Partial Reduction and Delaunay/Deprit Variables

0.18 MB·

by  Lei Zhao

PARTIAL REDUCTION AND DELAUNAY/DEPRIT VARIABLES
LEIZHAO
Abstract. Basedonaconceptuallinkbetweenthepartialreductionprocedureofthereductionoftherotational
symmetryoftheN-bodyproblemwiththesymplecticcross-sectiontheoremofGuillemin-Sternberg,wepresent
4 alternativeproofsofthesymplecticityofDelaunayandDepritcoordinatesincelestialmechanics.
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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
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6 6.3. DepritcoordinatesfortheN-Bodyproblem 9
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References 9
6
.
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4 1. Introduction
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:
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.
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JohannBernoulliInstituteforMathematicsandComputerScience,UniversityofGroningen,TheNetherlands
E-mailaddress:[email protected]