Integration of the Euler-Poinsot Problem in New Variables

In tegra tion of the Euler -P oinso t Problem in New V aria bles Martin Lara a, ∗ , Sebasti´ an F errer b a R e al Observator io de la Armada, 11110 San F erna ndo, Sp ain b Dep art amento de Matem´ atic a Aplic ada, Universidad de Mur cia, 30100 Mur cia, Sp ain Abstract The essen tially un iqu e reduction of the Euler-Po insot p roblem ma y b e p er- formed in different sets of v ariables. Action-angle v ariables are us ually pre- ferred b ecause of their suitabilit y for approac h ing p erturb ed rigid-b o dy m o- tion. B ut they are just one among the v ariety of s ets of canonical co ord inates that int egrate the pr ob lem. W e presen t an alternate set of v ariables that, while allo wing for similar p erformances than action-angles in th e study of p ertur b ed problems, sho w an imp ortant adv ant age o v er them: Th eir trans- formation from and to An d o y er v ariables is giv en in explicit form. Keywor ds: Euler-P oinsot reduction, Hamilton-Jacobi equation, elliptic in tegrals and functions, action-angle v ariables 1. In tro duction The Euler-P oinsot problem is a three degrees-of-freedom (DOF) prob- lem whose sup er-integ rable c h aracter limits the solutions to qu asi-p erio dic orbits on tw o-torus (F ass´ o, 2005). Because of the symmetry with resp ect to rotations ab out the angular momentum v ector, the p roblem is form ulated as a 1-DOF Hamilt onian when using Ando y er (1923) v ariables. Then, the essen tially u n ique complete reduction that pro vides the in tegration of the problem can b e p erformed in different v ariables, and it is usually done by solving the Hamilton-Jaco bi equation. In the stud y of the rigid b o dy rotation un der external torques the u s e of suitable v ariables reve als crucial to the solution by p ertur bation method s. A common trend is to use action-angle v ariables (Sado v, 1970a,b; Kinoshita, 1972), but other v ariables can b e used ins tead (Hitzl and Br eakwell, 1971). ∗ Corresponding au t hor Email addr esses: mlara@roa.es (Martin Lara), sferrer @um.es (Sebasti´ an F errer) Pr eprint submitte d to Me chanics R ese ar ch Communic ations Octob er 2, 2018 The Hamilton-Jacobi equation of the Euler-P oinsot prob lem in Andoy er v ariables can b e solv ed form ally , without need of sp ecifying the n ew Hamil- tonian (F errer a nd Lara, 2010). W e sho w h o w this new, unsp ecified Hamil- tonian can b e cast in to a standard form in which the mo dulus of the elliptic in tegrals that app ear in the solution of the transformation, remains as an undetermined state fu nction of the n ew m omenta. Und er certain conditions imp osed to th e formal tr an s formation, the mod ulus is determin ed by solv- ing a system of p artial differential equations. The condition we require is “simplification” and find a new set of v ariables that while sho w in g similar p erformances than action-angle s (Sado v, 1970a,b), has the b enefit of not requiring the computation of implicit fu n ctions. In addition, we demon- strate that Sad ov’s transform ation is also a mem b er of the general family of Euler-P oinsot transform ations to Andoy er v ariables. 2. Complete Reduction of the Euler-Poins ot Problem The Hamiltonian of the torqu e-free rotatio n is (Deprit, 1967) H =  sin 2 ν / A + cos 2 ν /B  ( M 2 − N 2 ) / 2 + N 2 / 2 C, (1) where A , B , and C are the principal moment s of inertia of the b o d y , and the An do y er v ariables are defined by thr ee pairs of conjugate v ariables: the rotation angle on the equatorial plane of the b o dy ν and the pro jection of the angular momen tum vec tor on the b o dy axis of maxima inertia N , the precession angle on the inv arian t plane µ and the mo d ulus of the angu lar momen tum v ector M , and the no d e angle on the inertial p lane λ and the pro jection of the angular momen tum vec tor on the axis p erp endicular to the inertial plane Λ. Because λ , Λ and µ are cyclic λ = λ 0 , Λ = Λ 0 , and M = M 0 are constant, and Eq. (1) is a Hamiltonian of 1-DOF. The integrat ion ma y b e done by complete r ed uction. T o this goal, we lo ok for canonical transformations T K : ( λ, µ, ν, Λ , M , N ) → ( ℓ, g , h, L, G, H ) that con ve rt Eq. (1) in a new Hamiltonian K that dep end s only on m omenta. Because of the tw o-torus top ology of the Euler-P oinsot problem only t wo momen ta are required in K , and in view of neither λ nor Λ app ear in Eq. (1), w e c ho ose h = λ , H = Λ and K ≡ K ( L, G ). 2.1. F orma l Solution of the Hamilto n-Jac obi Equation In the Hamilton-Jacobi approac h , the transformation T K is d eriv ed f rom a generating function in mixed v ariables S = S ( µ, ν , L , G ) su c h that ( ℓ, g , M , N ) = ∂ S ∂ ( L, G, µ , ν ) (2) 2 Because µ is cyclic in E q. (1), S is c hosen in separate v ariables S = G µ + W ( ν, L, G ). T h en, from Eq. (1) we form the Hamilton-Jacobi equation sin 2 ν 2 A + cos 2 ν 2 B ! " G 2 −  ∂ W ∂ ν  2 # + 1 2 C  ∂ W ∂ ν  2 = K (3) where W ma y be solved b y qu adrature. T h en, calling β = L/G , the trans- formation Eqs. (2) are ℓ = I 2 G 2 ∂ K ∂ β , (4) g = µ + I 1 − I 2 G 2  2 K + β ∂ K ∂ β  , (5) N = G p Q, (6) M = G, (7) where I 1 = Z ν ν 0 p Q d ν, I 2 = Z ν ν 0 1 √ Q ∂ Q ∂ (1 / ∆) d ν, (8) Q = sin 2 ν / A + cos 2 ν /B − 1 / ∆ sin 2 ν / A + cos 2 ν /B − 1 /C , (9) and 1 / ∆ = 2 K /G 2 . (10) W e only discuss the general ca se A < B < C . As √ Q m ust b e real for all ν , w e get B ≤ ∆ ≤ C an d 1 2 G 2 /C ≤ K ≤ 1 2 G 2 /B , thus constraining the motion to rotatio ns ab out th e axis of maxima inertia. The discussion of other cases is left to the reader. The transformation equations for ℓ and g , Eqs. (4)–(5), dep end on the in tegration of th e t wo quad r atures in Eq. (8). Ho wev er, as far as K d e- p ends only on the momenta G and L , the quadratures in Eq. (8) can b e solv ed with ou t need of sp ecifying the formal dep endence of K on the new momen ta, thereb y giving rise to a whole family of canonical transf ormations (F errer and Lara, 2010). The closed form solution of Eq. (8) relies on w ell kn o w n changes of v ar- iables. Thus, in tro du cing th e p arameter f > 0 and the function 0 ≤ m ≤ 1 f = C ( B − A ) ( C − B ) A , m = ( C − ∆) ( B − A ) ( C − B ) (∆ − A ) , (11) 3 and the auxiliary v ariable ψ defined as cos ψ = √ 1 + f sin ν q 1 + f sin 2 ν , sin ψ = cos ν q 1 + f sin 2 ν , (12) then, the quadratures in Eq. (8) are solv ed to giv e I 1 = γ  m f + m F ( ψ | m ) − Π( − f , ψ | m )  , (13) I 2 = γ A C C − A F ( ψ | m ) , (14) where γ = q (1 + f ) ( f + m ) /f = s B ∆ ( C − A ) ( C − A ) A C ( C − B ) (∆ − A ) , (15) F ( ψ | m ) is the elliptic integral of the fir st kind of mo du lus m and amplitude ψ , and Π( − f , ψ | m ) is the elliptic integral of th e third kind of mo d ulus m , amplitude ψ , an d c haracteristic − f . It is w orth ment ioning that for the definition of the elliptic inte gral of third kind we adhere to the con v en tion in (Byrd and F riedman, 1971). 2.2. The Stand ar d Hamiltonian F rom E qs. (10 ) and (11) w e note that K is characte rized b y the identi t y K = G 2 2 A  1 − C − A C f f + m  , (16) whic h can b e take n as a definition by assuming that m = m ( L, G ) in Eq. (16). Then , Eqs. (4)–(7) are rewritten ℓ = 1 2 γ 1 + f f + m ∂ m ∂ β F ( ψ | m ) , (17) g = µ + γ  1 f + m  m − f f + m β 2 ∂ m ∂ β  F ( ψ | m ) − Π( − f , ψ | m )  , (18) N = G s f f + m q 1 − m sin 2 ψ , (19) M = G. (20) T rans f ormations in the literature can b e obtained as particula r ca ses o f the general form Eqs. (16 )–(20). T hus, the new Hamiltonian selected by Hitzl and Breakw ell (1971 ) is the a v erage of the Ando y er Hamiltonia n Eq. (1), which is a lso the inte rmediate Hamiltonian of Kinoshita (1972), w h ile a previous p rop osal of ours (F errer and Lara, 2010) transforms the And o yer Hamiltonian to the axisymmetric case. 4 3. New v ariables Searc hing for simplification in Eqs. (17) and (18) w e p rop ose to c ho ose 1 2 γ 1 + f f + m ∂ m ∂ β = − 1 , 1 f + m  m − f f + m β 2 ∂ m ∂ β  = 1 . (21) Equations (21) can b e solv ed for β = β ( m ) without need of solving any partial differenti al equation. F u rthermore, by squaring β w e can express m as a function of L/G m = f h (1 + f ) G 2 /L 2 − 1 i . (22) Corresp ond ingly , the new Hamiltonian in n ew v ariables is K = G 2 2 A −  1 B − 1 C  L 2 2 , (23) whose Hessian never v anishes, and w h ic h is formally equ al to the uniaxial case for a n ew maxim um momentum of inertia, s a y P , such that 1 /P = 1 / A + 1 /C − 1 /B . Then, the direct tr ansformation is ℓ = − F ( ψ | m ) (24) g = µ + q (1 + f ) ( f + m ) /f [ F ( ψ | m ) − Π( − f , ψ | m )] (25) L = N q (1 + f ) / (1 − m sin 2 ψ ) (26) G = M (27) where ψ is defi ned in E q. (12) and m is easily computed in Ando y er v ariables from its definition in Eq. (11 ) b y noting that ∆ = G 2 / (2 K ) = M 2 / (2 H ), where H is giv en in Eq. (1). The in v erse transformation requires using th e J acobi amplitude am to in v ert Eq. (24) ψ = − am ( ℓ | m ) , (28) where m is c omputed from Eq. (22). Th en, from Eq. (12) we get cos ν = − √ 1 + f sn( ℓ, m ) p 1 + f sn 2 ( ℓ, m ) , sin ν = cn( ℓ, m ) p 1 + f sn 2 ( ℓ, m ) . (29) where sn, cn, dn, stand for the usu al Jacobi elliptic fu nctions. Finally , the in v erse trans formation of Eqs . (24)–(2 7 ) is completed with µ = g + (1 + f ) ( G/L ) [ ℓ + Π( − f , am ( ℓ | m ) | m )] , (30) N = L d n( ℓ, m ) / p 1 + f , (31) M = G. (32) 5 4. T ransformation to action-angle v ariables Note in Eq. (28) that the v ariable ℓ is 4 K ( m )-p erio dic, with K ( m ) the complete elliptic in tegral of the first kind. With a view on p erturbations of the Euler-P oinsot problem, where elliptic functions would b e expanded in F ourier series, it could b e desired that ℓ b e 2 π -p erio dic (an angle). The new v ariable ℓ ′ = − π 2 K ( m ) F ( ψ | m ) , (33) will b e obtained b y requiring to Eq. (17) th at 1 2 γ 1 + f f + m ∂ m ∂ β ′ = − π 2 K ( m ) , (34) where β ′ = L ′ /G ′ . Equation (34) is in separate v ariables and is in tegrated b y quadr atur e to giv e, up to an in tegration constant, β ′ = 2 π q (1 + f ) ( f + m ) /f  Π( − f , m ) − m f + m K ( m )  , (35) where Π( − f , m ) is th e complete elliptic in tegral of the third kind. Equation (35) defines m as implicit fu nction of L ′ /G ′ . No w , w e rep lace β ′ in Eq. (18) by its v alue fr om Eq . (35) to g et g ′ = µ + q (1 + f ) ( f + m ) /f  Π( − f , m ) K ( m ) F ( ψ | m ) − Π( − f , ψ | m )  . (36) Remark ably , Eqs. (20), (33), (35) and (36) reco v er the original transfor- mation to action-a ngle v ariables (Sado v, 1970a,b) without need of relying on their classical definition. 1 Finally , we n ote that the inv erse transf ormation from action-angles to Ando y er v ariables requires the computation of m from th e implicit function Eq. (35). 1 In (Sad ov, 1970a,b) , f ≡ κ 2 and m ≡ λ 2 , Andoy er v ariables are ( h, ψ , φ, L, G, G ζ ) ≡ ( λ, µ, ν, Λ , M , N ), and the action-angles are ( f , ν , h, I , G, L ) ≡ ( ℓ ′ , g ′ , h ′ , L ′ , G ′ , H ′ ). Be- sides, Sado v ’s auxiliary angle is ξ = − ψ . Note t h at there is a typo in the defin ition of λ 2 in Eq. (4) of ( S adov , 1970a ), whic h should b e multiplie d by A/C . This typo is easily traced in Eq. (2.20) of (Sadov, 1970b), but it still remains in (Sadov, 1984; Kozlo v, 2000). 6 5. Conclusions Action-angle v ariables are not necessarily th e b etter option for dealing with p erturb ed motion. This fact is v ery well known for p ertur b ed Kep- lerian motion where Delaunay v ariables are used cu s tomarily . Th e same happ en s to rotational motion where act ion-angle v ariables ha v e the incon- v enience of b eing related to An d o y er v ariables through implicit relations. But t he complete redu ction of the Euler-Poi nsot problem may b e ac hiev ed in a v ariet y of canonical v ariables. Indeed, w e demons trate that when solving the Hamilton-Jacobi equ ation of the Euler-P oinsot problem in Andoy er v ari- ables, the new Hamiltonian can b e cast into a s tand ard form as a fu nction of the mo du lus of th e elliptic integral s required in th e solution, a qu an tit y that is consubstantial to the p r oblem. Then, the solution of the Hamilton-Hacobi equation can b e written formally as a function of the mo dulu s and its partial deriv ativ es with resp ect to the new momenta. Solving these partial deriv a- tiv es according to certain criteria p ro vides the desired transformation. In ou r case, we require “simplification” and find a new transformation of v ariables that, wh ile ha ving similar c haracteristics than actio n-angles v ariables, do es not rely on implicit fun ctions. Besides, we sho w that the transformation to action-angle v ariables p ertains also to this general family . 6. Ac knowledgemn ts W e thank supp ort f rom th e Spanish Min istr y of Science and Inno v ation, projects A Y A 2009-118 96 (M.L.) and MTM 2009-10 767 (S.F.), and f r om F und aci´ on S ´ eneca of the autonomous region of Murcia (grant 12006/ PI/09). W e are indebt with Prof. Sado v, Russian Academy of Sciences, for sending us copies of his p reprints in R u ssian. References Ando y er, M.H., 1923. Cour s de M ´ ecanique C ´ eleste, Gauthier-Villars et cie , P aris, p. 57. Byrd, P .F., F r iedman, M.D., 1971 . Hand b o ok of Elliptic I n tegrals for Engi- neers and Scien tists, Springer-V erlag, Berlin Heidelb erger New Y ork. Deprit, A., 1967. F ree Rotation of a Rigid Bo d y S tudied in th e Ph ase Space. Am. J. Ph ys. 35, 424– 428. F ass` o, F., 2005, S up erin tegrable Hamiltonian Systems: Geometry and P er- turbations. 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