ) dep ending only on the momen t a. The tra nsformation will b e defined throug h a generating function in mixed v a r ia bles S = S ( r, θ , ν , t, F , G, H , U ) suc h that f = S F , g = S G , h = S H , u = S U , R = S r , Θ = S θ , N = S ν , T = S t . (68) Then, from (67) w e set the H-J equation  1 2  S 2 r + 1 r 2 S 2 θ  + V ( r ) + S t  χ ( r , S θ , S ν , S t ) = Φ( F , G, H , U ) (69) Because t , θ , and ν are not presen t in (67), the generating function ma y b e c hosen in separate v ariables S = U t + H ν + G θ + W ( r, F , G, H , U ) (70) 12 Then, 1 2 W 2 r + G 2 r 2 ! + V ( r ) + U = Φ( F , G, H , U ) χ ( r , G, H , U ) (71) that can b e solv ed for W by a simple quadrature: W = Z r r 0 q Q ( r , F , G, H , U ) d r , (72) where Q = 2 Φ( F , G, H , U ) χ ( r , G, H , U ) − 2 V ( r ) − G 2 r 2 − 2 U ≥ 0 . (73) Therefore, the tr a nsformation is f = Φ F I 3 , (74) g = θ + G I 1 + Φ G I 3 − Φ Z r r 0 χ G χ 2 √ Q d r , (75) h = ν + Φ H I 3 − Φ Z r r 0 χ H χ 2 √ Q d r , (76) u = t − I 2 + Φ U I 3 − Φ Z r r 0 χ U χ 2 √ Q d r , (77) R = W r = q Q, (78) Θ = G, N = H, T = U. (79) where I 1 = Z r r 0 1 √ Q d  1 r  , I 2 = Z r r 0 d r √ Q , I 3 = Z r r 0 d r χ √ Q . (80) The case χ = χ ( r ) . T ransformatio ns a re clearly simplified when we c ho ose the reg- ularizing factor χ indep enden t of the momen ta G , H , and U . Th us, if χ = χ ( r ), we ha ve χ G = χ H = χ U = 0 , (81) and the rightmost terms on the right side of (74)– (77) v anish, which turn these equations in to f = Φ F I 3 , (82) g = θ + G I 1 + Φ G I 3 , (83) h = ν + Φ H I 3 , (84) u = t − I 2 + Φ U I 3 . (85) Note that χ = 1 a nd χ = r 2 mak e I 2 = I 3 and I 3 = −I 1 , resp ectiv ely . W e plan to discuss these c hoices in detail b elo w, fo cusing on a tw o-parametric family V = V ( r ; µ, b ) of ra dial p oten tials. 13 4.2 F amilies of the iso c hronal p oten tial In what f ollo ws we only deal with the transformatio n (82)–(85) a nd, in a ddition, limit our study to the sp ecific case of H ´ enon’s iso ch ronal p oten t ia l [21, 6] V = − µ b + √ b 2 + r 2 , (86) where b and µ are parameters. A relev an t , particular case of the iso c hrona l is the Keplerian p oten tial V = − µ/r that o ccurs fo r b = 0. F or the iso c hrona l p oten tial (86) w e find con v enien t to write (73) a s Q = r 2 + b 2 r 2 Q ≥ 0 , Q = − α p r 2 + b 2 − 2 √ r 2 + b 2 + 1 a ! , (87) where α , p , a nd a , are certain functions of the momen ta and parameters, whic h only will b e determined after the regularizing parameter χ has b een chosen. Then, Q = α p  1 s − 1 s 1   1 s 2 − 1 s  , (88) where s 1 ≥ ( s = √ r 2 + b 2 ) ≥ s 2 are the t w o p ossible ro ots of the conic Q = 0: s 1 , 2 = p 1 ± q 1 − p/a = a (1 − e 2 ) 1 ± e = a (1 ± e ) , e 2 = 1 − p a < 1 . (89) 4.2.1 In t ro ducing auxiliary v ariables The ro ots s 1 , 2 , the extreme v alues of s , mak e natural the introduction of the auxiliary v ariables ψ and φ , defined through √ r 2 + b 2 = a (1 − e cos ψ ) , d √ r 2 + b 2 = a e sin ψ d ψ , (90) √ r 2 + b 2 = p 1 + e cos φ , d 1 √ r 2 + b 2 ! = − e p sin φ d φ, (91) in order to ease the solution of quadratures in (80), whic h w e find con ve nien t to write I 1 = Z r r 0 r 2 + b 2 r 2 √ Q d 1 √ r 2 + b 2 ! , (92) I 2 = Z r r 0 1 √ Q d  √ r 2 + b 2  , ( 9 3) I 3 = − Z r r 0 r 2 + b 2 χ √ Q d 1 √ r 2 + b 2 ! = Z r r 0 1 χ √ Q d  √ r 2 + b 2  . (94) F or giv en v alues of the regularizing factor χ , t he quadratures ab o v e, (92)–(9 4), may b e in tegra ted in closed form without need of sp ecifying the new Hamiltonian, whic h remains as a formal function of the momen ta Φ = Φ( F , G, H , U ). Details on the families of canonical tr ansformations generated by the cases χ = 1, χ = b + √ b 2 + r 2 , and χ = r 2 are given b elow. Besides , fo r the Keplerian case b = 0 we reco ve r Delauna y’s [8], Levi-Civita’s [25, 26], and Hill’s [22] canonical tr ansformations, as particular examples of our families. An elegan t alternat ive for the deriv ation of these historic t r a nsformations ma y b e found in [2]. 14 4.3 Case I: Delauna y’s family of transformations χ = 1 . F rom (73) and (87) w e obtain α = µ, a = µ 2( U − Φ) , p = G 2 µ + 2 b − b 2 a . (95) Then, in tro ducing the change of (90) in (9 3), I 2 is easily solve d to giv e I 2 = I 3 = µ q 8( U − Φ) 3 ( ψ − e sin ψ ) . (96) Analogously , the change of (91) is introduced in (92) to solv e the quadrature I 1 . W e obtain I 1 = − φ 1 2 G − φ 2 2 √ G 2 + 4 b µ , (97) where w e hav e in t r o duced tw o new auxiliary v ariables φ 1 , φ 2 , defined by means of the trigonometric relations [37] tan φ 1 2 = v u u t 1 + e − b/a 1 − e − b/a s 1 − e 1 + e tan φ 2 = v u u t 1 + e − b/a 1 − e − b/a tan ψ 2 , (98) tan φ 2 2 = v u u t 1 + e + b/a 1 − e + b/a s 1 − e 1 + e tan φ 2 = v u u t 1 + e + b/a 1 − e + b/a tan ψ 2 . (99) Substitution o f I 1 and I 2 ≡ I 3 , giv en b y (97) a nd (96), resp ective ly , in (82)–(85) pro vides the family of canonical transformatio ns of the iso c hrona l p oten tial to whic h w e attac h the name of Delauna y . Among the v ariet y of p ossible c hoices of the new Hamiltonian Φ, a simplifying o ptio n is to take Φ = U + Ψ( F , G , H ; µ , b ), tha t make s u to remain as the original time. In addition, the choice Ψ H = 0 , Ψ F = Ψ G = µ − 1 ( − 2Ψ) 3 / 2 , ma y b e solv ed for Ψ, to giv e Φ = U − 1 2 µ 2 ( F + G ) 2 (100) that maximally simplifies the transformatio n equations: f = ψ − e sin ψ , (101) g = θ − φ 1 2 − G √ G 2 + 4 b µ φ 2 2 + f , (102) h = ν , (103) u = t, (104) while retaining the top ology of the original Hamiltonian. Delauna y’s selec tion Ψ H = 0 , Ψ G = 0 leads to Φ = U − µ 2 2 F 2 (105) 15 that dep ends on few er momenta and, therefore, constrains the top ology of the original system to p erio dic solutions only —whic h may b e adequate fo r a p erturba tion study lik e [37]. In the K eplerian case b = 0, V = − µ/r , the selection of the new Hamiltonia n Φ = Φ( U, F ) do es not constrain the range of solutions and the sp ecific selection of (1 05) pro vides the p opular Delaunay transformation that , ta king in to accoun t that φ 1 = φ 2 = φ , is f = ψ − e sin ψ , g = θ − φ, h = ν , u = t, (106) where the most extended notation writes L ≡ F , ℓ ≡ f . 4.4 Case I I: Levi-Civita’s family of transformations χ = b + √ b 2 + r 2 No w, w e write α = Φ + µ, a = Φ + µ 2 U , p = G 2 Φ + µ + 2 b − b 2 a . (107) The c ha ng e ( 91) is used to in tegrate I 1 , (92), and the change (90) to integrate I 2 and I 3 , (93) and (94). W e get I 1 = − φ 1 2 G − φ 2 2 q G 2 + 4 b ( µ + Φ) , (108) I 2 = µ + Φ √ 8 U 3 ( ψ − e sin ψ ) , (109) I 3 = ψ √ 2 U − b φ 2 q G 2 + 4 b ( µ + Φ) . (110) where φ 1 , φ 2 , ar e the same auxiliary v ariables defined in (98) and ( 9 9) resp ectiv ely . Then, substitution of I 1 , I 2 , I 3 , (1 0 8)–(110), in (82)–(85) prov ides the family of canonical trans- formations o f the iso chronal p oten t ia l to whic h w e tie the name of Levi-Civita. The, Keplerian case b = 0, φ 1 = φ 2 = φ , giv es f = Φ F 1 √ 2 U ψ , (111) g = θ − φ + Φ G 1 √ 2 U ψ , (112) h = ν + Φ H 1 √ 2 U ψ , (113) u = t − µ + Φ √ 8 U 3 ( ψ − e sin ψ ) + Φ U 1 √ 2 U ψ . (114) The canonical tr a nsformation still r emains undefined. W e note in (111) that f = ψ when Φ F = √ 2 U , a partial differen tial equation t hat may b e solv ed to giv e Φ = √ 2 U h F + C 1 ( G, H , U ) i , (115) 16 where C 1 is an arbitrary function o f G , H , and U . The simple c hoice C 1 = µ pro vides the “first” Levi-Civita [25 ] transfor ma t io n 3 f = ψ , g = θ − φ, h = ν, u = t − µ √ 8 U 3 ( f − e sin f ) . (116) Other p ossibilit y is to force Φ U = ( µ + Φ) / ( 2 U ) so that only p erio dic o scillations a re in tro duced in t he time scale (114). Then, Φ = − µ + √ U C 2 ( F , G, H ) . (11 7 ) with C 2 arbitrary . The com bination of b oth conditions (115) and (117 ) leads to Φ = √ 2 U F − µ √ 2 U + C ( G, H ) ! , (118) with C a r bit r a ry . If we further tak e C ≡ 0, w e get f = ψ , g = θ − φ, h = ν , u = t + F 2 U e sin f . (119) that is the famous “second” Levi-Civita [2 6 ] transformation. 4.5 Case I I I: Hill’s family of transformations: χ = r 2 W e g et now α = µ, a = µ 2 U , p = G 2 − 2 Φ µ + 2 b − b 2 a . (120) The required quadratures are solve d with the same c hanges of v ariables as b efore, giving I 1 = −I 3 = − φ 1 2 √ G 2 − 2 Φ − φ 2 2 √ G 2 − 2Φ + 4 b µ , (121 ) I 2 = µ √ 8 U 3 ( ψ − e sin ψ ) , (122) whic h substitution in (82)–(85) provides the family of canonical transformations of the iso c hronal p oten tial to which w e assign the name of Hill. F o cusing on the K eplerian case b = 0, φ 1 = φ 2 = φ , a simplification choice that mak es f = φ is t o select Φ in suc h a w ay that Φ F I 1 = − φ , then Φ = 1 2 h G 2 − ( F + C ) 2 i , (123) where C ≡ C ( G, H , U ) is an arbitrary function. Then, Φ F = − ( F + C ) , Φ G = G − ( F + C ) C G , Φ H = − ( F + C ) C H , Φ U = − ( F + C ) C U . (124) 3 Note that Levi-Civita pro cee ds by scaling r b y 1 / √ 2 U , or cho osing χ = r/ √ 2 U in our notation. This is equiv alent to s caling our new Hamiltonian by the sa me factor, and yields to Levi-Civita’s original Hamiltonian Φ = F while the tr ansformatio n (116) re main identical 17 The trivial c hoice C = 0 makes g = θ , h = ν , and u = t − µ ( ψ − e sin ψ ) / √ 8 U 3 . Cho osing C as a linear com bination of the momen ta, simply ads φ to the angle v ariables. Sp ecifically , when w e set C = − G w e obtain 4 Φ = F  G − 1 2 F  (125) and reco v er the TR-mapping [34, 1 1] f = φ, g = θ − f , h = ν , u = t − µ √ 8 U 3 ( f − e sin f ) . (126) 5 Conclus ions A deep insigh t in the computat io n of canonical transformations is obta ined throug h a general form ulation of the Hamilto n- Jacobi equation in t he extended phase space that, b esides, includes a regularizing function. As far as the transformed Hamiltonian ma y re- main formal, one can obtain fa milies of canonical transformations instead of single sets of canonical v ariables. Then, particular transformations that meet the user’s required c ha r- acteristics are systematically computed from the solution of partial differen t ial equations that may enjoy v ery simple solutio ns. Our fo r m ulatio n sho ws that differen t transformations in the literature b elong t o the same fa milies. F urthermore, it reve als fundamen tal features of the transfor ma t ions. Th us, among the w o rk ed examples, w e identify families ha ving as mem b ers kno wn canonical c hanges of v ariables that originally we re defined through implicit relations. Ac knowledgemen ts P artia l supp ort is recognized from pro jects MTM 2006 - 06961 (S.F.) and A Y A 200 9-1189 6 (M.L.) of the G o v ernmen t of Spain, and a gran t from F undaci´ on S ´ eneca of the autonomous region of Murcia. References [1] Abraha m, R., Marsden, J.E., 19 85, F undatio ns of Mec hanics, P erseus Bo oks, Cam- bridge Mass. [2] Andoy er, M.H., 1913 , “Sur l’Anomalie Excen trique et l’Anomalie V r aie comme ´ El ´ emen ts Canoniques du Mouv emen t Elliptique d’apr ` es MM. T. Levi-Civita et G.-W. Hill,” Bul letin astr onomique , V ol. 30 , pp. 425–429. [3] Andoy er, M.H., 1923 , Cours de M´ ec anique C´ eleste , t. 1, Gauthier-Villars et cie, P aris, p. 57. 4 Note that Hill pro ceeds by s caling r by 1 / q G − 1 2 F , or cho osing χ = r 2 / ( G − 1 2 F ) in o ur notatio n. This is equiv a lent to scaling our Φ new Hamiltonian b y G − 1 2 F , what yields to Hill’s Hamiltonian Φ = F while the trans fo rmation (126) r e mains unalter e d. 18 [4] Benettin, G., 2004, “The elemen ts of Hamiltonian p erturba tion theory ,” in Hamilto- nian Systems an d F ourier Analysis: New Pr osp e cts for Gr avitational Dynamics , D. Benest and C. F ro esc hle (ed.), pp. 1–98 . Cam bridge Scientific Publ., Cambridge, UK. [5] Bla nes, S., Budd, C., 20 0 4, “Explicit Adaptiv e Symplectic ( Easy) In tegrato rs: A Scaling In v arian t Generalisation of the Levi-Civita and KS Regularisations,” Celestial Me chanics and Dyna m ic al Astr onomy , V o l. 89, No. 4, pp. 383–4 0 5. [6] Bo ccaletti, D ., Pucacco, G ., 2001, The ory of Orbits. 1: In te gr able Systems and Non- p erturb ative Metho ds , Springer-V erlag, Berlin Heidelb erg, pp. 316 and ff. [7] Chang D. E. and Marsden, J. E., 2003, Geometric deriv ation of the D elauna y v ari- ables and geometric phases, Celes Me ch & Dyn Astr on 86 : 1 85–208. [8] D elauna y , Ch.E., 1867 , “Th ´ eorie du Mouv emen t de la Lune,” M´ emoir es de l’A c ademie des Sc i enc es de l’Institut I mp´ erial de F r anc e , V ol. 28 , See Chap. 1, par a . 5, pp. 9–11. [9] D eprit, A., 1967, F ree Rota tion o f a Rigid Bo dy Studied in the Phase Space, American J Ph ysics 35, 424–428. [10] Deprit, A. and Rom, A., 1970, Celest Me ch. 2 , 1 66. [11] Deprit, A., 198 1, “A Note Concerning the TR- T ra nsformation,” Celestial Me chanics , V ol. 23, pp. 299 – 305. [12] Deprit, A., 1993, “Complete Reduction o f the Euler-P oinsot Problem,” Journal of the Astr onautic al Scienc es , V ol. 4 1, No. 4, pp. 603–628. [13] F ass` o, F., 1996 , The Euler-P o insot top: a non-commu tativ ely in tegrable system with- out global a ction-angle co ordinates. J. Appl. Math. Phys. (ZAMP) 47, 953 –976. [14] F ass` o, F., 2005, “Sup erintegrable Hamiltonian Systems: G eometry and P erturba- tions,” A cta Applic andae Mathem a tic ae , V ol. 87, pp. 93–121 [15] F errer, S., Lara , M., 2009, “Orbital Canonical T ransformations. The Iso c hrona l F am- ily”, XI I Jornadas Mec´ anica Celeste , Lal ´ ın, Spain 1 - 3 July 2009 . [16] F ukushima, T., 2008, “Simple, Regular, and Efficien t Numerical Integration of Ro- tational Motion,” The Astr onomic al Journal , V ol. 13 5 , pp. 2298– 2322. [17] F ukushima, T., 2008, “ Canonical and Univ ersal Elemen ts of the Rotationa l Motion of a T riaxial Rigid Bo dy ,” The Astr onomic al Journal , V ol. 136, pp. 1 728–173 5 . [18] Goldstein, H., Poole, C., and Safko, J., 2002, Classic al Me chanic s , 3nd e di tion , P ear- son Addison-W esley . [19] Grandati, Y., B´ erard, A., and Mohrbac h, H., 2 0 08, “Bohlin-Arnold-V a ssiliev’s dualit y and conserv ed quantities ,” arXiv:0803.2 610v2 [math-ph] 19 [20] Gurfil, P ., 20 05, Canonical F ormalism for Modelling and Con trol of Rigid Bo dy Dynamics, in New T rends in Astro dynamics and Applications, Ann. N.Y. A c ad. Sci. , V o l. 1065: 3 91–413. [21] H ´ enon, M., 1959, “L ’a mas iso c hrone I,” Annales d’Astr ophysique , V ol. 22, pp. 126–39. [22] Hill, G. W., 1913, “Motion of a system of material p oints under the action of gravi- tation,” Astr onomic al Journal , V ol. 27, iss. 646– 647, pp. 1 7 1–182. [23] Classic al Mec hanics,..... [24] Kinoshita, H., 1972 , “F irst-Order P erturbatio ns of t he Tw o Finite Bo dy Problem,” Public ations of the Astr onomic al So ciety of Jap an , V o l. 24, pp. 423–457. [25] Levi-Civita, T., 1913, “Nouvo sistema canonico di elemen ti ellittici,” Annali di Matematic a , serie I I I, V ol. XX, pp. 153 –170. [26] Levi-Civita, T., “ Sur la r´ egularization du probl` eme des tro is corps,” A cta mathemat- ic a , V ol. 42 , 1918, pp. 99–144. [27] Lum, K-Y, Blo c h, A.M., 19 9 9, Generalized Serret-Andoy er T ransformation and Ap- plications for the Controlled Rig id Bo dy , Dynamics and Contr ol , V o l. 9, 39– 66. [28] Marsden, J.E., and Ratiu, T. 1999, Intr o duction to Me cha n ics and Symmetry , T AM 17, 2nd Ed, Springer, New Y ork. [29] Ortega, J. P . and Ratiu, T. 200 4, Mom entum Maps and Hamiltonian R e duction , Birkh¨ auser, Boston. [30] P oincar ´ e, H., 1 893, L es m´ e tho de s m ouvel les de l a m´ ec anique c´ eleste , V ol. 2, Ga uthier- Villars et Fils, P aris, pp. 315–342 . [31] Struc kmeier, J., 2005, “Hamiltonian Dynamics o n the symplec tic extended phase space for autonomous and non-auto nomous systems,” Journal of Physics A: Mathe- matic al and Gener al , V ol. 38, No . 6, pp. 1257–127 8. [32] Sado v, Y u. A., 1 970, “The Action-Angles V aria bles in the Euler-Poins ot Problem,” Prikladnaya Matematika i Mekhanika , V ol. 34, pp. 962-964. [ Journal of Applie d Math- ematics and Me chanic s (English translation), V ol. 34, No. 5, 1970 , pp. 922–925] [33] Sado v, Y u. A., 1970, “The Action-Angle V ariables in the Euler-Poinsot Problem,” Preprin t No. 22 of the Keldysh Institute of Applied Mathematics of the Academ y o f Sciences of the USSR, Mosco w (in Russian). [34] Sc heifele, G., 1 970, “G´ en ´ eralisation des ´ el ´ emen ts de Delauna y en m ´ ecanique c ´ eleste. Application a u mouv emen t d’un satellite a rtificiel”, L es Comptes r endus de l’A c ad´ emi e de s sc ienc es de Paris , S´ er. A, T ome 271, pp. 729–732. [35] Sussm an, G.J., Wisdom, J., 2001, Structur e and Interpr etation of Cla ssic al Me ch a n- ics , The MIT Press, Cam bridge, pp. 403 and ff.. 20 [36] Whittak er, E. T., A treatise on the analytical dynamics of particles and rig id b o dies, Cam bridge Mathematical Library , Cam bridge Univ ersit y Press, Cam bridge, 1 988, Reprin t of the 193 7 edition. [37] Y anguas, P ., 2001 , “P erturbations o f the iso c hrone mo del,” Nonline arity , V o l. 14 , pp. 1–34. 21