Symplectic integrators in sub-Riemannian geometry: the Martinet case

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Symplectic integrators in sub-Riemannian geometry: the Martinet case

Monique Chyba, Ernst Hairer, Gilles Vilmart

To cite this version:

Monique Chyba, Ernst Hairer, Gilles Vilmart. Symplectic integrators in sub-Riemannian geometry:

the Martinet case. [Research Report] RR-6017, INRIA. 2006, pp.10. �inria-00113372v2�

inria-00113372, version 2 - 13 Nov 2006

a p p o r t

d e r e c h e r c h e

ISSN0249-6399ISRNINRIA/RR--6017--FR+ENG

Thème NUM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Symplectic integrators in sub-Riemannian geometry:

the Martinet case

Monique Chyba — Ernst Hairer — Gilles Vilmart

N° 6017

Novembre 2006

Unité de recherche INRIA Rennes

IRISA, Campus universitaire de Beaulieu, 35042 Rennes Cedex (France)

Téléphone : +33 2 99 84 71 00 — Télécopie : +33 2 99 84 71 71

the Martinet ase

Monique Chyba

∗

,Ernst Hairer

†

, GillesVilmart

† ‡

ThèmeNUMSystèmesnumériques

ProjetIpso

Rapport dereherhe n°6017Novembre200610pages

Abstrat: Weompare theperformanes ofsympletiand non-sympletiintegratorsfortheomputation

of normalgeodesisand onjugate points in sub-Riemanniangeometry at the example of the Martinetase.

For this ase study we onsider rst the atmetri, and then a one parameter perturbation leading to non

integrablegeodesis. Fromouromputationswededuethatneartheabnormaldiretionsasympletimethod

ismuhmoreeientforthisoptimalontrolproblem. Theexplanationreliesonthetheoryofbakwarderror

analysis ingeometrinumerialintegration.

Key-words: sub-Riemanniangeometry,Martinet,abnormalgeodesi,sympletiintegrator,bakwarderror

analysis.

Thisworkwas partially supportedbythe Fonds NationalSuisse,projet No. 200020-109158 and bythe NationalSiene

FoundationgrantDMS-030641.

∗

UniversityofHawaii,DepartmentofMathematis,2565MCarthytheMall,Honolulu96822,Hawaii

†

UniversitédeGenève,Setiondemathématiques,2-4rueduLièvre,CP64,1211Genève4,Switzerland

‡

INRIARennes,projetIPSO,CampusdeBeaulieu,35042RennesCedex,Frane

le problème de Martinet

Résumé: Onomparelesperformanesd'intégrateurssympletiquesetnonsympletiquespourlealulde

géodésiquesnormales et depointsonjuguésdans unexemplesous-riemannien, le problèmede Martinet. On

étudie leproblèmed'abordave unemétrique plate,puisave une perturbation àunparamètreonduisantà

des géodésiquesnon intégrables. Deetteétude,ondéduit queprohedesdiretionsanormales, uneméthode

sympletiqueestbienpluseaepoureproblèmedeontrleoptimal. L'expliationreposesurlathéoriede

l'analyse rétrogradeenintégrationnumériquegéométrique.

Mots-lés : géométriesous-riemannienne,Martinet,geodésiqueanormale,intégrateursympletique,analyse

rétrograde

Introdution

This paper is a follow-up to a series of artiles that were published in the past deades, see [1, 2℄ and the

referenes therein. There, the authors provide an analyti study of the singularity of the sub-Riemannian

sphere in the Martinet ase. They omplementtheir analysis with numerial omputations to represent the

geodesisandthesphere,andtoloatetheonjugatepoints. Theintegratorusedforthese omputationsisan

expliit Runge-Kuttamethod oforder 5(4). Thegoalof thepresentpaperis to omparetheperformanesof

asympletiintegratorversusanon-sympletioneforthis optimalontrol problem. Tobemorepreise, let

(U,∆, g)^beasub-Riemannianstruture whereU ^isanopenneighborhoodofR^3,∆ ^adistribution ofonstant rank 2and g ^{a Riemannian metri. When} ∆ ^{is aontat}distribution, there are noabnormalgeodesis,and anon-sympletiintegratorisaseient asasympletione. However,whenthe distributionis takenasthe

kerneloftheMartinetone-form,weshowthatasympletiintegratorismuhmoreeientfortheomputation

ofthenormalgeodesisandtheironjugatepointsneartheabnormaldiretions.

Bothproblems,theMartinetataseandanonintegrableperturbation,areintroduedinSet.1together

with the orresponding dierentialequations. Numerial experiments with an expliit RungeKutta method

and withthesympletiStörmerVerletmethod arepresentedin Set.2andillustratedwith gures. Closeto

abnormalgeodesis,theresultsarequitespetaular. Forarelativelylargestepsize,thesympletiintegrator

providesasolution withthe orretqualitative behaviorand asatisfatory auray, whilefor thesame step

size the non-sympleti integrator gives a ompletely wrong numerial solution with an inorret behavior,

partiularly for the non integrable ase. The explanation relies on the theory of bakward error analysis

(Set.3). Itis relatedto thegeometristruture oftheproblemanditssolutions.

1 A Martinet type sub-Riemannian struture

In this setion, we briey reall someresults of [1℄ for a sub-Riemannianstruture (U,∆, g)^{. Here,} U ^{is an}

openneighborhoodoftheorigininR^{3 withoordinates}q= (x, y, z)^,andg^{isaRiemannianmetriforwhiha}

graduatednormalform,atorder0^,isg= (1 +αy)dx²+ (1 +βx+γy)dy^{2. The}distribution∆^{is generatedby}

thetwovetoreldsF1= _∂x^∂ +^y₂²_∂z^{∂ and}F2=_∂y^{∂ whih orrespondto}∆ = kerω ^whereω =dz−^y2²dx^{is the}

Martinetanonialone-form. Tothisdistributionweassoiatetheaneontrolsystem

˙

q=u1(t)F1(q) +u2(t)F2(q),

where u1(t), u2(t)^{aremeasurableboundedfuntionswhihatasontrols.}

Weonsidertwoases,theMartinet ataseg =dx²+dy^{2, anintegrablesituation,andaoneparameter}

perturbationg=dx²+ (1 +βx)²dy^{2forwhih thesetofgeodesisisnon}integrable.

1.1 Geodesis

It follows from the Pontryagin maximum priniple, see [1, 2℄, that the normal geodesis orresponding to

g=dx²+ (1 +βx)²dy^{2aresolutionsofan}Hamiltoniansystem

˙ q=∂H

∂p (q, p), p˙=−∂H

∂q (q, p), ⁽¹⁾

where q= (x, y, z)^isthestate,p= (px, py, pz)^{istheadjointstate,andthe}Hamiltonianis

H(q, p) =1 2

px+pz

y² 2

²

+ p²_y (1 +βx)²

.

Inotherwords,thenormalgeodesisaresolutionsofthefollowingequations:

˙

x = px+pz

y² 2

˙

y = py

(1 +βx)²

˙

z =

px+pzy² 2

y² 2

˙

px = β p²_y (1 +βx)³

˙

py = −

px+pzy² 2

pzy

˙

pz = 0.

(2)

Notiethatthevariablesz^andpz^{donotinuenetheotherequations(exeptviatheinitialvalue}pz(0)^),so

thatweareatuallyonfrontedwithaHamiltoniansystemindimensionfour. FortheMartinetatase(β= 0^),

theinterestingdynamistakesplaein thetwo-dimensionalspaeofoordinates(y, py)^{. The}Hamiltonianis

H(y, py) = p²_y 2 +1

2

px+pz

y² 2

² ,

where px^andpz ^{havetobeonsideredasonstants. Thisisaone-degreeoffreedommehanialsystemwitha}

quartipotential. Forpx<0< pz^,theHamiltonianH(y, py)^{hastwoloalminimaat}(y=±p

−2px/pz, py= 0)^,

whihorrespondtostationarypointsofthevetoreld. Inthisase,theorigin(y= 0, py= 0)^{isasaddlepoint.}

Whereasnormalgeodesisorrespondtoosillatingmotion,itisshownin[1,2℄thattheabnormalgeodesis

are thelines z=z0 ^{ontainedin the plane}y = 0^{. Fortheonsidered metris,theabnormalgeodesisanbe}

obtained asprojetions of normal geodesis, we say that they are not stritly abnormal. In [2℄, the authors

introdue a geometri framework to analyze the singularities of the sphere in the abnormal diretion when

β 6= 0^{. Seealso[3,4℄forapreisedesriptionoftheroleoftheabnormalgeodesisin}sub-Riemanniangeometry in the generalnon-integrablease,i.e., when theabnormalgeodesisanbestrit. Themajorresultof these

papersistheproofthatthesub-Riemanniansphereis notsub-analytibeauseoftheabnormalgeodesis.

Interestingphenomenaarisewhen thenormalgeodesisareloseto theseparatriesonnetingthe saddle

point. Therefore,weshallonsiderinSet.2theomputationofnormalgeodesiswithy(0) = 0^andpy(0)>0

but small.

1.2 Conjugate points

FortheHamiltoniansystem(1) weonsidertheexponentialmapping

exp_q0,t:p07−→q(t, q0, p0)

whih, for xed q0 ∈ R^{3, is the projetion}q(t, q0, p0) ^{onto the state spae of the solution of} (¹) ^{starting at} t= 0^from(q0, p0)^{. Followingthedenition in[1℄wesaythatthepoint}q1 ^isonjugateto q0 ^alongq(t)^ifthere

exists (p0, t1)^, t1 >0^{, suh that} q(t) = exp_q0,t(p0) ^with q1 = exp_q0,t1(p0)^{, and themapping}exp_q0,t1 ^isnotan

immersion at p0^{. We say that} q1 ^{is the rst onjugate point if} t1 ^{is minimal. Firstonjugate points play a}

majorrolewhenstudyingoptimalontrolproblemssineitisawellknownresultthatageodesiisnotoptimal

beyondtherstonjugatepoint.

For the numerial omputation of the rst onjugatepoint, weompute the solution of the Hamiltonian

system(1)togetherwithitsvariationalequation,

˙

y=J⁻¹∇H(y), Ψ =˙ J⁻¹∇²H(y)Ψ. ⁽³⁾

Here,y= (q, p)^andJ ^{istheanonialmatrixfor}Hamiltoniansystems. Itanbeshownthat forRunge-Kutta methods, thederivativeofthenumerialsolutionwithrespetto theinitial value,Ψn=∂yn/∂y0^,istheresult

ofthesamenumerialintegratorappliedtotheaugmentedsystem(3),see[5,LemmaVI.4.1℄. Here,thematrix

Ψ =

∂q/∂q0 ∂q/∂p0

∂p/∂q0 ∂p/∂p0

hasdimension6×6^{. Theonjugatepointsareobtainedwhen}∂q/∂p0 ^{beomessingular,i.e.,}det(∂q/∂p0) = 0^.

2 Comparison of sympleti and non-sympleti integrators

For the numerial integration of the Hamiltonian system (1), where we rewrite

∂H

∂q(q, p) = Hq(q, p) ^and

∂H

∂p(q, p) =Hp(q, p)^{,weonsidertwo}integratorsofthesameorder2^:

• ^anon-sympleti,expliitRungeKuttadisretization,denoted rk2 ^{(see[5,Set.II.1.1℄),} qn+1/2 = qn+h

2Hp(qn, pn)

qn+1 = qn+hHp(qn+1/2, pn+1/2)

pn+1/2 = pn−h

2Hq(qn, pn)

pn+1 = pn−hHq(qn+1/2, pn+1/2)

(4)

• ^thesympletiStörmerVerletsheme(seee.g. [5,Set.VI.3℄),

pn+1/2 = pn−h

2Hq(qn, pn+1/2) qn+1 = qn+h

2

Hp(qn, pn+1/2) +Hp(qn+1, pn+1/2)

(5)

pn+1 = pn+1/2−h

2Hq(qn+1, pn+1/2)

where qn = (xn, yn, zn)^andpn= (px,n, py,n, pz,n)^{. Here,}qn≈q(nh), pn≈p(nh)^andh^{is thestepsize.}

Fortheomputationoftheonjugatepoints,weapplythenumerialmethodstothevariationalequation(3).

Notie that only the partial derivatives with respet to p0 ^{haveto be omputed. Conjugate points are then}

deteted whendet(∂qn/∂p0)^{hangessign. We} approximate themby linearinterpolationwhih introduesan errorofsizeO(h²)^{. Thisisomparabletotheaurayofthehosen}integratorswhiharebothofseondorder.

Remark 2.1 The StörmerVerlet sheme (5) is impliit in general. A few xed point iterations yield the

numerialsolutionwiththedesiredauray. NotiehoweverthatthemethodbeomesexpliitintheMartinet

at aseβ = 0^{. One simplyhas to omputethe omponentsin a suitableorder, forinstane} px,n+1^, pz,n+1^,

py,n+1/2^,yn+1^,xn+1^,zn+1^,py,n+1^.

2.1 Martinet at ase

Weonsiderrsttheataseβ = 0^intheHamiltoniansystem(2). Asinitialvalueswehoose(f. [1℄)

x(0) =y(0) =z(0) = 0, px(0) = cosθ0, py(0) = sinθ0, pz(0) = 10, ^where θ0=π−10⁻³, ⁽⁶⁾

−6 −4 −2

−.5 .5

−6 −4 −2

−.5 .5

−6 −4 −2

−.5 .5

−6 −4 −2

−.5 .5

−6 −4 −2

−.5 .5

−6 −4 −2

−.5 .5 x

y

non-sympletimethod,h= 0.1

x

y

StörmerVerlet,h= 0.1

x

y

non-sympletimethod,h= 0.05

x

y

StörmerVerlet,h= 0.05

x

y

non-sympletimethod,h= 0.02

x

y

StörmerVerlet,h= 0.02

Figure1: Trajetoriesin the(x, y)^{-planefortheatase}β= 0^.

sothatwestartlosetoanabnormalgeodesis,andweintegratethesystemovertheinterval[0,9]^.

Figure1displaystheprojetionontothe(x, y)^{-planeofthenumerialsolutionobtainedwithdierentstep}

sizesh^bythetwointegrators. Theinitialvalueisattheorigin,andthenalstateisindiatedbyatriangle. The irlesrepresenttherstonjugatepointdetetedalongthenumerialsolution,whilethestarsgivetheposition

oftherstonjugatepointontheexatsolutionoftheproblem. Thereisanenormousdierenebetweenthetwo

numerial integrators. The sympleti (StörmerVerlet) method (5) providesa qualitatively orret solution

already with a large step size h = 0.1^{, and it gives an exellent} approximation for step sizes smaller than

h = 0.05^{. On the other hand,the} non-sympleti, expliitRungeKutta method (4) givesompletely wrong results, andstepsizes smallerthan10^{−3 areneededto provide anaeptablesolution. An} explanationof the dierentbehaviorof thetwointegratorswillbegivenin Set.3below.

−.5 .5

−1 1

.5

−1 1

y py

non-sympletimethod h= 0.05

y py

StörmerVerlet h= 0.05

zoom×100

Figure2: Phaseportraitsinthe(y, py)^{-planefortheatase}β = 0^.

As notied in Set.1, the normal geodesis in the at ase are determined by a one-degree of freedom

Hamiltonian systemin thevariables y ^and py^{. Wethereforeshowin Figure2the projetionontothe} (y, py)^-

spae of the solutions previouslyomputed with step size h = 0.05^{. The exat solution starts at} (0,sinθ0)

above the saddle point, turns around the positive stationary point, rossesthe py^-axisat (0,−sinθ0)^{, turns}

around the negative stationary point, and then ontinues periodially. The numerial approximation by the

non-sympleti method overs morethan one and a half periods, whereas the StörmerVerlet and the exat

solutionoverlessthanoneperiodforthetimeinterval[0,9]^{. Sinetheonjugatepointisnotverysensiblewith}

respetto perturbationsin the initialvalue forpy^{, the}(y, py)^{oordinates ofthe onjugatepoint obtainedby}

thenon-sympletiintegratorareratheraurate,buttheorrespondingintegrationtimeisompletelywrong.

Table 1 lists the onjugate time obtained with the two integrators using various step sizes. There is a

signiantdierene between thetwo methods. We ansee that with theStörmerVerlet method (5) a step

sizeoforderh= 10^{−2providesasolutionwith}4^{orretdigits. Astepsizea}100^{timessmallerisneededtoget}

thesamepreisionwiththenon-sympletimethod.

2.2 Non integrable perturbation

Forournextnumerialexperimentwehoosetheperturbationparameterβ=−10^{−4inthedierentialequation}

(2). We onsider thesameinitial valuesand the sameintegrationintervalasin Set.2.1. The exatsolution

isnolongerperiodiand, duetothefat thatβ ^{ishosennegative,itsprojetionontothe}(y, py)^-spaeslowly

spiralsinwardsaroundthepositivestationarypoint(seerightpitureinFigure4).

Figures3and4andTable1displaythenumerialresultsobtainedbythetwointegratorsforthedierential

equation (2) with β = −10^{−4. The} interpretation of the symbols (triangles, irles, and stars) is the same as before. Theexellentbehaviorof thesympleti integratoris evenmorespetaular thanin theatase,

and thepitures obtainedfor the StörmerVerlet method agree extremely well with the exatsolution. The

Table1: Aurayfortherstonjugatetime.

Martinetatase

h ^{rk2 Verlet}

10⁻¹ 4.504945 8.504716 10⁻² 6.748262 8.416622 10⁻³ 8.360340 8.416412 10⁻⁴ 8.416349 8.416410

exatsolution: t1≈8.416409

Nonintegrablesituation

h ^{rk2 Verlet}

10⁻¹ 4.511294 4.883832 10⁻² 7.380322 4.877056 10⁻³ 4.877183 4.876998 10⁻⁴ 4.876997 4.876997

exatsolution: t1≈4.876997

−6 −4 −2

−.5 .5

−6 −4 −2

−.5 .5

−6 −4 −2

−.5 .5

−6 −4 −2

−.5 .5 x

y

non-sympletimethod,h= 0.05

x

y

non-sympletimethod,h= 0.02

x

y

non-sympletimethod,h= 0.009

x

y

StörmerVerlet,h= 0.1

Figure3: Trajetoriesinthe(x, y)^{-planeforthenonintegrablease}β =−10^−4.

−.5 .5

−1 1

.5

−1 1

y py

non-sympletimethod h= 0.05

y py

StörmerVerlet h= 0.05

zoom×10

Figure 4: Phaseportraitsin the(y, py)^{-planeforthenonintegrablease}β=−10^−4.

non-sympleti method givesqualitativelywrong solutionsforstep sizes largerthanh= 0.01^{. Inthe} (y, py)^-

spaeitalternativelyspiralsaroundtherightandleftstationarypointswhereastheexatsolutionspiralsonly

aroundthepositivestationarypoint. InontrasttotheMartinetatase,theonjugatepointobtainedbythe

non-sympletimethodisherewrongalsointhe(y, py)^-spae.

2.3 An asymptoti formula on the rst onjugate time in the Martinet at ase

Now that we have shown the eieny of sympletiintegrators, we an make more preise the asymptoti

behaviourstudiedin [1℄. Fortheinitialvaluesof(6)andβ = 0^{,onsidertheratio} R= t1√pz

3K(k),

where t1 ^{istherstonjugatetimeforthenormalgeodesi,and}K(k)^{isanelliptiintegraloftherstkind,} K(k) =

Z π/2 0

p 1

1−k²sin²udu, k= sin(θ0/2).

Bystudyinganalytisolutionsforthenormalgeodesis,itisprovedin[1℄thatthisratiosatisestheinequality

2/3≤R≤1^{. It followsfromaresalingoftheequations(2)that}R^is independentofpz^.

InFigure5,werepresentthevaluesof1−R ^{asafuntion of}ε=π−θ0^{, forvariousinitialvalues}θ0^{. The}

numerialresultsindiate thattheratioR^dependsonθ0^,andR−→1^{− slowlyfor}θ0−→π^−.