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PARTIAL REDUCTION AND DELAUNAY/DEPRIT VARIABLES LEIZHAO Abstract. Basedonaconceptuallinkbetweenthepartialreductionprocedureofthereductionoftherotational symmetryoftheN-bodyproblemwiththesymplecticcross-sectiontheoremofGuillemin-Sternberg,wepresent 4 alternativeproofsofthesymplecticityofDelaunayandDepritcoordinatesincelestialmechanics. 1 0 2 ul Contents J 1. Introduction 1 2 1 2. TheThree-bodyProblemandtheJacobiDecomposition 2 3. Reductions: fromJacobitoDeprit 3 ] S 3.1. Jacobi’seliminationofthenodesofthethree-bodyproblem 3 D 3.2. Reductionofthethree-bodyproblemintheDepritvariables 4 . h 3.3. DepritcoordinatesforN-bodyproblem 6 t a 4. AConceptualViewofthePartialReductionProcedure 6 m 5. SymplecticComplementoftheSymplecticCross-Sections 7 [ 6. SymplecticityofDelaunayandDepritcoordinates 7 2 6.1. Delaunaycoordinates 8 v 6.2. Depritcoordinatesforthethree-bodyproblem 9 8 6 6.3. DepritcoordinatesfortheN-Bodyproblem 9 0 References 9 6 . 1 0 4 1. Introduction 1 : v Inthisarticle, weaimtopresentaconceptuallinkbetweentheideaofthepartialreductionprocedure i X [1]inthereductionoftheSO(3)-symmetryofthethree-bodyand N-bodyproblems(whosephasespaces, r after reduction by the translation symmetries, are denoted indifferently by Π) with the symplectic cross- a sectiontheoremofGuillemin-Sternberg,andpresentitsroleinthedeductionofseveralimportantDarboux coordinates,theDelaunayandtheDepritcoordinatesofcelestialmechanics. FollowingJacobi,weknowthatthe(full)reductionoftheSO(3)-symmetrycanbeachieved,byexample, byfixingthetotalangularmomentumC(cid:126)ofthesystemandruleouttheSO(2)-symmetryofrotationsaround the direction of C(cid:126). The method of partial reduction proposed in [1] is to only fixing the direction of the angular momentum. The resulting submanifold of the phase space is symplectic, and the restriction of theSO(3)-symmetryinthissubmanifoldbecomesthesymmetryofSO(2),amaximaltorusofSO(3). We see that the action of the maximal torus SO(2) has a non-trivial dynamical effect (a periodic orbit in the SO(3)-reducedsystemareingeneralonlyquasi-periodicinthesystemwiththeSO(3)-symmetry,withan addition frequency corresponds to the action of the maximal torus), while the rotation of the direction of Date:July15,2014. 1 2 LEIZHAO thetotalangularmomentumonlysendorbitstootherorbitsdonotinterferetheessenceofthedynamics. We remark that more generally, this procedure can be achieved for a Hamiltonian action of an arbitrary compact connected Lie group Gr on a symplectic manifold (M,ω) with moment map µ. In this general context,thepartialreductionprocedureisachievedbyfixingaCartansubalgebrah∗ing∗(wheregdenotes theLiealgebraofGr),fixingaWeylchambert∗ inh∗andconsiderthesetµ−1(t∗). AtheoremofGuillemin + + andSternbergstatesthatthissetisasymplecticmanifold. TherestrictionofGr-actiontoµ−1(W+)(which iscalledasymplecticcross-sectionoftheG action)isthustheHamiltonianactionofoneofitsmaximal r torus. (Thebook[2]providesanicepresentationofalltheinvolvednotionsinthetheoryofLiegroupsand Liealgebra.) Beingabelian,atorussymmetricgroupisingeneralmucheasiertohandle. Withthehelpofthisconstruction,weshalldealwithsomeconcreteproblemofdeterminingaction-angle coordinatesfor N −1uncoupledKeplerianellipses. AgenericSO(3)-coadjointorbitishomeomorphicto S2, which only admits one invariant symplectic form up to multiplication of a constant. By determining thisconstantinconcretecircumstances,wecanthusrecoverthesymplecticformonΠfromitsrestriction tothesymplecticcrosssectionandtheKirillov-Konstantsymplecticformofthecoadjointorbits. Insucha way,weobtainalternativeproofsofthesymplecticityoftheimportantDelaunayandDepritcoordinatesin celestialmechanics,avoidingtheuseofHamilton-Jacobimethods. We organize this article as the following: In Section 2, we recall the Hamiltonian formulation of the three-bodyproblemandthereductionofthetranslation-invarianceusingtheJacobicoordinates. InSection 3, we recall the reduction of the rotation-invariance of Jacobi and Deprit. In Section 4, we indicate the link of these reduction procedures with the symplectic cross-section theorem of Guillemin-Sternberg. In Section5weproveatheoremontheformofthecomplementarypartofthesymplecticform,whichisthen appliedinSectionSymplecticityofDelaunayandDepritcoordinatesto(re-)establishthesymplecticityof theDelaunayandDepritcoordinates. 2. TheThree-bodyProblemandtheJacobiDecomposition Thethree-bodyproblemisaHamiltoniansystemwithphasespace (cid:110) (cid:111) (pj,qj)j=0,1,2 =(p1j,p2j,p3j,q1j,q2j,q3j)∈(R3×R3)3|∀0≤ j(cid:44)k≤2,qj (cid:44)qk , (standard)symplecticform (cid:88)2 (cid:88)3 ω = dpl ∧dql, 0 j j j=0 l=1 andtheHamiltonianfunction 1 (cid:88) (cid:107)p (cid:107)2 (cid:88) m m F = j − j k , 2 m (cid:107)q −q (cid:107) 0≤j≤2 j 0≤j<k≤2 j k inwhichq ,q ,q denotethepositionsofthethreeparticles,and p ,p ,p denotetheirconjugatemomenta 0 1 2 0 1 2 respectively. The physical space R3 is equipped with the usual Euclidean norm (cid:107)·(cid:107). The gravitational constanthasbeensetto1. TheHamiltonian F isinvariantundertranslationsinpositions. Tosymplecticallyreducethesystemby thissymmetry,wemayswitchtotheJacobi(baricentric)coordinates(P,Q),i=0,1,2,with i i    PP0 == pp0++σp1p+p2  QQ0 ==qq0−q  P1 = p1 1 2  Q1 =q1−σ0q −σ q , 2 2 2 2 0 0 1 1 PARTIALREDUCTIONANDDELAUNAY/DEPRITVARIABLES 3 inwhich 1 m 1 m =1+ 1, =1+ 0. σ m σ m 0 0 1 1 The Hamiltonian F is thus independent of Q due to the symmetry. We fix P = 0 and reduce the 0 0 translation symmetry by eliminating Q . In the (reduced) coordinates (P,Q),i = 1,2, the function 0 i i F = F(P ,Q ,P ,Q )describesthemotionsoftwofictitiousparticles. 1 1 2 2 Inthesamefashion(c.f. [?, n.385]), wemayreducethetranslationsymmetryoftheN-bodyproblem, andtostudythe(reduced)dynamicsofN−1fictitiousparticles. 3. Reductions: fromJacobitoDeprit ThegroupSO(3)actsonΠ,thereducedphasespaceofthethree-bodyproblembythetranslationsym- metry, by simultaneously rotating the two relative positions Q ,Q and the two relative momenta P ,P . 1 2 1 2 ThisactionisHamiltonianunderthestandardsymplecticformonΠ,theHamiltonianF isinvariantunder thisSO(3)-action,anditsmomentmapisthetotalangularmomentum1C(cid:126) =C(cid:126) +C(cid:126) ,inwhichC(cid:126) := Q ×P 1 2 1 1 1 andC(cid:126) := Q ×P . ThereductionprocedurecanthenbeachievedbyfixingthemomentmapC(cid:126) (equiva- 2 2 2 lently, the direction ofC(cid:126) andC = |C(cid:126)|) to a regular value (i.e. C(cid:126) (cid:44) (cid:126)0) and then reducing the system from theSO(2)-symmetryaroundC(cid:126). AsSO(3)alsoactsonthespaceof(oriented)directionsofC(cid:126),thereduced systemoneobtainsmustbeindependentofthedirectionofC(cid:126),andthereforehas4degreesoffreedom. The plane perpendicular to the total angular momentumC(cid:126) is invariant. It is called the Laplace plane. ChoosingtheLaplaceplaneasthereferenceplane2(i.e. fixC(cid:126) vertical)shallgiveusaveryconvenientway of calculating the reduced Hamiltonian, as was obtained by Jacobi. Nevertheless, we can also fixC(cid:126) non- orthogonaltothereferenceplane. Inthiscase,theDepritcoordinatesshallprovideusanexplicitreduction procedure. 3.1. Jacobi’s elimination of the nodes of the three-body problem. As the angular momentaC(cid:126) ,C(cid:126) of 1 2 thetwoKeplerianmotionsandthetotalangularmomentumC(cid:126) = C(cid:126) +C(cid:126) mustlieinthesameplane, the 1 2 nodelinesoftheLaplaceplanewiththeorbitalplanesofthetwoellipsesmustcoincide. WenowdescribethetwoKeplerianmotionsinDelaunayvariables. Leta ,a bethesemimajoraxesof 1 2 theinnerandouterellipsesrespectively. TheDelaunaycoordinates (L,l,G,g,H,h),i=1,2 i i i i i i forbothellipsesarethusdefinedas:  √ √  GlLiii==µLii(cid:113)M1i−ea2ii cmainregcauunllaaarrnmaonmogmaulleyanrtummomentum  ghHii =Gicosii avlreogrnutgimciateulndcteoomoffpptoehnreiecanestncoterfnetdhienganngoudlea,rmomentum i in which e , e are the eccentricities and i ,i are the inclinations of the two ellipses respectively. We 1 2 1 2 shallwrite(L,l,G,g,H,h)todenotetheDelaunaycoordinatesforabodymovingonangeneralKeplerian elliptic orbit. From their definitions, we see that these coordinates are well-defined only when neither of 1Wehaveidentifiedso∗(3),thespaceof3×3anti-symmetricmatrices,withR3inthestandardway. 2i.e.thehorizontalplane. 4 LEIZHAO Figure1. SomeDelaunayVariables the ellipses is circular, horizontal or rectilinear. We refer to [3], [4] or [5, appendix A] for more detailed discussionsofDelaunaycoordinates. BychoosingtheLaplaceplaneasthereferenceplane,wecanexpressH ,H asfunctionsofG ,G and 1 2 1 2 C :=(cid:107)C(cid:126)(cid:107)as: C2+G2−G2 C2+G2−G2 H = 1 2,H = 2 1. 1 2C 2 2C SinceC(cid:126) isvertical, wehavedH ∧dh +dH ∧dh = dC ∧dh . Wecanthenreducethesystembythe 1 1 2 2 1 SO(2)-symmetryaroundthedirectionofC(cid:126). Thedegreesoffreedomofthesystemisthenreducedfrom6 to4. ThisreductionprocedurewasfirstcarriedoutbyJacobiandisthuscalled“Jacobi’seliminationofthe nodes”. Denote by Π(cid:48) the subspace of Π one gets by posingC (cid:44) 0 and fix the direction ofC(cid:126) to the vertical vert direction(0,0,1).ThespaceΠ(cid:48) isaninvariantsymplecticsubmanifoldofΠ.Jacobi’seliminationofnode vert impliesthatthecoordinates (L ,l ,G ,g ,L ,l ,G ,g ,C,h ) 1 1 1 1 2 2 2 2 1 areDarbouxcoordinatesonadenseopenset3ofΠ(cid:48) . vert 3.2. Reductionofthethree-bodyproblemintheDepritvariables. Letusconsideraninvariantsubman- ifoldΠ(cid:48)ofΠbyproperlyfixingthedirectionofC(cid:126) (cid:44)0.ThedenseopensetofΠwithnon-vanishingC(cid:126)isthus theunionofsuchinvariantsymplecticmanifolds, andanytwoofthemcanbetransformedbetweenthem byarotation. InΠ(cid:48),theSO(3)-symmetryofthesystemF isrestrictedtoa(Hamiltonian)SO(2)-symmetry, and is easier to handle. As the standard symplectic form on Π is invariant under the SO(3)-action, Π(cid:48) is alsoaninvariantsymplecticsubmanifoldofΠ. WecannowchooserestrictthedynamicalstudyofFtoΠ(cid:48). Following[1],thisrestrictionprocedureiscalledpartialreduction. ForC(cid:126)non-vertical,thereductionprocedureisconvenientlyunderstoodintheDepritcoordinates4 (L ,l ,L ,l ,G ,g¯ ,G ,g¯ ,Φ ,ϕ ,Φ ,ϕ ), 1 1 2 2 1 1 2 2 1 1 2 2 3onwhichallthevariablesarewell-defined,i.e.theellipsetheydescribearenon-degenerate,non-circular,non-horizontal. 4Theterminologyfollowsfrom[6]. PARTIALREDUCTIONANDDELAUNAY/DEPRITVARIABLES 5 Figure2. SomeDepritVariables defined as follows: Let ν be the intersection line of the two orbital planes5, ν be the intersection of the L T Laplaceplanewiththehorizontalreferenceplane. Weorientν bytheascendingnodeoftheinnerellipse, L andchooseanyorientationforν . Let T • g¯ ,g¯ denotetheanglesfromν 6tothepericentres; 1 2 L • ϕ denotestheanglefromν toν ; 1 T L • ϕ denotestheanglefromthefirstcoordinateaxisinthereferenceplanetoν ; 2 T • Φ =C =(cid:107)C(cid:126)(cid:107),Φ =C =theverticalcomponentofC(cid:126). 1 2 z Proposition3.1. (Chierchia-Pinzari[6])DepritcoordinatesareDarbouxcoordinates. Intheopendense subsetofΠwherealltheDepritvariablesarewell-defined,wehave: ω =dL ∧dl +dG ∧dg¯ +dL ∧dl +dG ∧dg¯ +dΦ ∧dφ +dΦ ∧dφ . 0 1 1 1 1 2 2 2 2 1 1 2 2 The variables (L ,l ,L ,l ,G ,g¯ ,G ,g¯ ,Φ ,ϕ ) form a set of Darboux coordinates on a dense open 1 1 2 2 1 1 2 2 1 1 set (on which all the variables are well-defined) of Π(cid:48), any of the subspaces of Π one gets by fixing the direction of C(cid:126) non-vertical. In these coordinates, the Hamiltonian can be written in closed form in the “planar variables” (L ,l ,G ,g¯ ,L ,l ,G ,g¯ ) andC. We can then fixC and reduce the system from the 1 1 1 1 2 2 2 2 SO(2)-symmetryaroundthedirectionofC(cid:126)tocompletethereductionprocedure. In[7], Depritestablishedasetofcoordinatescloselyrelatedtothesetofcoordinatespresentedabove. The actual form of our Deprit coordinates was independently discovered and first presented by Chierchia andPinzariin[6]. Notethatinbothofthesereferences,DepritcoordinatesarebuiltforthegeneralN-body problem, with the aim to generalize Jacobi’s elimination of nodes, or to conveniently reduce the SO(3)- symmetryofthe N-bodyproblemfor N ≥ 4,whichisofsignificantimportancefortheperturbativestudy oftheN-bodyproblem(c.f.[8]). 5ThisisthecommonnodelineofthetwoplanesintheLaplaceplane. 6Aconventionalchoiceoforientationofthenodeline,isgivenbytheirascendingnodes,whichleadstooppositeorientationsof νLinthedefinitionofg¯1andg¯2. 6 LEIZHAO Remark3.1. InΠ(cid:48) ,wehaveg¯ =g ,g¯ =g andΦ =Φ .Theanglesφ ,φ arenotdefinedindividually. vert 1 1 2 2 1 2 1 2 Nevertheless,theirsumφ +φ remainswelldefined.OnecanthenrecoverJacobi’seliminationofthenode 1 2 fromtheDepritvariablesbyalimitprocedure,see[6]fordetails. 3.3. DepritcoordinatesforN-bodyproblem. LetuspresenttheDepritcoordinatesinN-bodyproblem, or for N −1 Keplerian ellipses7, by induction on N: Divide the N −1 Keplerian ellipses into a group of N − 2 Keplerian ellipses and another group consists of only one Keplerian ellipse (whose elements are written with an subscript N − 1). Denote the total angular momentum of the N − 2 Keplerian ellipses in the first group by C(cid:126) and the total angular momentum of the whole system by C(cid:126). Then the Deprit N−2 coordinatesforthegroupofN−2Keplerianellipses,excepttheconjugatepairofC anditsconjugate N−2,z angle,togetherwithL ,l ,G ,g¯ ,C,φ ,C ,φ aretheDepritcoordinatesfortheN−1Keplerian N−2 N−2 N−2 N−2 1 z 2 ellipses,inwhichC istheprojectionofC(cid:126) toC(cid:126),andg¯ istheargumentoftheperihelionfromthe N−2,z N−2 N−2 nodelineofthisKeplerianellipsewiththeLaplaceplane,i.e. theplaneorthogonaltoC(cid:126). Moreexplicitand precisedefinitionsofthesevariablescanbefoundin[6]. As mentioned above, the symplecticity of these set of coordinates is proven by Chierchia and Pinzari [6](Depritprovedthesymplecticityofhissetofcoordinatesin[7]). Weshallgiveanalternativeproofin Section6. 4. AConceptualViewofthePartialReductionProcedure Nowletusmakeanremarkonthegeneralizationoftheideaofpartialreduction[1]forarbitrarycompact connectedgroupG ,whichsimultaneouslygivesaconceptualwayofunderstandingthisprocedure. r LetG beacompactconnectedLiegroupwhichactsinaHamiltonianwayonaconnectedsymplectic r manifold (M,ω) and let µ : M → g∗ be the associated moment map, in which g∗ is the dual of the Lie algebragofG . SinceG iscompact,thereexistaninvariantinnerproductong,whichpermitstoidentify r r gwithitsdualg∗. ForanyfixedCartansubalgebrah ⊂ g,denotebyTˇ thecorrespondingCartansubgroup (i.e. a maximaltorus)inG . Let uschoosea (positive)Weyl chambert∗ in h∗ ⊂ g∗. It turnsoutthat the r + pre-imageµ−1(t∗)isa“symplecticcross-section”(inthewordsof[9])oftheG actionon(M,ω): + r Theorem 4.1. (Guillemin-Sternberg [9]) The pre-image µ−1(t∗) of the positive Weyl chamber is a Tˇ- + invariant symplectic submanifold of (M,ω). The restriction of theG action on µ−1(t∗) is a Hamiltonian r + torusactionofTˇ. ForanyclosedsubgroupTˇ(cid:48) ⊂ Tˇ,thesubsetofµ−1(t∗)containingpointsfixedbyTˇ(cid:48) isa + Tˇ symplecticsubmanifoldofµ−1(W+). SinceG is a compact connected Lie group, the Cartan subalgebras in g∗ are conjugate to each other. r AsµinterwinestheG actionon(M,ω)andthecoadjointactionofG ong∗,anytwoofthese“symplectic r r cross-sections”istheimageundertheG -actionofeachother. r Remark4.1. Theoriginalstatementalsorequires M tobecompact. Nevertheless,inorderonlytogetthe citedstatements,thecompactnessisnotnecessary. Inthethree-bodyorN-bodyproblemsinR3,thegroupSO(3)actsinaHamiltonianwayonthe(translation- reduced) phase space, whose moment map is just the angular momentum vector C(cid:126) ∈ so(3) (cid:27) R3. Any Cartansubalgebraisthevectorspaceofinfinitesimalgeneratorsofrotationswithfixedrotationaxis,which is a 1-dimensional vector subspace (homeomorphic to R) in R3. A positive Weyl chamber is therefore a 7ThereaderunderstandsthatmorepreciselythismeansKeplerianellipticmotions. PARTIALREDUCTIONANDDELAUNAY/DEPRITVARIABLES 7 connectedcomponentofthis1-dimensionalvectorsubspaceminustheorigin,formedbyinfinitesimalgen- eratorsgeneratingrotationswiththesameorientation. Thepre-imageofthepositiveWeylchamberisthe submanifoldonegetsbyfixingthedirectionofC(cid:126),whichiseasilyseentobeinvariantundertheHamiltonian flowoftheN-bodyproblem. Theorem4.1showsthatthissubmanifoldissymplecticandtherestrictionof theSO(3)-actiontothissubmanifoldistheSO(2)-actionaroundthefixeddirectionofC(cid:126). Thisisexactlythe “partialreduction”proceduredescribedin[1]. Moreover, as already mentioned in Subsection 3.2, Jacobi explicitly establishes a set of action-angle coordinatesonΠ(cid:48) fromtheDelaunaycoordinates. Westateatheoreminthenextsection,whichallowsus toeasilydeduceDeprit’scoordinatesfromthosefoundbyJacobi,constructtheDepritcoordinatesformore bodies,andprovethesymplecticityofthesecoordinates. 5. SymplecticComplementoftheSymplecticCross-Sections Theorem5.1. SupposethatacompactconnectedLiegroupG actsinaHamiltonianwayonthesymplectic r manifold (M,ω) with moment map µ : M → g∗. Let us fix a Weyl chamber t∗ in g∗. Suppose that ∀x ∈ + µ−1(t∗), µinducesanisomorphismbetweentheω-orthogonalspaceofµ−1(t∗)at xandthetangentspace + + atµ(x)ofacoadjointorbiting∗,andsupposethat,uptomultiplicationofaconstant,thereexistsonlyone G -invariant symplectic form on the coadjoint orbit, so that there exists only one symplectic form on the r normal space of µ−1(t∗) at x which can be extended to aG -invariant form along aG -orbit. Then there + r r existsaconstantD ∈R,suchthat µ0 ω=ω +µ∗ω˜ , 0 µ0 where ω is the restriction of ω on µ−1(t∗) and D µ∗ω˜ are seen as extended to twoG -invariant two- 0 + µ0 µ0 r forms,andforx ∈ M,ω istheKirillov-Konstantsymplecticformonthecoadjointorbitpassingthrough 0 µ0 µ =µ(x )andD dependsonlyonthecoadjointorbitofµ . 0 0 µ0 0 Proof. We fix a point x ∈ µ−1(t∗). For any two vectors v ,v ∈ T M, we may decompose them as 0 + 1 2 x0 v = u +w ,v = u +w , such that u ,u ∈ T µ−1(t∗) and w ,w ∈ (T t∗)⊥, in which (T t∗)⊥ is the 1 1 1 2 2 2 1 2 x0 + 1 2 x0 + x0 + orthogonalspaceofT µ−1(t∗)withrespecttotheformω. Thestatementisequivalentto x + ∀w ,w ∈(T t∗)⊥,ω (w ,w )= Dµ∗ω˜ (w ,w ). 1 2 x0 + x0 1 2 µ0 1 2 Restrictedtotheω-orthogonalspaceofµ−1(t∗)at x ,bothformsωandµ∗ω˜ arebilinear,anti-symmetric, + 0 µ0 non-degenerate, and they can be extended to two G -invariant forms. Therefore after being extended in r suchways,theyagreeuptoaG -invariantfactorD(andhenceDisconstantonthepre-imageofacoadjoint r orbit). Hencewehave ω=ω +Dµ∗(ω˜ ), 0 µ0 inwhichD= D dependsonlyonthecoadjointorbitofµ . µ0 0 (cid:3) 6. SymplecticityofDelaunayandDepritcoordinates Inthespatialcases,thesymmetricgroupisalwaysSO(3). TheSO(3)coadjointorbitsweshallconsider are homeomorphic to S2, which admit only one SO(3)-invariant symplectic form up to multiplication of constants. TheSO(3)-momentmapisthetotalangularmomentumC(cid:126)andtheformµ∗ω˜ isseentobeequal µ0 todC ∧dh (bypassingtosymplecticcylindralcoordinates). Ourmaintaskinthissectionistodetermine z C(cid:126) thefactorDindifferentcontexts. 8 LEIZHAO 6.1. Delaunaycoordinates. 6.1.1. PlanarDelaunaycoordinates. WefirstanalyzetheplanarDelaunaycoordinates(L,l,G,g). Let K be the energy of the planar Kepler problem. When K is negative, we know that all its orbits are closed. Consider two commuting SO(2)-actions on the phase space, one by shifting the phase along the elliptic orbits,andanotheronebyrotatingtheorbitsintheplane. Claim6.1. GisthemomentmapassociatedtotheHamiltonianactionofthegroupSO(2)actingbysimul- taneousrotationsinpositionsandinmomentaonthephasespace. AnSO(2)-orbitisparametrizedbythe argumentoftheperiheliong(whenthisangleiswell-defined). Proof. ThisisastandardcalculationofaSO(2)-momentmap. (cid:3) Claim6.2. ListhemomentmapassociatedtotheHamiltonianactionofthegroupSO(2)onthephasespace byphaseshiftsontheKeplerianellipticorbits. AnSO(2)-orbitsisparametrizedbythemeananomalyl. Proof. ThesecondandthirdlawsofKeplerimpliesthatthemomentmapassociatedtothisSO(2)-action isindependentoftheeccentricityoftheellipticorbit. Itisthusenoughtocalculatethismomentmapalong orbitswithzeroeccentricity,i.e.,thecircularorbits,alongwhichtheSO(2)-actionisjustsimultaneousrota- tionsinpositionsandmomenta,andthemomentmapiseasilyseentobethe(circular)angularmomentum L. (cid:3) Itisdirecttoverifythat(L,l,G,g)arefunctionallyindependent. Moreover,similarlyasin[10],interms ofthePoissonbrackets,wehave • {L,l}={G,g}=1,bydefinitionofthemomentmap. • {L,g} = {G,l} = 0, by definition of the moment map and the commutativity of the two SO(2)- actions. • {L,G}=0,bythefactthatGisafirstintegralfortheKeplerproblem. • {l,g}=0,asaresultofthefirstthreePoissonbrackets,thesymplecticformmayonlybewrittenin theformdL∧dl+dG∧dg+f dl∧dg.byclosednessofthis2-form, f = f(l,g)dependsonlyonl,g. astheSO(2)-actionoftheanglelisHamiltonian,the1-formdL+fdgmustbeexact,whichimplies f = f(g)onlydependsong. Let f(g)dg = dF(g),thenL+F(g)isamomentmapassociatedtol. AstwoSO(2)-momentmapsmayonlydifferbyaconstant,F(g)mustbeaconstant,whichinturn impliesthat f =0. 6.1.2. Spatial Delaunay coordinates. Based on the symplecticity of the planar Delaunay coordinates, a directapplicationofTheorem5.1confirmsthesymplecticityofthespatialDelaunaycoordinatesuptoan indeterminedfactor D = D(G). Todetermine D,wegothroughalimitingprocedurebylettingtheorbital plane tends to horizontal. Some care must be taken since the angles g and h in the Delaunay variables are not well-defined for horizontal ellipses. We thus restrict to the submanifold for which all the spatial Delaunayvariablesarewelldefinedandsuchthatg = 0, i.e. thedirectionoftheperihelionoftheellipse agreeswiththedirectionoftheascendingnode. The2-formdL∧dl+dG∧dg+DdH∧dhisthusrestricted todL∧dl+DdH∧dhonthissubmanifold. Thankstotherestriction,theanglehisnowexactlytheangle between the direction of the perihelion and the first coordinate axis, which remain well-defined when the orbitalplaneishorizontal. ThustheformdL∧dl+DdH∧dhcanbeextendedcontinuously(andactually smoothly)tohorizontalorbitalplaneaftertherestriction. However,forhorizontalellipse,wehaveH =G, and the angle h agrees with the planar argument of the perihelion (the angle g in the planar Delaunay coordinates). BycomparingwiththeplanarDelaunaycoordinates,wefindD=1. PARTIALREDUCTIONANDDELAUNAY/DEPRITVARIABLES 9 6.2. Depritcoordinatesforthethree-bodyproblem. ItisnothardtodeducefromJacobi’seliminationof nodethatasetofDarbouxcoordinatesonthepartiallyreducedspaceis(L ,l ,G ,g¯ ,L ,l ,G ,g¯ ,C,φ ). 1 1 1 1 2 2 2 2 1 Therefore, from Theorem 5.1 we know that the symplectic form on a dense open set of Π takes the form dL ∧dl +dG ∧dg¯ +dL ∧dl +dG ∧dg¯ +dC∧dφ +DdC ∧dφ . Nowletusdeterminethefactor 1 1 1 1 2 2 2 2 1 z 2 D= D(C)bysomelimitingprocedures. WeseethatthetermDdC ∧dφ doesnotdependspecificallyon z 2 L . 1 To determine the constant D, we restrict the form dL ∧dl +dG ∧dg¯ +dL ∧dl +dG ∧dg¯ + 1 1 1 1 2 2 2 2 dC ∧ dφ + DdC ∧ dφ to the symplectic subspace of identical Keplerian motions, in which we have 1 z 2 C C L = L ,l = l ,G =G = ,g¯ +φ = g¯ +φ = g = g ,H = H = z,φ = h = h . Bycomparing 1 2 1 2 1 2 1 1 2 2 1 2 1 2 2 1 2 2 2 withtherestrictionofthe(decoupled)Delaunaycoordinates,wefindD=1. 6.3. DepritcoordinatesfortheN-Bodyproblem. TheproofisinductivebytakingtheDepritcoordinates forthefirstN−2KeplerianellipsesandDelaunaycoordinatesforthelastKeplerianellipse.Apartialreduc- tionprocedureagaingivessymplecticcoordinatesontheinvariantsubspaceobtainedbyfixingthedirection ofthetotalangularmomentum. Theorem5.1thusconfirmsthedesiredresultexceptforthedetermination of the constant D. We can now take the first 2 Keplerian elliptic motions as identical (therefore we can consider one (fictitious) Keplerian elliptic motion with twice of the circular angular momentum and the angularmomentuminsteadofthem)andfinishtheargumentbycomparingtheresultingcoordinateswith theDepritcoordinatesforthefirstN−2Keplerianellipses. WefindD=1. Acknowledgements. ManythankstoAlainChencinerandJacquesFéjozfortheirconstanthelpduringyears, andtothem,togetherwithHenkBroer,forcommentsandsuggestions. References [1] F.Malige,P.Robutel,andJ.Laskar.Partialreductioninthen-bodyplanetaryproblemusingtheangularmomentumintegral. 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[10] JFéjoz.On“Arnold’stheorem”incelestialmechanics-asummarywithanappendixonthePoincarécoordinates.Discreteand ContinuousDynamicalSystems,33:3555–3565,2013. JohannBernoulliInstituteforMathematicsandComputerScience,UniversityofGroningen,TheNetherlands E-mailaddress:[email protected]