cartan meets chaplygin kurt m. ehlers and jair koiller
TRANSCRIPT
THEORETICAL AND APPLIED MECHANICSVolume 46 (2019) Issue 1, 15โ46 DOI: https://doi.org/10.2298/TAM190116006E
CARTAN MEETS CHAPLYGIN
Kurt M. Ehlers and Jair Koiller
Abstract. In a note at the 1928 International Congress of MathematiciansCartan outlined how his โmethod of equivalenceโ can provide the invariants
of nonholonomic systems on a manifold ๐ with kinetic lagrangians [29]. Car-tan indicated which changes of the metric outside the constraint distribution
๐ธ โ ๐๐ preserve the nonholonomic connection ๐ท๐๐ = Proj๐ธ โ๐๐ , ๐,๐ โ ๐ธ,
where โ๐๐ is the Levi-Civita connection on ๐ and Proj๐ธ is the orthogonalprojection over ๐ธ. Here we discuss this equivalence problem of nonholonomic
connections for Chaplygin systems [30,31,62]. We also discuss an example-a
mathematical gem!-found by Oliva and Terra [76]. It implies that there ismore freedom (thus more opportunities) using a weaker equivalence, just to
preserve the straightest paths: ๐ท๐๐ = 0. However, finding examples that
are weakly but not strongly equivalent leads to an over-determined system ofequations indicating that such systems should be rare. We show that the two
notions coincide in the following cases: i) Rank two distributions. This implies
for instance that in Cartanโs example of a sphere rolling on a plane withoutslipping or twisting, a (2,3,5) distribution, the two notions of equivalence coin-
cide; ii) For a rank 3 or higher distribution, the corank of D in D+[D,D] mustbe at least 3 in order to find examples where the two notions of equivalence
do not coincide. This rules out the possibility of finding examples on (3,5)
distributions such as Chaplyginโs marble sphere. Therefore the beautiful (3,6)example by Oliva and Terra is minimal.
1. Introduction: dโAlembert, Hertz, Cartan, Chaplygin
A constrained mechanical system consists of a Lagrangian ๐ฟโถ๐๐ โ R and adistribution ๐ธ โ ๐๐ of rank ๐ < ๐ = dim๐. DโAlembertโs principle gives
(1.1)๐
๐๐ก
๐๐ฟ
๐๐โ ๐๐ฟ
๐๐= ๐, with ๐ = ฮ๐ถ(๐), such that ๐ธ โถ ๐ถ(๐) โ ๐ = 0.
Equations (1.1) are given in column form, one entry for each ๐๐ , with ๐ถ an ๐ร๐matrix, and ฮ = (๐1, . . . , ๐๐) a row ๐-vector, ๐ = ๐ โ ๐ is the number of contraints.When cast in Hamiltonian form, with ๐ = ๐๐ฟโ๐๐, ๐ป = ๐๐ โ๐ฟ (see eg. [11]) then
๐๐โ = (๐, ๏ฟฝ๏ฟฝ) = (๐๐ปโ๐๐ โ ๐๐ปโ๐๐ + ๐) =๐๐ป + (0, ๐), ๐ โ ๐๐ปโ๐๐ = 0.
2010 Mathematics Subject Classification: 37J60; 53D15; 70F25; 70G45; 70H45.Key words and phrases: nonholonomic systems, reduction, hamiltonization, Cartan
equivalence.
15
16 EHLERS AND KOILLER
The multipliers ๐1,โฏ, ๐๐ can be eliminated by brute force, more or less as follows.Matrix ๐ถ has an invertible ๐ ร ๐ block. One uses the corresponding ๐ equations tosolve for all the ๐โs. Splitting ๐ถ(๐) โ ๐ = 0 in two blocks, one could solve for ๐ of the๐โs in terms of ๐ = ๐ โ ๐ others, the coefficients being functions of all the ๐โs. Also,by differentiating, one eliminates ๐ of the ๐โs.
Eventually one gets a system of first order ODEs for the ๐-coordinates ๐ andfor ๐ of the velocities, totalizing a system of ๐ + ๐ ordinary differential equations.This procedure is very awkward. This is why geometric approaches are so helpful.
This paper is also intended as a review for a non-expert, familiar with basicdifferential geometry, at the level of the first chapters of [52]. For a crash course onGeometric Mechanics, with emphasis on control, of interest to mechanical engineersworking in Robotics, see [70]. A geometric mechanics approach to nonholonomicsystems was anticipated by A. Baksa [8], see also [66]. For a short survey onnonholonomic systems, see [10]. For a comprehensive introduction, see [3]. Ourgoal is to discuss the equivalence problem of Cartanโs nonholonomic connection๐ท๐๐ under changes of metric. We pursue a recent observation by Terra andOliva [76] that there is more freedom if one is happy by keeping the same straightestpaths ๐ท๐๐ = 0. We show nonetheless that in many cases the notions coincide.
1.1. From dโAlembert to Hertz. Jean le Rond dโAlembert asserted in hisTraite de Dynamique that, due to the constraints imposed on the system, a forceappears in the right hand side of EulerโLagrange equations. This force producesno resultant work in the system1. It was noticed by H. Hertz in his Prinzipiender mechanik ( [51], 1894) that dโAlembertโs principle yields different equationsthan those obtained by the calculus of variations with constraints (the so-calledโvakonomicโ paths), except when the distribution is integrable (holonomic); in thiscase both lead to geodesics of the induced metric on every leaf ๐ฟ of the foliation.Inspired by Gaussโ least constraint principle, Hertz proposed the principle of leastcurvature: the path followed by an inertial mechanical system is as straight aspossible, given that the velocity vector must satisfy the constraints. This gives thecorrect equations for nonholonomic systems (hereafter shortened as nh)2.
1.2. Appell, Whittaker, Maggi, Hamel. We will skip, for brevity, theaccomplishments of their generation at the beginning of the 20th century. In dangerof being superficial, we take the nerve to say that their contribution was mainlyon the practical side: โquasivelocitiesโ. Let ๐ โR๐ be local coordinates on ๐๐, andconsider a local moving frame, defined by an ๐ ร ๐ invertible matrix ๐ (๐),
๐๐ฝ = ๐
๐๐๐ฝ=
๐
โ๐=1
๐๐ผ๐ฝ๐
๐๐๐ผ, โ ๏ฟฝ๏ฟฝ๐ฝ๐๐ฝ =โ ๐๐ผ
๐
๐๐๐, ๏ฟฝ๏ฟฝ = (๐ (๐))โ1๐.
1Part II, Chapter 1 (1743, 1758). For an appraisal of dโAlembertโs work, see [33]. Thevalidation of dโAlembertโs principle has been experimentally established (see eg. [54,55,68], as
the limit of appropriate damping forces.2An inertial system is governed exclusively by its kinetic energy. Poincare appreciated Hertzโs
view that Mechanics could be founded on the concept of constraints rather than forces: โil nous
force a reflechir, a nous affranchir de vieilles associations dโideesโ [78]. We prefer to call thesolutions of (1.1) straightest paths. The terminology nonholonomic geodesic would be more ap-
propriate for the vakonomic setting.
CARTAN MEETS CHAPLYGIN 17
In books on mechanical engineering the keyword quasicoordinates refers to nonex-isting entities ๐ such that
๐๐
๐๐๐ฝ=โ
๐ผ
๐๐
๐๐๐ผ
๐๐๐ผ๐๐๐ฝ
=โ๐ผ
๐๐
๐๐๐ผ๐๐ผ๐ฝ = ๐๐ฝ(๐).
Using such quasicoordinates, the multipliers can be eliminated automatically, andthe number of ODEs would be just ๐ + ๐ . Moreover, they realized that whensymmetries were present, a proper choice of quasicoordinates would yield reducedequations, which in some cases could be explicitly integrated in closed form.
1.3. Nh connections: Cartan and contemporaries. In the late 1920โsCartan, Synge, Schouten, Vranceanu and Wagner provided a differential geometricdescription of nh systems. For historical information on that period we refer to [35].They considered only purely inertial Lagrangians (no potential energy) and we willfollow suit here.
Let ๐ denote the Riemannian metric on ๐ and ๐ธ โ ๐๐ a distribution.
Definition 1.1. The (partial) connection on ๐ธ โ ๐๐, encoding dโAlembertโsprinciple, is the combination of two operations,
i) computing the Levi-Civita connection โ๐๐ , of ๐ for ๐,๐ โ ๐ธii) projecting โ๐๐ orthogonally back to ๐ธ.
This is analogous to the induced Levi-Civita connection on a submanifold of aRiemannian manifold. The nh connection is therefore
(1.2) ๐ท๐๐ = Proj๐ธ โ๐๐
and the straightest path equation is given by
๐ท๐๐ = 0, ๐ โ ๐ธ.
Affine connections on ๐๐ are classical objects. According to [35], the study ofaffine connections on subbundles of ๐๐ (partial connections) started with Schouten,and was further developed by Vranceanu, Synge and Wagner. Without this knowl-edge, nonholonomic connections were discussed in [63] in which one of the authorswas a participant. See [45] for a recent work in which the notion is used.
1.4. Chaplygin and Hamiltonization. The idea of Hamiltonization of nhsystems goes back to Chaplygin himself. In the first decade of the 20th century hedeveloped the method of reducing multipliers [31], applied to systems with abeliansymmetries. In some cases, under a time reparametrization depending on the basevariables ๐ โ ๐, the reduced system can be cast in Hamiltonian form, and in severalexamples the reduced system is integrable (more information in section 6).
The jurisprudence was established by Chaplyginโs sphere. It is a dynamicallyunbalanced sphere, but with the center of mass at the geometric center. Theconfiguration space is ๐ = ๐๐(3) ร R2. The sphere rolls without slipping on ahorizontal plane, twisting about the vertical axis allowed. The problem can beviewed as an abelian (๐บ = R2) Chaplygin nh system [30,31]. See also [36]. Theno slip contraints define a distribution ๐ธ of rank 3 that is strongly nonholonomic(i.e. 1-step, 3-5). It was shown to be Hamiltonizable when reduced to ๐ โ๐2 [13].
18 EHLERS AND KOILLER
1.5. Organization of the paper. In sections 2 and 3 we discuss E. Cartanโsnote at the 1928 ICM. Cartanโs method to obtain the dynamics, in the case of Liegroups, is equivalent to ArnoldโEulerโs [2]. Section 3 gives the initial ๐บ-structurefor the equivalence method. Section 4 builds on recent work by Terra and Oliva [76].They consider a weaker notion of equivalence, the preservation of the acceleration๐ท๐๐ instead of the nonholonomic connection ๐ท๐๐ . We present cases where theweak and strong notions coincide. The special case of Chaplygin systems is outlinedin section 5, and section 6 presents final remarks and directions for further research.
We finish the introduction with a historical note3.
1.6. Chaplygin and Cartan. They were born 3 days apart, Cartan on April9, and Chaplygin April 5, 1869. What more in common? Both came from poorfamilies and were discovered and protected by their elementary school teachers.They endured hardships and tragedies in WW2.
Chaplygin was very much involved in the revolutionary movement in the 1910โs.He protested persecutions in the 1930โs, defending Luzin fiercely. Still, he wasawarded in most prestigious medals due to his achievements, specially during theWar: Hero of Socialist Labour, Order of Lenin (twice), and Order of the RedBanner.
Cartanโs son Louis, a member of the Resistance, was murdered by the Nazis.Henri Cartan, the oldest, was a leader in the creation of the European Math Unionafter the war and was influential in bringing Germany back.
Cartan can be described as a pure, pure mathematician, and Chaplygin avery applied one4, but this may be misleading; they shared a common interestin gravitation and cosmology. Chaplygin gas is still a basic ingredient for darkmatter theory. As Chern and Chevalley [32] wrote about Cartan, and Lyusternikon Chaplygin, both were excellent teachers and were all for Women in Mathematics.
Lyusternik tells the following anecdote in [71]: โI was examined by SergeiAlekseevich on particle mechanics [entering the Moscow State University]. Myanswer was in no way remarkable. I was surprised several months later when takingthe [course on] Mechanics of a System that he remembered my name. โThereโsnothing surprising about thatโ, I was told, โChaplygin has a phenomenal memoryโ.When he was Rector of the Higher Courses for Women, he knew all the studentsby nameโ.
3 Information from โThe Early Years of the Moscow Mathematical Schoolโ (Lyusternik,[71]) and the websites http://www-history.mcs.st-andrews.ac.uk/Biographies/Chaplygin.html,
http://www-history.mcs.st-andrews.ac.uk/Biographies/Cartan.html4Chaplyginโs career was mostly in aeronautical engineering. We quote from the MacTutor
site: The Central Aerohydrodynamic Institute, or TsAGI, was founded in 1918 and Chaplyginhelped organize the Institute from that time. In fact from the time that TsAGI was founded, hedevoted almost all his energies to the project. On the death of Zhukovsky in 1921, Chaplyginbecame Chairman of the Board, a position he held until 1930. He was Executive Director of the
Institute from 1928 to 1931, then he became head of the scientific work of the Institute (...) In theautumn of 1941, the Central Aerohydrodynamic Institute was evacuated from Moscow to Kazan
and Novosibirsk. Chaplygin took charge of the Novosibirsk branch of the Institute and rapidlyorganised the building of a wind tunnel and research laboratories. However, the hard work anddifficult circumstances told on his health and he died from a brain haemorrhage in October 1942.โ
CARTAN MEETS CHAPLYGIN 19
1.7. Hilbert: โMathematics Knows No Racesโ [82]. There was a strongpolitical aspect at the ICM in Bologna, held from September 3 to September 101938. Hilbert intended to deliver a speech (but it did not happen, not clear why)with that timely title. But Hilbertโs dream is becoming true (and hopefully willremain so). At ICM 2014 and 2018 Fields medals were awarded to mathematiciansborn in Third World countries, one of them Maryam Mirzakhani. She died so early.
2. Cartanโs equivalence for nh connections
It is not a surprise that Cartan used adapted moving frames5 in his approachfor nh mechanics in [29]. He considered only kinematical Lagrangians ๐ฟ = ๐ on aconfiguration space ๐, where ๐ is the kinetic energy of a Riemannian metric andthe constraints defined by a distribution ๐ธ โ ๐๐ with rank ๐ < ๐.
Cartan advocated his equivalence method to obtain invariants characterizingthe nonholonomic connection (1.2). See [50,75,77] for modern description of theequivalence method with applications.
Cartanโs note was revisited in [63]. As pointed out by Terra and Oliva [76],there is an inaccuracy. It was asserted that Cartanโs equivalence criterion (seesection 3.2) provided necessary and sufficient conditions for a change of metric in๐ to keep the straightest paths. Cartanโs criterion gives sufficient conditions, butthey are not always necessary, and this brings more opportunities. We will shownonetheless in section 4 that the exceptions are rare.
The difference ๐ด(๐,๐ ) = โ๐๐ โโ๐๐ between two affine connections in ๐ is a2-tensor (very easily seen to be bilinear). Clearly โ and โ have the same geodesics ifand only if ๐ด is skew-symmetric. This implies (obviously) that โ๐๐ = โ๐๐. Onecan make the same observation about partial connections, the difference between๐ท๐๐ and ๏ฟฝ๏ฟฝ๐๐ being a โpartialโ 2-tensor. Terra and Oliva propose a broadernotion of equivalence: the preservation of the straightest path equation ๐ท๐๐ = 0.
We start with the stronger notion of equivalence, preservation of ๐ท๐๐ . Wewill consider the weaker notion ๐ท๐๐ in section 4. We will show that for rank 2distributions, preservation of ๐ท๐๐ is equivalent to preservation of ๐ท๐๐. We alsoshow that the notions coincide when dim(๐ธ + (๐ธ,๐ธโ) โ dim๐ธ < 3.
Definition 2.1. ๐ธ โ ๐๐ is bracket generating if any local frame ๐๐ for ๐ธtogether with its iterated Lie brackets (๐๐, ๐๐โ, (๐๐, (๐๐ , ๐๐โโ, . . . spans ๐๐.
The only case Cartan discussed in some detail were the 1-step distributions,those satisfying ๐ธ+(๐ธ,๐ธโ = ๐๐. We call those distributions strongly nonholonomic(Snh). In this case, Cartan proved that, preserving the connection ๐ท๐๐ , one can
5For a thorough account of moving frames in mechanics, see [73], where there is an interestinghistorical observation: at the same time as Cartan (in France), Schouten (in Holland) was also
interested in moving frames for nh mechanics [80]. In fact, nonholonomic mechanics was trendy
at that time! Vranceanu, then a student of Levi-Civita in Rome, also made a contribution tonh geometry at the 1928 ICM [87]. In Ireland, Synge was also studying nh systems [84]. A
few years later, Wagner studied the geometry of nonholonomic systems in his doctoral thesisunder V. F. Kagan at Lomonosov Moscow State University (see [35] for an account of Wagnercurvature tensor).
20 EHLERS AND KOILLER
modify the metric in the orthogonal complement ๐น of ๐ธ in ๐๐, as long as ๐นremains orthogonal to ๐ธ. This seems uneventful in view of dโAlembertโs principle.
The simplest example of a strongly nonholonomic distribution is a contactdistribution (2-3) in R3. Their invariants are given in [37]. Chaplyginโs ball (3-5)is also Snh. For a ๐ โฉพ 2-step distribution, Cartan observed something unexpected.Denote by ๐ธ(1) = ๐ธ + (๐ธ,๐ธโ. As in the Snh case, one can change the metric
in the orthogonal complement ๐น of ๐ธ inside ๐ธ(1). ๐น remains orthogonal to ๐ธ.Surprisingly, there is total flexibility in changing the metric outside ๐ธ(1), as if thenh geometry becomes โsaturatedโ at the level of ๐ธ(1). The proof is in section 3.
About the invariants, Cartan indicated that for distributions with ๐ โฉพ 2-steps,implementing the equivalence method would become cumbersome. We have theimpression that at this point Cartan lost interest.
โlโinteret geometrique sevanouirait rapidement a mesure queles cas envisages deviendraient plus compliques.โ
In fact, as far as we know, he did not pursue this study later. We took thenerve to do the equivalence problem for nh connections on 2-3 [37], the 2-step 2-3-5distribution [38,65], and the Engel 2-3-4 in [64].
Cartan was right in his warning: it can be difficult to interpret geometrically theinvariants. At any rate, in spite of Cartanโs somewhat pessimistic statements, withthe help of computer algebra further cases may be done nowadays. Cartanโs methodof equivalence is algorithmic and (almost) unsupervised. We wish to advertise thisstudy for the special case of Chaplygin systems, both in the strong and weak senses.Some thoughts are presented in section 5.
2.1. Derived ideal. The bracket generating condition can be refined in termsof a filtration of ideals of differential forms. To do this let โ be the ideal of alldifferential forms annihilating ๐ธ. It is locally generated by ๐ โ ๐ one-forms.
Definition 2.2. The set of forms โ โฒ = {๐ โ ๐ผ โ ๐๐ โก 0 mod (๐ผ)} is called thederived ideal of โ.
Set โ = โ0 and define โ๐ inductively by โ๐ = (โ๐โ1)โฒ. The condition that๐ธ is bracket generating can be redefined in terms of the sequence of inclusions๐ผ0 โ ๐ผ1 โ ๐ผ2 โ โฏ terminating with the 0 ideal. In the vectorfields side, we denote๐ธโฒ = ๐ธ + (๐ธ,๐ธโ and define inductively ๐ธ(0) = ๐ธ, ๐ธ๐ = (๐ธ๐โ1)โฒ. Then โ(๐) is the
annihilator of ๐ธ(๐). This comes from the following lemma, a simple consequence ofCartanโs magic formula:
Lemma 2.1. โ(1) is the annihilator of ๐ธ(1) = ๐ธ + (๐ธ,๐ธโ.
Proof. Just stare at
๐๐(๐,๐ ) = ยฑ๐ ๐(๐ ) ยฑ๐๐(๐ ) ยฑ ๐(๐,๐ โwhere the correct signs do not matter. ๏ฟฝ
Remark 2.1. The filtration of ideals โ(๐) forms the annihilators of ๐ธ(๐) where๐ธ๐+1 = ๐ธ๐ + (๐ธ๐ ,๐ธ๐โ which form the bigโ growth vector. If, instead, we take๐ธ = ๐ธ1 = ๐ป = ๐ป1 and ๐ป๐+1 = ๐ธ + (๐ธ,๐ป๐โ we get the โsmallโ growth vector. In
CARTAN MEETS CHAPLYGIN 21
general ๐ธ๐ contains ๐ป๐. In the examples discussed here the big and the smallsequences are the same. As we will see shortly, nonholonomic geometry (meaning๐ท๐๐ ) sees only the first derived ideal.
2.2. The Levi-Civita connection. Once a frame in ๐ is chosen, it definesa riemannian metric ๐ in ๐ for which it is an orthonormal basis. Let (๐๐ผ) be anorthonormal frame and (๐๐ผ) its dual coframe satisfying ๐๐ผ(๐๐ฝ) = ๐ฟ๐ผ๐ฝ . The metricwrites in terms of the coframe as
๐ = (๐1)2 + โ โ โ + (๐๐)2.Recall that the Levi-Civita connection of the metric is given by (see eg. [52])
(2.1) โ๐๐๐ฝ = ๐๐ผ๐ฝ(๐)๐๐ผ ,where the connection forms ๐๐ผ๐ฝ satisfy (uniquely) the structure equations
๐๐๐ผ + ๐๐ผ๐ฝ โง ๐๐ฝ = 0, ๐๐ผ๐ฝ = โ๐๐ฝ๐ผ .
In the traditional Christoffel symbols notation one writes
(2.2) โ๐๐พ๐๐ฝ = ฮ๐ผ๐พ๐ฝ๐๐ผ , ฮ๐ผ
๐พ๐ฝ = ๐๐ผ๐ฝ(๐๐พ), ฮ๐ผ๐พ๐ฝ = โฮ๐ฝ
๐พ๐ผ ,
Proposition 2.1 (Cartanโs moving frames approach for geodesics).
โ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ = 0 โ ๏ฟฝ๏ฟฝ๐พ = โ๐ฃ๐ผ๐ฃ๐ฝฮ๐พ๐ผ๐ฝ(๐), ๏ฟฝ๏ฟฝ = ๐ฃ๐ฟ๐๐ฟ, ฮ๐พ
๐ผ๐ฝ = ๐๐พ๐ฝ(๐๐ผ).
2.3. Euler-Arnold equations: ฮ๐ผ๐พ๐ฝ
versus ๐๐พ๐ผ๐ฝ
. In control problems, it isvery useful to prepare a table of the expansions of the commutators (๐๐ฝ , ๐๐พโ in itsown moving frame {๐๐ฟ}
(๐๐ฝ , ๐๐พโ = ๐๐ฟ๐ฝ๐พ(๐)๐๐ฟ.Thus it is important to relate the Christoffel symbols ฮ๐ผ
๐พ๐ฝ = ๐๐ผ๐ฝ(๐๐พ) with thevalues of ๐๐พ๐ผ๐ฝ(= ๐๐พ(๐๐ผ , ๐๐ฝโ).
Proposition 2.2. We have the relations
(2.3) ๐๐ผ๐พ(๐๐ฝ) โ ๐๐ผ๐ฝ(๐๐พ) = ๐๐ผ(๐๐ฝ , ๐๐พโ
(2.4) 2ฮ๐ฝ๐ผ๐พ = 2๐๐ฝ๐พ(๐๐ผ) = ๐๐ผ(๐๐ฝ , ๐๐พโ โ ๐๐ฝ(๐๐พ , ๐๐ผโ โ ๐๐พ(๐๐ผ , ๐๐ฝโ.
Proof. ๐๐๐ผ(๐๐ฝ , ๐๐พ) = โ๐๐ผ(๐๐ฝ , ๐๐พโ = โโ๐ฟ ๐๐ผ๐ฟ โง ๐๐ฟ)(๐๐ฝ , ๐๐พ) = ๐๐ผ๐ฝ(๐๐พ) โ๐๐ผ๐พ(๐๐ฝ). Hence (2.3) follows. These relations can be inverted with the usualtrick. If we write the cyclic permutations of (2.3), we get three equations relatingsix quantities on the right hand side, but in reality they are just three objects due tothe anti-symmetry of ๐๐ผ๐ฝ(๐๐พ) in the lower indices ๐ผ, ๐ฝ . Solving this linear systemyields (2.4). ๏ฟฝ
Proposition 2.3 (EulerโArnold equations). The geodesic equations can alsobe written as
(2.5) โ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ = 0 โ ๏ฟฝ๏ฟฝ๐พ = โ๐ฃ๐ผ๐ฃ๐ฝ((๐๐ผ๐พ๐ฝ + ๐๐ฝ๐พ๐ผ)โ2โ = โ๐ฃ๐ผ๐ฃ๐ฝ๐๐ผ๐พ๐ฝ(๐), ๏ฟฝ๏ฟฝ = ๐ฃ๐ฟ๐๐ฟ.
Proof. Immediate from the previous Proposition. One of the three terms in(2.4) does not contribute due to the skew symmetry with respect to ๐ผ, ๐ฝ . ๏ฟฝ
22 EHLERS AND KOILLER
Example 2.1. For a Lie group, with say, a left invariant frame and left invariantmetric the ๐๐ผ๐พ๐ฝ are constants. The expert reader will recognize that (2.5) becomesthe ArnoldโEuler equations in the Lie Algebra, in the case of an orthonormal frame.For a general non-orthonormal frame (see eg. [62])
(2.6) ฮฉ๐ผ = ๐๐ผ๐พ๐๐ฝ๐ฟ๐พ๐๐ฝ๐ฮฉ๐ฮฉ๐ฟ,
where we changed to the usual notation ๐ฃ๐ผ = ฮฉ๐ผ and we define a left invariantmetric by 2๐ = (๐บฮฉ,ฮฉ), with ๐บ = (๐บ๐ผ๐ฝ) is a symmetric positive definite matrix.The vector ฮฉ represents a Lie algebra element expanded in the basis ๐๐พ . Thepresence of entries ๐๐ผ๐พ of the cometric matrix ๐บโ1 is morally a โnonlocalโ effect.
2.4. Straightest paths (a la Cartan or a la EulerโArnold). We will usethe following conventions and un-conventions: capital roman letters ๐ผ, ๐ฝ, . . . runfrom 1 to ๐. Lower case roman ๐, ๐, . . . run from 1 to ๐ (The dimension of theconstraint distribution). Greek characters ๐ผ,๐ฝ, ๐พ, from ๐ + 1 to ๐. Summation overrepeated indices is assumed unless otherwise stated. We apologize if we are some-times careless in using upper or lower indices, and sometimes we put ฮฃ to indicatea summation more explicitly. We will further subdivide the Greek characters intolower and upper case to account for forms belonging to the derived ideal โ โฒ ofโ = span{๐๐ผ}.
Cartan proposed studying nonholonomic systems using adapted coframes
๐ = โ๐๐
๐๐ผ} , where 1 โฉฝ ๐ โฉฝ ๐ , ๐ + 1 โฉฝ ๐ผ โฉฝ ๐.
This coframe is dual to an adapted orthonormal moving frame {๐๐, ๐๐ผ}, the first ๐ vectors being tangent to ๐ธ.
Proposition 2.4. The Nh connection of Definition 1.1 is obtained simply byrunning the indices only up to ๐ :
๐ท๐๐๐๐ = ฮ๐๐๐๐๐ (1 โฉฝ ๐, ๐, ๐ โฉฝ ๐ ).
A curve ๐(๐ก) is a straightest path if ๏ฟฝ๏ฟฝ = โ๐ ๐=1 ๐ฃ๐๐๐ (๐ฃ๐ = ๐๐(๏ฟฝ๏ฟฝ)quasivelocities)
with (see (2.2) and (2.5))(2.7) ๏ฟฝ๏ฟฝ๐ = โ๐ฃ๐๐ฃ๐ฮ๐
๐๐ or equivalently, ๏ฟฝ๏ฟฝ๐ = โ๐ฃ๐๐ฃ๐๐๐๐๐ .
In the sequel we will sometimes use the ฮ๐๐๐ = ๐๐๐(๐๐) (that we call the Cartan
format) and sometimes the ๐๐๐๐ = ๐๐(๐๐, ๐๐โ (that we call the ArnoldโEuler format),whichever is more convenient in the situation being considered.
Corollary 2.1. Given a left invariant distribution ๐ธ, and a left invariantmetric (it may be given by a matrix ๐บ), the nonholonomic equations are obtainedby the orthogonal projection with respect to this metric, of the Euler equations(2.5), (2.6) for non-orthonormal frames for ฮฉ on the distribution ๐ธ.
Nota bene. We stress that although ฮ๐๐๐ โ ๐๐๐๐ , the difference is skew symmetric
in the indices ๐, ๐. For the straightest path equations the difference will disappearafter the summations over ๐, ๐ of the corresponding โป๐ฃ๐๐ฃ๐ , so we may take the ๐๐๐๐ .
But for the nh connection we need to stick to the ฮ๐๐๐ .
CARTAN MEETS CHAPLYGIN 23
2.5. Example: maximally symmetric 2-3-5 distribution. Let
๐1 =๐
๐๐ฅ1โ ๐ฅ2
๐
๐๐ฅ3โ ๐ฅ1๐ฅ2
๐
๐๐ฅ4โ ๐ฅ2
2
๐
๐๐ฅ5
๐2 =๐
๐๐ฅ2+ ๐ฅ1
๐
๐๐ฅ3+ ๐ฅ2
1
๐
๐๐ฅ4+ ๐ฅ1๐ฅ2
๐
๐๐ฅ5
๐3 = (๐1, ๐2โ = 2๐
๐๐ฅ3+ 3๐ฅ1
๐
๐๐ฅ4+ 3๐ฅ2
๐
๐๐ฅ5
๐4 = (๐1, ๐3โ = 3๐
๐๐ฅ4,
๐5 = (๐2, ๐3โ = 3๐
๐๐ฅ5
which form a basis for a 5-dimensional nilpotent Lie algebra onR5. The exponentialmap gives a diffeomorphism between the nilpotent algebra and the group. Usingthe BakerโCampbellโHausdorf formula one determines the cubic polynomial grouplaw which is:
โโโโโโโ
๐ฅ1
๐ฅ2
๐ฅ3
๐ฅ4
๐ฅ5
โโโโโโโ
โ
โโโโโโโ
๐ฆ1๐ฆ2๐ฆ3๐ฆ4๐ฆ5
โโโโโโโ
=
โโโโโโโ
๐ฅ1 + ๐ฆ1๐ฅ2 + ๐ฆ2
๐ฅ3 + ๐ฆ3 + 12(๐ฅ1๐ฆ2 โ ๐ฅ2๐ฆ1)
๐ฅ4 + ๐ฆ4 + 12(๐ฅ1๐ฆ3 โ ๐ฅ3๐ฆ1) + 1
12(๐ฅ2๐ฆ
21 โ ๐ฅ1๐ฆ1๐ฆ2 + ๐ฅ2
1๐ฆ2 โ ๐ฅ1๐ฅ2๐ฆ1)๐ฅ5 + ๐ฆ5 + 1
2(๐ฅ2๐ฆ3 โ ๐ฅ3๐ฆ2) + 1
12(๐ฅ2๐ฆ1๐ฆ2 โ ๐ฅ1๐ฆ
22 + ๐ฅ1๐ฅ2๐ฆ2 โ ๐ฅ2
2๐ฆ1)
โโโโโโโ
.
By construction the vector fields ๐๐ provide a left invariant frame on ๐บ. Thedual coframe is
โโโโโโโ
๐1
๐2
๐3
๐4
๐5
โโโโโโโ
=
โโโโโโโ
๐๐ฅ1
๐๐ฅ212(โ๐ฅ2๐๐ฅ1 + ๐ฅ1๐๐ฅ2 โ ๐๐ฅ3)
16(โ๐ฅ1๐ฅ2๐๐ฅ1 + ๐ฅ2
1๐๐ฅ2 โ 3๐ฅ1๐๐ฅ3 + 2๐๐ฅ4)16(โ๐ฅ2
2๐๐ฅ1 + ๐ฅ1๐ฅ2๐๐ฅ2 โ 3๐ฅ2๐๐ฅ3 + 2๐๐ฅ5)
โโโโโโโ
.
Proposition 2.5. Let ๐ธ = span(๐1, ๐2) be a rank two distribution on an ๐-dimensional Riemannian manifold ๐. In terms of an orthonormal coframe ๐,
๏ฟฝ๏ฟฝ1 = โ๐112๐ฃ1๐ฃ2 โ ๐2
12(๐ฃ2)2, ๏ฟฝ๏ฟฝ2 = ๐112(๐ฃ1)2 + ๐2
12๐ฃ1๐ฃ2
where
(2.8) ๐212 = โ
๐๐1 โง ๐3 โง โ โ โ โง ๐๐
๐1 โง โ โ โ โง ๐๐and ๐1
12 = โ๐๐2 โง ๐3 โง โ โ โ โง ๐๐
๐1 โง โ โ โ โง ๐๐.
Proof. Since ๐ provides a basis for the cotangent space, we can express eachcomponent of the Levi-Civita connection one-form ๐๐๐ as ๐
1๐๐๐
1+โ โ โ +๐๐๐๐๐
๐. Because
๐ = 2 we only need ๐12 and since the path ๐ is tangent to ๐ธ we only need ๐112๐
1 and๐212๐
2. Using the structure equations we compute ๐๐1โง๐3โงโ โ โ โง๐๐ = โ๐112๐
1โงโ โ โ โง๐๐
and ๐๐2 โง ๐3 โ โ โ โง ๐๐ = ๐212๐
1 โง โ โ โ โง ๐๐. ๏ฟฝ
[This is easily modified for rank ๐ distributions with ๐ > 2. One obtains a linearsystem of equations for the required coefficients of the Levi-Civita connection.]
24 EHLERS AND KOILLER
We take ๐ = (๐1)2 + โ โ โ + (๐5)2 making the given frame orthonormal. Since๐๐1 = ๐๐2 = 0, the previous proposition implies that the nonholonomic geodesicequations are ๏ฟฝ๏ฟฝ1 = ๏ฟฝ๏ฟฝ2 = 0. Hence
(๏ฟฝ๏ฟฝ1, ๏ฟฝ๏ฟฝ2, ๏ฟฝ๏ฟฝ3, ๏ฟฝ๏ฟฝ4, ๏ฟฝ๏ฟฝ5) = ๐ด๐1 +๐ต๐2
which are equivalent to the system of equations
โโโโโโโ
๏ฟฝ๏ฟฝ1
๏ฟฝ๏ฟฝ2
๏ฟฝ๏ฟฝ3
๏ฟฝ๏ฟฝ4
๏ฟฝ๏ฟฝ5
โโโโโโโ
=
โโโโโโโ
๐ด๐ต
๐ฅ1๏ฟฝ๏ฟฝ2 โ ๐ฅ2๏ฟฝ๏ฟฝ1
๐ฅ21๏ฟฝ๏ฟฝ2 โ ๐ฅ1๐ฅ2๏ฟฝ๏ฟฝ1
โ๐ฅ22๏ฟฝ๏ฟฝ1 + ๐ฅ1๐ฅ2๏ฟฝ๏ฟฝ2
โโโโโโโ
whose solutions give the nonhonolomic geodesics in terms of polynomials of up tothird degree (for ๐ฅ4, ๐ฅ5).
Remark 2.2. Cartan introduced in his thesis [26], and developed in two veryfamous papers [27,28], the simplest exceptional complex Lie algebra ๐ข2. It turnsout that ๐ธ is the maximally symmetric 2-3-5 distribution and that its Lie algebraof symmetries is precisely the 14-dimensional, non-compact real form of ๐ข2.
Cartanโs work has been discussed in the 1990โs by Bryant and Hsu [22,23] andmore recently in Agrachev, [1], Bor and Montgomey [12], and Baez [4,5]. Two wayshave been found to realize โconcretelyโ a Lie group for ๐ข2. One is via octonionsand the other as the symmetries of the rolling distribution, without slip or twist, oftwo spheres in the ratio 1:3. It seems that Cartan never explicitly discussed rolling,but it is implicit in his work. Bryant pointed out that for the 1:3 (or 3:1, it doesnot matter) rolling distribution, Cartanโs tensor F (the lowest order invariant for2-plane fields in R5) vanishes, thus it is locally equivalent to โflatโ distribution fromCartanโs 1910 paper-the nilpotent 2-3-5 as we presented above.
3. G-structures in nonholonomic geometry
At the 1928 ICM Elie Cartan described the transformations of coframes ona manifold leading to the same nh connection. In modern language, a family ofcoframes together with a matrix group ๐บ of transformation is called a ๐บ-structure.It is a sub-bundle of the coframe bundle ๐โถ๐น โ(๐)โ ๐, as a ๐บ๐(๐) principal bundle[50,75]. Both for subriemannian (SR) or nonholonomic (NH) geometry, the storybegins with the change of coframes
(3.1) ( ๏ฟฝ๏ฟฝ๐
๏ฟฝ๏ฟฝ๐ผ) = (๐ถ ๐ต0 ๐ด
) (๐๐
๐๐ผ)
with๐ถ โ ๐(๐ ), ๐ด โ ๐บ๐(๐ โ ๐ ), ๐ต โ๐(๐ , ๐ โ ๐ ).
3.1. โMaster packagesโ. We now derive the conditions for strong and weakequivalence. Dually to (3.1), the frames change via
(๐๐, ๐๐ผ) = (๐๐, ๐๐ผ)(๐ถ ๐ต0 ๐ด
) .
CARTAN MEETS CHAPLYGIN 25
We can take ๐ถ = identity without loss of generality. They correspond to changingthe metric in ๐ but keeping it unaltered inside ๐ธ. In the case of SR geometry,there are no restrictions in these matrices, this is the initial G-structure. For nhsystems things are more subtle. Proposition 2.4 implies
Proposition 3.1. i) Strong equivalence: the nh connection is preservedunder the change of metric if and only if for all 1 โฉฝ ๐, ๐, ๐ โฉฝ ๐ .
ฮ๐๐๐ = ฮ๐
๐๐ .
ii) Weak equivalence: the trajectories are preserved under the change of met-ric if and only if
๐๐๐๐ = ๐๐๐๐ + skew symmetric stuff in ๐, ๐.
We thank Prof. Waldyr Oliva for calling our attention to the additional flexi-bility in the weak notion. It is due to the โnota beneโ in section 2.4.
To find the transformation rules we take ๐ถ = identity, so ๐๐ = ๐๐, 1 โฉฝ ๐ โฉฝ ๐ andre-compute the structure constants:
(๐โ, ๐๐โ = (๐โ, ๐๐โ = ๐๐โ๐๐๐ + ๐๐ผโ๐(๐ต๐๐ผ๐๐ + stuff ๐๐) = (๐๐โ๐ + ๐๐ผโ๐๐ต๐๐ผ)๐๐ + stuff ๐๐.
We get immediately that for 1 โฉฝ ๐, ๐, โ โฉฝ ๐ , and sum over ๐ + 1 โฉฝ ๐ผ โฉฝ ๐:
Proposition 3.2.
๐๐โ๐ โ ๐๐โ๐ = ๐๐โ๐ + ๐๐ผโ๐๐ต๐๐ผ
(which clearly is still skew symmetric with respect to indices โ, ๐.)
ฮ๐๐๐ โ ฮ๐
๐๐ = ฮ๐๐๐ +
1
2(๐๐ผ๐๐๐ต๐๐ผ โ ๐๐ผ๐๐๐ต๐๐ผ + ๐๐ผ๐๐๐ต๐๐ผ)
(which clearly is still skew symmetric with respect to indices ๐, ๐.)
Proposition 3.3 (Master equations). The โpackage of conditionsโ for nh con-nection equivalence is
(3.2) ๐๐ผ๐๐๐ต๐๐ผ โ ๐๐ผ๐๐๐ต๐๐ผ + ๐๐ผ๐๐๐ต๐๐ผ = 0.
For the straightest paths equivalence the conditions are
(3.3) ๐๐ผโ๐๐ต๐๐ผ + ๐๐ผโ๐๐ต๐๐ผ = 0.
For a Chaplygin system where ๐โ ๐ is a principal bundle, ๐ต and the ๐๐ผโ๐ willbe functions of the base variable.
Since the preservation of the nh connection implies that the trajectories aremaintained, (3.2) implies (3.3).
A question then arises: in what circumstances does (3.3) in turn imply (3.2),i.e. the two equivalence notions coincide?
We present situations where the notions coincide in the next section 4. Thecounter example by Terra an Oliva [76] is reviewed in section 4.6.
26 EHLERS AND KOILLER
3.2. Change of metric preserving the nh connection (Cartanโs way).Both sets of conditions as given by the above proposition 3.3 seem difficult toanalyze. The package (3.2) for nh connection equivalence looks specially diabolical.Probably, this is why Cartan chose to work with coframes structure equations. Hefocused only on the nh connection equivalence. Let us now review his approach.
In the previous section 2.1 we defined the first derived ideal โ(1). Recall thatit is formed by combinations ๐ = โ๐๐ผ๐
๐ผ (coefficients are functions in ๐) of the๐๐ผ such that ๐๐ โ โ. This means that the 2-form ๐๐, as a 2-form, can be writtenin terms of wedge products where every term contains at least one of the ๐๐ผโs; inother words, there are no terms of the form ๐๐โง๐๐ . Equivalently, ๐๐ vanishes whenapplied to a pair of vectors in ๐ธ. This is clear from Cartanโs magic formula.
We mentioned in the introduction that Cartan observed that for nh systems,there must be restrictions on ๐ต in order to preserve the nh connection. He provedthat for strongly nonholonomic (Snh) distributions, ๐ต โก 0. Thus, in that case, ๐นmust remain perpendicular to ๐ธ in the new metric, although one can change themetric at will in ๐น . Probably the reason that Cartan was only interested in thenonholonomic connection (and not the trajectories) is because he showed that thereis a unique (intrinsically defined!) special metric in ๐น . A more general result was
presented in [63]. Let ๐ = dim(โ(1)), sodim(๐ธ + (๐ธ,๐ธโ) = ๐ โ ๐.
As we mentioned before, we subdivide the ๐ โ ๐ Greek indices into ๐ capital Greekindices ฮฆ and ๐ก = ๐โ๐ โ๐ lower case Greek (those listed as the last batch of objects).In the sequel, when we say โall the Greekโ, it means both lower and upper case.
Definition 3.1. The specially adapted frames for ๐๐
(3.4) (๐๐, ๐ฮฆ, ๐๐), 1 โฉฝ ๐ โฉฝ ๐ , ๐ + 1 โฉฝ ฮฆ โฉฝ ๐ + ๐, ๐ + ๐ + 1 โฉฝ ๐ โฉฝ ๐
are those that satisfy the following requirements:
i) The ๐๐ span ๐ธ.ii) The ๐๐ generate a complement ๐น of ๐ธ in ๐ธ + (๐ธ,๐ธโ (thus (๐๐, ๐๐) generate
๐ธ + (๐ธ,๐ธโ, which is annihilated by โ(1)).iii) The ๐ฮฆ are chosen to complete the full frame for ๐๐ (we placed them in the
middle sector for convenience).
The dual coframe of the frame just constructed is denoted
(๐๐, ๐ฮฆ, ๐๐)โ , 1 โฉฝ ๐ โฉฝ ๐ , ๐ + 1 โฉฝ ฮฆ โฉฝ ๐ + ๐, ๐ + ๐ + 1 โฉฝ ๐ โฉฝ ๐.
Notice that by duality, the ๐ฮฆ โ โ(1). The last forms, in lower case Greek, arein โ but not in โ(1). The most general change of frames among specially adaptedones is of the form
(3.5) (๐๐, ๐ฮฆ, ๐๐) = (๐๐, ๐ฮฆ, ๐๐)โโโ
๐ถ ๐ต1 ๐ต2
0 ๐ด๐ 02,30 ๐ด1 ๐ด2
โโโ
where the 023 block is zero by construction, that is: the first and last batches stillgenerate ๐ธ + (๐ธ,๐ธโ. Also ๐ถ โ ๐(๐ ), ๐ด๐ โ ๐บ๐(๐), ๐ด2 โ ๐บ๐(๐ โ ๐ โ ๐).
CARTAN MEETS CHAPLYGIN 27
It is not hard to check that all possible metrics in ๐ are contemplated bythis construction: given any frame {๐๐ผ}๐ผ=1,...๐ in ๐, there is an orthogonal matrix๐ โ ๐(๐) such that the rotated frame {๐๐ผ} โ ๐ has the first ๐ vectors in ๐ธ and thefirst ๐ + ๐ vectors in ๐ธ + (๐ธ,๐ธโ.
The change of coframes writes as
(3.6)โโโ
๏ฟฝ๏ฟฝ๐
๏ฟฝ๏ฟฝฮฆ
๏ฟฝ๏ฟฝ๐ผ
โโโ =โโโ
๐ถ ๐ต1 ๐ต2
0 ๐ด๐ 02,30 ๐ด1 ๐ด2
โโโ
โโโ
๐๐
๐ฮฆ
๐๐ผ
โโโ .
Theorem 3.1. [63] In the notation of (3.5) and (3.6), make ๐ถ = identitywithout loss in generality. The condition for the nh connection to be the same forthe two metrics, determined by the corresponding coframes, is
๐ต2 = 0.
For the proof, we must compare the structure equations
(3.7) ๐๐๐ = โ๐๐๐ โง ๐๐ โ ๐๐๐ผ โง ๐๐ผ, ๐๏ฟฝ๏ฟฝ๐ = โ๏ฟฝ๏ฟฝ๐๐ โง ๏ฟฝ๏ฟฝ๐ โ ๏ฟฝ๏ฟฝ๐๐ผ โง ๏ฟฝ๏ฟฝ๐ผ.
Definition 3.2. We say that forms (on any degree) are ๐ธ-equivalent if theirdifference is in โ and denote โผ๐ธ . So anything containing one of the Greeks (the๐๐ผโs or the ๐ฮฆโs) can be dropped in terms of the equivalence relation.
Lemma 3.1. The nh connection is preserved if and only if ๐๏ฟฝ๏ฟฝ๐ โผ๐ธ ๐๐๐.
Proof. By (3.6) with ๐ถ = identity, we have ๐๐ = ๐๐. In view of (2.1) and (2.7)the connection is preserved by the change of metrics if and only if
๐๐๐ โผ๐ธ ๏ฟฝ๏ฟฝ๐๐ .
So in view of (3.7) we get ๐๏ฟฝ๏ฟฝ๐ โผ๐ธ ๐๐๐. ๏ฟฝ
Proof of Theorem 3.1. Since ๏ฟฝ๏ฟฝ๐ = ๐๐+(๐ต1)๐ฮฆ๐ฮฆ+(๐ต2)๐๐ผ๐๐ผ it follows fromthe lemma that ๐((๐ต1)๐ฮฆ๐ฮฆ + (๐ต2)๐๐ผ๐๐ผโ โผ๐ธ 0. Now,
๐((๐ต1)๐ฮฆ๐ฮฆโ = ๐(๐ต1)๐ฮฆ โง ๐ฮฆ + (๐ต1)๐ฮฆ๐๐ฮฆ โผ๐ธ 0
as ๐๐ฮฆ โผ๐ธ 0 (because by choice ๐ฮฆ โ โ(1)). It follows that
๐((๐ต2)๐๐ผ โง ๐๐ผโ = (๐ต2)๐๐ผ โง ๐๐ผ โผ๐ธ 0.
This means that the 1-form given by the combination (๐ต2)๐๐ผ๐๐๐ผ is in โ(1). Butunless all the coefficients vanish this is impossible. We exhausted the generators ofโ(1) with the upper case Greek indices. Hence ๐ต2 = 0. It was once said that whenbasic theorems are interesting they are trivial to prove. ๏ฟฝ
Summary. In subriemannian geometry one can add any combination of formsin โ to the ๐๐. In nonholonomic geometry we can still add forms in โ(1) andpreserve the connection. Thus nh geometry has some degree of โnear-sightednessโ.
28 EHLERS AND KOILLER
3.3. Initial ๐บ-structure for Nh connection equivalence. The condition๐ต2 = 0 means geometrically that the orthogonal complement ๐น of ๐ธ in ๐ธ(1) cannotbe changed. But the metric in ๐น can be altered. Any complement ๐ป of ๐ธ(1) to๐๐ can be declared orthogonal to ๐ธ(1) without affecting the connection. From adifferent perspective, look back at the Levi-Civita connection of ๐, restricted tothe latin indices
(3.8) โ๐๐๐๐ = ๐๐๐(๐๐)๐๐ + ๐ฮฆ๐(๐๐)๐ฮฆ + ๐๐ผ๐(๐๐)๐๐ผ.
In general the ๐ฮฆ๐(๐๐)๐ฮฆ โ๐ป do not vanish, and they are outside ๐ธ + (๐ธ,๐ธโ. Thisis not harmful, since the nh connections are obtained by dropping out all the Greeks๐ฮฆ and ๐๐ผ terms in (3.8). The first terms, defining the nh connections are the samein both metrics.
Theorem 3.2. The initial ๐บ-structure for the equivalence problem for Nh con-nections ๐ท๐๐ is the sub-bundle on the frame bundle consisting of coframes
๐ = (๐๐, ๐ฮฆ, ๐๐)โ , 1 โฉฝ ๐ โฉฝ ๐ , ๐ + 1 โฉฝ ฮฆ โฉฝ ๐ + ๐, ๐ + ๐ + 1 โฉฝ ๐ โฉฝ ๐
where ๐ฮฆ, ๐๐ annihilate ๐ธ and ๐ฮฆ annihilate ๐ธ1 = ๐ธ+(๐ธ,๐ธโ. The initial ๐บ-groupacting on ๐ consists of matrices of the form
โโโ
๐ถ ๐ต1 00 ๐ด๐ 00 ๐ด1 ๐ด2
โโโ
with ๐ถ โ ๐(๐ ), ๐ต1 โ๐(๐ , ๐), ๐ด๐ โ ๐บ๐(๐, ๐), ๐ด1 โ๐(๐ โ ๐ โ ๐, ๐), ๐ด2 โ ๐บ๐(๐ โ ๐ โ ๐).
Dimension count. Let dim๐ธ = ๐ , dim๐น = ๐ก, dim๐ป = ๐, so that ๐ = dim๐ = ๐+ ๐ + ๐ก.The number of independent functions is
๐ (๐ + 1)โ2 + ๐ก(๐ก + 1)โ2 + ๐(๐ + ๐ก).
Revisiting example 2.5. Here โ = โ0 = span{๐3, ๐4, ๐5} and we can compute
๐๐1 = ๐๐ฅ1 โง ๐๐ฅ2 โโก 0, ๐๐4 = ๐๐ฅ1 โง ๐3 โก 0, ๐๐5 = ๐๐ฅ2 โง ๐3 โก 0 (all mod โ),
hence the uppercase Greeks are โ1 = span{๐4, ๐5}. Furthermore we can check that๐๐4 and ๐๐5 โโก 0 mod ๐ผ1 so that โ2 = 0.
The column vector (๐1, ๐2, ๐3, ๐4, ๐5)๐ก๐ is thus an adapted coframe, where forconvenience we kept the uppercase Greeks ๐4, ๐5 at the end of the list. The initial๐บ0 โ ๐บ๐(5) consists of all matrices of the form
โโโโโโโ
๐ถ11 ๐ถ12 0 ๐14 ๐15๐ถ21 ๐ถ22 0 ๐24 ๐250 0 ๐33 ๐34 ๐350 0 0 ๐44 ๐450 0 0 ๐54 ๐55
โโโโโโโ
.
In light of proposition 2.8, Cartanโs condition is easy to appreciate. If we let ๐ถbe the identity matrix and replace ๐1 with ๐1 = ๐1 + ๐14๐4 + ๐15๐5 and ๐2 with
CARTAN MEETS CHAPLYGIN 29
๐2 = ๐2 + ๐24๐4 + ๐25๐5, then the coefficients ๐ผ212 and ๐ผ1
12 in the motion equations(2.5) remain unchanged since
๐๐4 โง ๐3 โง ๐4 โง ๐5 = ๐๐5 โง ๐3 โง ๐4 โง ๐5 = 0.
4. Remarks on weak equivalence-preserving the acceleration ๐ท๐๐
Cartanโs equivalence criterion-to preserve the nh connection, obviously impliesthat their nonholonomic paths are the same. Much to our surprise, Terra and Olivahave recently found an example of a change of metric that does not preserve thenh connection but still preserves the paths. We will discuss their example at theend of this section. First, we will make some considerations indicating that thisis an โextreme eventโ. We start by showing that when dim๐ธ = 2, preservation oftrajectories implies, conversely, preservation of the connection. In [37,38,64], thecases of, respectively, 2-3, 2-3-5, and 2-3-4 distribution growth were studied.
4.1. Rank two distributions.
Proposition 4.1. For any nonintegrable distribution ๐ธ of rank 2 on a manifold๐ of dimension ๐ endowed with an arbitrary Riemannian metric, the straightestpath equations are preserved if and only if ๐ต2 = 0.
Proof. Theorem 3.1 yields sufficiency. To show necessity, let ๐ = (๐1, . . . , ๐๐)โ ,where โ indicates transpose, be an specially adapted coframe (3.4) in Definition 3.1,
with โ = span{๐3, . . . , ๐๐}, โ(1) = span{๐3, . . . , ๐๐โ1}, and ๐ = (๐1)2 + โ โ โ + (๐๐)2.Here โ(1) is the annihilator of the three-dimensional distribution ๐ธ(1) = ๐ธ+ (๐ธ,๐ธโ,and ๐๐ โ โ(1). Note that in the special case ๐ = 3, โ(1) = 0 and ๐3 โ ๐ผ(1).
The structure equations for ๐ are
๐
โโโโโโโโโโ
๐1
๐2
๐3
โฏ๐๐โ1
๐๐
โโโโโโโโโโ
โก
โโโโโโโโโโ
๐1๐1 โง ๐2
๐2๐1 โง ๐2
0โฏ0
๐๐๐1 โง ๐2
โโโโโโโโโโ
(mod ๐ผ).
Note that ๐๐ โ 0 since ๐๐ โ ๐ผ(1). Applying Proposition 2.3, the straightest pathsequations for the quasivelocities ๐ฃ1 and ๐ฃ2 are
(4.1)๏ฟฝ๏ฟฝ1 = ๐2๐ฃ1๐ฃ2 + ๐1(๐ฃ2)2
๏ฟฝ๏ฟฝ2 = โ๐2(๐ฃ1)2 + ๐1๐ฃ1๐ฃ2.
Now suppose that in equation (3.6) ๐ต2 = (๐1, ๐2)โ and, without loss of general-ity, ๐ถ, ๐ด0, and ๐ด2 are identity matrices, and that ๐ด1 = 0. The modified coframe is
thus ๏ฟฝ๏ฟฝ = (๐1, . . . , ๏ฟฝ๏ฟฝ๐)โ = (๐1+๐1๐๐, ๐2+๐2๐๐, ๐3, . . . , ๐๐)โ with associated modifiedmetric (๏ฟฝ๏ฟฝ1)2 + โ โ โ + (๏ฟฝ๏ฟฝ๐)2. Note that the quasivelocities ๐ฃ1 and ๐ฃ2 are unchanged.
The structure equations for ๏ฟฝ๏ฟฝ are
30 EHLERS AND KOILLER
๐
โโโโโโโโโโ
๏ฟฝ๏ฟฝ1
๏ฟฝ๏ฟฝ2
๏ฟฝ๏ฟฝ3
โฏ๏ฟฝ๏ฟฝ๐โ1
๏ฟฝ๏ฟฝ๐
โโโโโโโโโโ
โก
โโโโโโโโโโ
(๐1 + ๐1๐๐)๏ฟฝ๏ฟฝ1 โง ๏ฟฝ๏ฟฝ2
(๐2 + ๐2๐๐)๏ฟฝ๏ฟฝ1 โง ๏ฟฝ๏ฟฝ2
0โฏ0
๐๐๏ฟฝ๏ฟฝ1 โง ๏ฟฝ๏ฟฝ2
โโโโโโโโโโ
(mod ๐ผ).
The straightest path equations for (the same) quasivelocities are now
๏ฟฝ๏ฟฝ1 = (๐2 + ๐2๐๐)๐ฃ1๐ฃ2 + (๐1 + ๐1๐๐)(๐ฃ2)2
๏ฟฝ๏ฟฝ2 = โ(๐2 + ๐2๐๐)(๐ฃ1)2 + (๐1 + ๐1๐๐)๐ฃ1๐ฃ2.
Comparing with the original equations (4.1) and recalling that ๐๐ โ 0, we can seethat it is necessary for ๐1 = ๐2 = 0 to preserve the nonholonomic paths. ๏ฟฝ
4.2. โMaster equationโ for keeping the straightest paths, revisited.Recall that the nonholonomic dynamics in the EulerโArnold format is given by
(4.2) ๏ฟฝ๏ฟฝ๐ = ๐๐โ๐(๐)๐ฃ๐๐ฃโ, 1 โฉฝ ๐, ๐, โ โฉฝ ๐ , ๐ = ๐ฃ๐๐๐,
where the ๐๐ โ ๐ธ (๐ = 1, . . . ๐ ) are part of a full orthonormal frame ๐๐ฝ (๐ฝ = 1, . . . ๐),with
(๐๐ผ , ๐๐ฝโ = ๐๐พ๐ผ๐ฝ๐๐พ .
Recall that we change metric as follows. An orthonormal basis {๐๐, ๐๐ผ} for the newmetric is related to {๐๐, ๐๐ผ} by
(4.3) ๐๐ = ๐๐๐ถ๐๐ , ๐๐ผ = ๐๐๐ต๐๐ผ + ๐๐๐ด๐๐ผ.
Here ๐ต is ๐ ร ๐ and ๐ด is ๐ ร ๐, functions of ๐ โ ๐. We take ๐ถ =identity, so ๐๐ = ๐๐.
Definition 4.1. For each ๐ = 1, . . . ๐ , denote by ๐ถ๐ the (๐ โ ๐ ) ร ๐ matrix
(๐ถ๐)๐ผโ = ๐๐ผโ๐.
We call them โweightsโ. Note that the last ๐ rows of those matrices are zerosince we are working with specially adapted frames.
We get immediately from the package of conditions (3.3):
Proposition 4.2 (Master Equation for Weak Equivalence). The necessary andsufficient condition for a change of metrics (4.3) to preserve the straightest pathsequation (4.2) is:
All the ๐ ร ๐ โweighted matricesโ ๐ต๐ถ๐ for ๐ = 1, . . . ๐ are skew symmetric.
Recall that this means
(4.4) โ๐ผ
(๐๐ผโ๐๐ต๐๐ผ + ๐๐ผ๐๐๐ตโ๐ผ) = 0, for all 1 โฉฝ ๐, โ โฉฝ ๐ and 1 โฉฝ ๐ โฉฝ ๐ .
As we mentioned before, the skew symmetry requirement is natural since any skewsymmetric bilinear form ๐ฃโ ๐ด๐ค vanishes when ๐ฃ = ๐ค. In hindsight this propositionresults from the Euler-Arnold form for the nonholonomic dynamics, using the ๐๐๐๐(๐)instead of the ฮ๐
๐๐(๐).
CARTAN MEETS CHAPLYGIN 31
4.3. There are more equations than unknowns. Let ๐ = dimโ(1), so๐ก = ๐โ๐ โ ๐ is the dimension of the complement ๐น of ๐ธ in ๐ธ(1). This is the numberof the ๐ผ indices that matter forming matrix ๐ต2. Note that for all the remaining๐ผโs we have ๐๐ผ๐๐ = 0. We differentiated those latter in Definition 3.1 by using uppercase Greek letters (and indeed there are ๐ of them).
Each matrix ๐ถ๐ has two blocks. The upper, which we call ๐ถ๐, is the relevantone, of size (๐ โ ๐ โ ๐) ร ๐ . The lower block is of size ๐ ร ๐ and is filled with zeros.Therefore, in computing the products ๐ต๐ถ๐, we should divide matrix ๐ต in twoblocks. The left one, which we call ๐ต2, is of size ๐ ร (๐โ ๐ โ ๐) and meets the upperblocks ๐ถ๐. The right one, which we call ๐ต1, is of size ๐ ร ๐.
We can see that there are no conditions for ๐ต1 since it meets the lower partof the ๐ถ๐ which are zero. The skew symmetry conditions (4.4) are limited to thesmall Greek indices ๐ผโs, and apply to block ๐ต2, of size ๐ ร (๐ โ ๐ โ ๐). The numberof unknowns is ๐ข = ๐ (๐ โ ๐ โ ๐). On the other hand, the number ๐ of equationsencoded in ๐ถ๐๐ต2 + (๐ถ๐๐ต2)๐ = 0 is
๐ = ๐ (๐ (๐ + 1)โ2) = ๐ 2(๐ + 1)โ2.
We could assert for sure that there are nonzero solutions for ๐ต2 if there were moreunknowns than equations, ๐ข > ๐, i.e., if ๐ ร(๐โ๐ โ๐) > ๐ 2(๐ +1)โ2, or equivalently, if
๐ > ๐ (๐ + 1)โ2 + ๐ + ๐ (โ).
However, this is never the case. Since dim๐ธ(1) โฉฝ ๐ + ๐ (๐ โ 1)โ2 = ๐ (๐ + 1)โ2 (themaximum number of new vectorfields that one can form with the brackets) we get
๐ โฉฝ ๐ + ๐ (๐ + 1)โ2
which precludes (*). So there are always more equations than unknowns.
4.4. Corank 1, strongly nh, distributions. Here ๐ = ๐ โ 1, ๐ = 0. Is itpossible to find an example of nh geodesics equivalence with ๐ต โ 0? We indicatewhy not using the Master Equations. We will give a more general argument in thenext subsection, using Cartanโs viewpoint. To begin with, note that there are farmore conditions (๐ = (๐ โ 1)2๐โ2) than unknowns (๐ข = ๐ โ 1).
Matrices ๐ถ๐,1 โฉฝ ๐ โฉฝ ๐ โ 1 are row-vectors of size ๐ โ 1
๐ถ๐ = (๐๐1๐, ๐๐2๐, . . . , ๐๐๐โ1,๐)
and the matrix ๐ต2 = ๐ต is a column vector of size ๐ โ 1
๐ต = (๐1, . . . ๐๐โ1)โ .
The product ๐ต๐ถ๐ is a (๐ โ 1) ร (๐ โ 1) matrix whose ๐-th row (1 โฉฝ ๐ โฉฝ ๐ โ 1) is
๐๐(๐๐1๐ , ๐๐2๐ , . . . , ๐๐๐โ1,๐).
For the nh connection equivalence we know already that all ๐๐, ๐ = 1, . . . , ๐ = ๐โ1must vanish since the distribution is Snh. But it would be interesting to have adirect argument using the conditions (3.2). Since there is only one ๐ผ = ๐, the usualalternating trick shows that every term in (3.2) vanishes, i.e. all ๐๐๐๐๐๐ = 0. Sincesome ๐๐๐๐ โ 0, we conclude that all ๐๐ = 0.
32 EHLERS AND KOILLER
It is a little bit more difficult to analyze the conditions for keeping the straight-est paths (i.e., weak equivalence)
(4.5) ๐๐๐๐๐๐ + ๐๐๐
๐๐๐ = 0
for all 1 โฉฝ ๐, ๐, ๐ โฉฝ ๐ โ 1. It is convenient to form the skew symmetric matrix
๐ถ = (๐๐๐๐).Making ๐ = ๐ in (4.5) kills automatically the first term and we get ๐๐๐
๐๐๐ = 0 (not a
sum over ๐, it is just a single product).
Proposition 4.3. If ๐๐๐๐ โ 0 then ๐๐ = ๐๐ = 0.
If there is an example with a ๐๐ โ 0, then the whole ๐-th column (so also the
๐-th row) of ๐ถ must vanish. Conversely, suppose the ๐-th colum of ๐ถ is formed byzeros (so also the ๐-th row). Then the equation (4.5) for that value ๐ is void. Itseems therefore that the most favorable case for the possibility of having a nonzero๐ต is when only one structure coefficient is nonzero.
At first sight it seems that that pair of ๐โฒ๐ vanishes and all the other couldbe arbitrary. We now show that (4.5) forces a lot of others (we think that thisargument gives all) to be zero too.
When one structure coefficient is nonzero we have the group R๐ ร๐ป3, where๐ป3 is the smallest Heisenberg group. Denote (๐ฆ1, . . . , ๐ฆ๐, ๐ฅ, ๐ฆ, ๐ง) the coordinates.The corank 1, strongly nh distribution ๐ธ is generated by the ๐โ๐๐ฆ๐, together with
๐1 = ๐โ๐๐ฅ, ๐2 = ๐โ๐๐ฆ + ๐ฅ๐โ๐๐ง.The only nonzero commutator is (๐1,๐2โ = ๐โ๐๐ง.๐ต is a column matrix with ๐ + 2 entries. Let us look at the first ๐ entries
๐๐, ๐ = 1, . . .๐. The second term in (4.5) vanishes because all the possible ๐โ๐โ = 0.But we may take ๐ = ๐ + 1, ๐ = ๐ + 2 and since ๐๐+3
๐+1,๐+2 = 1 we conclude that๐๐ = 0, ๐ = 1, . . .๐. In order to show that ๐๐+1 and ๐๐+2 also vanish in the nhgeodesic equivalence we now look directly at ๐ป3. More generally, the reasoning isvalid for all Heisenberg algebras ๐ป๐, ๐ = 2๐ + 1.
The last coordinate is called ๐ง. The distribution of rank ๐ โ 1 = 2๐ (even)is generated by vectorfieds ๐๐โs and ๐๐ โs with (๐๐,๐๐โ = ๐ฟ๐๐๐โ๐๐ง, 1 โฉฝ ๐, ๐ โฉฝ ๐.Conditions (4.5) lead to ๐ต = 0. This is because when we make ๐ = ๐, we get2๐๐๐
๐๐๐ = 0 (again, a plain product). But for every ๐ there exists one (unique) ๐
with ๐๐๐๐ โ 0.Therefore for the standard metric in the Heisenberg distribution, there is no
difference between preserving the connection and preserving the straightest paths.
4.5. Corank of ๐ธ in ๐ธ(1) = ๐ธ + (๐ธ,๐ธโ < 3. The reasoning we now presenttakes care of the previous case ๐ = ๐ โ 1 and many others. It implies for instancethat to find examples where the straightest paths are the same in non-equivalentnH connections, the corank of ๐ธ in ๐ธ + (๐ธ,๐ธโ must be at least 3, no matter whatthe rank of ๐ธ is. So such examples can happen in (3,6), (4,7,. . . ),(4,8,. . . ), etc. butnot in (2,3), (2,3,4), (2,3,5), (3,5), etc. The minimal example is in (3,6), preciselythe situation in [76].
CARTAN MEETS CHAPLYGIN 33
We present the argument for a rank 3 distribution just to write things outexplicitly. We show that to preserve the straightest paths, covectors that are notin โ1 cannot be added to the ๐๐, ๐ = 1,2,3 if ๐ธ + (๐ธ,๐ธโ has dimension 4 or 5.
Proposition 4.4. Suppose ๐ธ is a totally nonholonomic, rank 3 distributionon a manifold ๐ of dimension ๐ > 3. If the corank of ๐ธ โ ๐ธ + (๐ธ,๐ธโ is one or two,then the block ๐ต2 in (3.6) is zero.
Proof. We slightly change the notational conventions as follows. Suppose ๐ isa specially adapted coframe for this distribution for which ๐๐, 4 โฉฝ ๐ โฉฝ ๐, span โ(1).When can we add multiples of ๐ฮฆ, ๐+1 โฉฝ ฮฆ โฉฝ ๐ to the ๐๐, 1 โฉฝ ๐ โฉฝ 3, while preservingthe straightest path equations?
Let (๐๐๐โ be the Levi-Civita connection forms matrix relative to the coframe
๐. Expand the components of ๐ as ๐๐๐ = โ๐๐=1 ๐
๐๐๐๐
๐. The nh trajectories are then
๏ฟฝ๏ฟฝ1 = โ๐ฃ2(๐112๐ฃ1 + ๐2
12๐ฃ2 + ๐312๐ฃ3) โ ๐ฃ3(๐1
13๐ฃ1 + ๐213๐ฃ2 + ๐3
13๐ฃ3)๏ฟฝ๏ฟฝ2 = ๐ฃ1(๐1
12๐ฃ1 + ๐212๐ฃ2 + ๐3
12๐ฃ3) โ ๐ฃ3(๐123๐ฃ1 + ๐2
23๐ฃ2 + ๐323๐ฃ3)(4.6)
๏ฟฝ๏ฟฝ3 = ๐ฃ1(๐113๐ฃ1 + ๐2
13๐ฃ2 + ๐313๐ฃ3) โ ๐ฃ2(๐1
23๐ฃ1 + ๐223๐ฃ2 + ๐3
23๐ฃ3).
Now suppose ๐ต2 โ 0 and we modify ๐ by adding linear combinations ๐๐ = โ๐ฮฆ๐ฮฆ
to the ๐๐, 1 โฉฝ ๐ โฉฝ 3, to obtain a new coframe ๏ฟฝ๏ฟฝ. The distribution ๐ธ is preserved,but the metric has changed. Denote the Levi-Civita connection form relative tothe modified coframe by ๏ฟฝ๏ฟฝ = (๏ฟฝ๏ฟฝ๐๐โ with components ๏ฟฝ๏ฟฝ๐๐ = โ๐
๐=1 ๏ฟฝ๏ฟฝ๐๐๐๐
๐.The equations for the quasivelocities (the same as above) are now
๏ฟฝ๏ฟฝ1 = โ๐ฃ2(๏ฟฝ๏ฟฝ112๐ฃ1 + ๏ฟฝ๏ฟฝ2
12๐ฃ2 + ๏ฟฝ๏ฟฝ312๐ฃ3) โ ๐ฃ3(๏ฟฝ๏ฟฝ1
13๐ฃ1 + ๏ฟฝ๏ฟฝ213๐ฃ2 + ๏ฟฝ๏ฟฝ3
13๐ฃ3)๏ฟฝ๏ฟฝ2 = ๐ฃ1(๏ฟฝ๏ฟฝ1
12๐ฃ1 + ๏ฟฝ๏ฟฝ212๐ฃ2 + ๏ฟฝ๏ฟฝ3
12๐ฃ3) โ ๐ฃ3(๏ฟฝ๏ฟฝ123๐ฃ1 + ๏ฟฝ๏ฟฝ2
23๐ฃ2 + ๏ฟฝ๏ฟฝ323๐ฃ3)(4.7)
๏ฟฝ๏ฟฝ3 = ๐ฃ1(๏ฟฝ๏ฟฝ113๐ฃ1 + ๏ฟฝ๏ฟฝ2
13๐ฃ2 + ๏ฟฝ๏ฟฝ313๐ฃ3) โ ๐ฃ2(๏ฟฝ๏ฟฝ1
23๐ฃ1 + ๏ฟฝ๏ฟฝ223๐ฃ2 + ๏ฟฝ๏ฟฝ3
23๐ฃ3).
Subtracting the expressions for ๏ฟฝ๏ฟฝ1 in (4.6) and (4.7) and equating the coefficientsin ๐ฃ๐๐ฃ๐ to zero we obtain
๐112 โ ๏ฟฝ๏ฟฝ1
12 = ๐212 โ ๏ฟฝ๏ฟฝ2
12 = ๐113 โ ๏ฟฝ๏ฟฝ1
13 = ๐313 โ ๏ฟฝ๏ฟฝ3
13 = 0
and
(4.8) ๐312 โ ๏ฟฝ๏ฟฝ3
12 = โ(๐213 โ ๏ฟฝ๏ฟฝ2
13).
Doing the same for ๏ฟฝ๏ฟฝ2 and ๏ฟฝ๏ฟฝ3 we find that
๐312 โ ๏ฟฝ๏ฟฝ3
12 = (๐123 โ ๏ฟฝ๏ฟฝ1
23)(4.9)
๐213 โ ๏ฟฝ๏ฟฝ2
13 = โ(๐123 โ ๏ฟฝ๏ฟฝ1
23)(4.10)
with the difference between all other pairs equal to zero.Equations (4.8), (4.9), and (4.10) are not independent and we can write
(4.11) ๐312 โ ๏ฟฝ๏ฟฝ3
12 = โ(๐213 โ ๏ฟฝ๏ฟฝ2
13) = ๐123 โ ๏ฟฝ๏ฟฝ1
23 = ๐.
34 EHLERS AND KOILLER
We know that connection forms ๐ and ๏ฟฝ๏ฟฝ = ๐+ฮ lead to the same nH trajectoriesif and only if
ฮ =โโโโโ
0 ๐๐3 โ๐๐2 โ โโ๐๐3 0 ๐๐1 โ โ๐๐2 โ๐๐1 0 โ โโโ โ โ 0 โ
โโโโโ
(mod ๐ผ).
Since ๐๏ฟฝ๏ฟฝ = โ๏ฟฝ๏ฟฝ โง ๏ฟฝ๏ฟฝ = โ(๐ +ฮ) โง ๏ฟฝ๏ฟฝ we must have
(4.12) ๐โโโ
๐1
๐2
๐3
โโโ โกโโโ
โ2๐๐2 โง ๐3
2๐๐1 โง ๐3
โ2๐๐1 โง ๐2
โโโ
(mod ๐ผ).
Since the ๐๐๐ are linearly independent, the ๐๐ must be linearly independent.This is only possible if the corank of ๐ธ in ๐ธ + (๐ธ,๐ธโ is three or more. ๏ฟฝ
The conclusion of this proposition is also true for distributions of rank greaterthan three. In this case (4.11) becomes
๐ = ๐๐พ๐๐ โ ๏ฟฝ๏ฟฝ๐
๐๐ = โ(๐๐๐๐ โ ๏ฟฝ๏ฟฝ๐
๐๐) = ๐๐๐๐ โ ๏ฟฝ๏ฟฝ๐
๐๐
and, modulo ๐ โ {๐๐, ๐๐ , ๐๐}, (4.12) becomes
๐โโโ
๐๐
๐๐
๐๐
โโโ โกโโโ
โ2๐๐๐ โง ๐๐
2๐๐๐ โง ๐๐
โ2๐๐๐ โง ๐๐
โโโ
(mod ๐ผ)
and the conclusion follows as before.
4.6. The example in ๐ = ๐๐(4) by Terra and Oliva [76]. In this ex-ample, ๐ = 6, ๐ = ๐ = 3. First we will discuss it using the Cartan approach, andafterwards by a modification of the โmaster equationsโ, which may be helpful inconcocting other families of examples.
4.6.1. Via Cartan equations. Let {๐ฅ1, ๐ฅ2, ๐ฅ3, ๐ฅ4, ๐ฅ5, ๐ฅ6} be a basis for ๐ ๐(4)with
๐ฅ1 =โโโโโ
0 0 0 โ10 0 0 00 0 0 01 0 0 0
โโโโโ , ๐ฅ2 =
โโโโโ
0 0 0 00 0 0 โ10 0 0 00 1 0 0
โโโโโ , ๐ฅ3 =
โโโโโ
0 0 0 00 0 0 00 0 0 โ10 0 1 0
โโโโโ ,
๐ฅ4 =โโโโโ
0 0 0 00 0 โ1 00 1 0 00 0 0 0
โโโโโ , ๐ฅ5 =
โโโโโ
0 0 1 00 0 0 0โ1 0 0 00 0 0 0
โโโโโ , ๐ฅ6 =
โโโโโ
0 โ1 0 01 0 0 00 0 0 00 0 0 0
โโโโโ .
Let ๐ = {๐1,๐2,๐3,๐4,๐5,๐6} be the corresponding left invariant frame for๐๐. Consider the bi-invariant metric ๐ on ๐๐(4) induced by the CartanโKillingform:
๐(๐๐,๐๐) = โ1
2tr(๐ฅ๐, ๐ฅ๐).
CARTAN MEETS CHAPLYGIN 35
It is easy to see that relative to ๐, ๐ is an orthonormal frame. The distribution๐ธ = span{๐1,๐2,๐3} is 1-step bracket generating with
(๐2,๐3โ =๐4, (๐3,๐1โ =๐5, (๐1,๐2โ =๐6.
Let ๐ = (๐๐โ be the dual coframe with ๐๐(๐๐) = ๐ฟ๐๐ . The structure equations are
๐๐1 = โ(โ๐3 โง ๐5 + ๐2 โง ๐6)๐๐2 = โ(โ๐1 โง ๐6 + ๐3 โง ๐4)๐๐3 = โ(๐1 โง ๐5 โ ๐2 โง ๐4)๐๐4 = โ(๐2 โง ๐3 + ๐5 โง ๐6)๐๐5 = โ(โ๐1 โง ๐3 โ ๐4 โง ๐6)๐๐6 = โ(๐1 โง ๐2 + ๐4 โง ๐5).
โ is spanned by {๐4, ๐5, ๐6} and โ1 = 0. The connection forms are
(๐๐๐โ =1
2
โโโโโโโโโโ
0 โ๐6 ๐5 0 โ๐3 ๐2
๐6 0 โ๐4 ๐3 0 โ๐1
โ๐5 ๐4 0 โ๐2 ๐1 00 โ๐3 โ๐2 0 โ๐6 ๐5
๐3 0 โ๐1 ๐6 0 โ๐4
โ๐2 ๐1 0 โ๐5 ๐4 0
โโโโโโโโโโ
.
If ๐ is a straightest path with ๏ฟฝ๏ฟฝ = ๐ฃ1๐1 + ๐ฃ2๐2 + ๐ฃ3๐3, the nonholonomic equationsare simply
๐ฃ1 = ๐ฃ2 = ๐ฃ3 = 0 (since ๐๐๐(๏ฟฝ๏ฟฝ) = 0, 1 โฉฝ ๐, ๐ โฉฝ 3).For 0 < ๐ < 1 let ๐บ be a deformation of ๐ given by (relative to the ordered basis ๐)
โโโโโโโโโโ
1 0 0 ๐ 0 00 1 0 0 ๐ 00 0 1 0 0 ๐๐ 0 0 1 0 00 ๐ 0 0 1 00 0 ๐ 0 0 1
โโโโโโโโโโ
.
Since the distribution ๐น spanned by ๐4, ๐5, ๐6 is not orthogonal to ๐ธ when๐ โ 0, the nh connections are different. Terra and Oliva observed in [76] that onegets the same nonholonomic paths.
In terms of the original coframe ๐, the coframe ๏ฟฝ๏ฟฝ for this metric is
โโโโโโโโโโ
๏ฟฝ๏ฟฝ1
๏ฟฝ๏ฟฝ2
๏ฟฝ๏ฟฝ3
๏ฟฝ๏ฟฝ4
๏ฟฝ๏ฟฝ5
๏ฟฝ๏ฟฝ6
โโโโโโโโโโ
=
โโโโโโโโโโ
๐1 + ๐๐4
๐2 + ๐๐5
๐3 + ๐๐6โ1 โ ๐2๐4
โ1 โ ๐2๐5
โ1 โ ๐2๐6
โโโโโโโโโโ
.
36 EHLERS AND KOILLER
Note that we have added to ๐1, ๐2, and ๐3 multiples of ๐4, ๐5, and ๐6, which arenot in the first derived ideal. So the connection form restricted to ๐ธ will change.True enough, the structure equations for ๏ฟฝ๏ฟฝ are
๐๐1 = โ(โ๐3 โง ๐5 + ๐2 โง ๐6) โ ๐(๐2 โง ๐3 + ๐5 โง ๐6)๐๐2 = โ(โ๐1 โง ๐6 + ๐3 โง ๐4) โ ๐(โ๐1 โง ๐3 โ ๐4 โง ๐6)๐๐3 = โ(๐1 โง ๐5 โ ๐2 โง ๐4) โ ๐(๐1 โง ๐2 + ๐4 โง ๐5)
๐๐4 = โโ1 โ ๐2(๐2 โง ๐3 + ๐5 โง ๐6)
๐๐5 = โโ1 โ ๐2(โ๐1 โง ๐3 โ ๐4 โง ๐6)
๐๐6 = โโ1 โ ๐2(๐1 โง ๐2 + ๐4 โง ๐5).
As expected the connection form, when restricted to ๐ธ changes: mod (โ) we get
๏ฟฝ๏ฟฝ12 = โ๐
2๏ฟฝ๏ฟฝ3, ๏ฟฝ๏ฟฝ13 =
๐
2๏ฟฝ๏ฟฝ2, ๏ฟฝ๏ฟฝ23 = โ
๐
2๏ฟฝ๏ฟฝ1.
The nonholonomic equations for the deformed metric are then
๏ฟฝ๏ฟฝ1 = โ๐ฃ1๐11(๏ฟฝ๏ฟฝ) โ ๐ฃ2๐12(๏ฟฝ๏ฟฝ) โ ๐ฃ3๐13(๏ฟฝ๏ฟฝ) = โ๐ฃ1(0) โ ๐ฃ2( โ๐
2๐ฃ3) โ ๐ฃ3(
๐
2๐ฃ3) = 0
๏ฟฝ๏ฟฝ2 = โ๐ฃ1๐21(๏ฟฝ๏ฟฝ) โ ๐ฃ2๐22(๏ฟฝ๏ฟฝ) โ ๐ฃ3๐23(๏ฟฝ๏ฟฝ) = โ๐ฃ1(๐
2๐ฃ3) โ ๐ฃ2(0) โ ๐ฃ3( โ
๐
2๐ฃ1) = 0
๏ฟฝ๏ฟฝ3 = โ๐ฃ1๐31(๏ฟฝ๏ฟฝ) โ ๐ฃ2๐32(๏ฟฝ๏ฟฝ) โ ๐ฃ3๐33(๏ฟฝ๏ฟฝ) = โ๐ฃ1( โ๐
2๐ฃ2) โ ๐ฃ2(
๐
2๐ฃ1) โ ๐ฃ3(0) = 0.
While the nonholonomic connection has changed, the changes cancel exactly in thestraightest path equations. A mathematical gem indeed!
4.6.2. Via Euler-Arnold equations (2.6). With the expectation to concoct otherexamples, we invoke Proposition 2.1 on a Lie algebra of dimension ๐. We denoteby ๐บ the matrix of the metric relative to a basis ๐๐ผ , ๐ = 1, . . . ๐, and we denoteby ฮฉ๐ผ , ๐ผ = 1, . . . , ๐ the coordinates in this basis. Denote by ๐๐บ the orthogonalprojection over ๐ธ, spanned by the first vectors ๐ = 1, . . . , ๐ . The equations for thenonholonomic system are obtained by applying ๐๐บ to the right hand side of ฮฉ inthe Euler-Arnold equations for geodesics in the Lie group.
We now recall a Linear Algebra formula to project over a subspace generatedby vectors ๐1, . . . , ๐๐ . Let ๐ด be the ๐ ร ๐ matrix whose columns are the ๐๐.
๐๐บ(ฮฉ) = ๐ด(๐ดโ ๐บ๐ด)โ1๐ดโ ๐บฮฉ.
This is often more convenient than a GramโSchmidt procedure on the ๐๐.When the ๐๐ = ๐๐ are the first ๐ vectors of the basis ๐๐ผ , then
๐ด = โ๐ผ๐ ร๐ 0๐ร๐
} , ๐ = ๐ โ ๐ .
A simple computation yields that the projection has matrix
๐๐บ = โ๐ผ (๐บ11)โ1๐บ12
0 0} ,
CARTAN MEETS CHAPLYGIN 37
or simplifying the notation, by cutting the rows of zeros
๐๐บ(ฮฉ) = (๐ผ โ (๐บ11)โ1๐บ12โฮฉ
where ๐บ is partitioned as (๐บ11 is ๐ ร ๐ and ๐บ12 is ๐ ร ๐)
๐บ = โ๐บ11 ๐บ12
โ โ } .
We thus get a recipe to compute the nh equations for the ฮฉ๐, ๐ = 1, . . . , ๐ : Addto each of the first ๐ equations ฮฉ๐ a suitable combination of the remaining ฮฉ๐ผ,๐ผ = ๐ + 1, . . . , ๐, and afterwards cross out all terms containing one of the ฮฉ๐ผโs(since they vanish on ๐ธ). The coefficients on the combination are ๐๐๐๐๐๐ผ, where
the tilde means that we are inverting the minor block ๐บ11 (that we denote ๏ฟฝ๏ฟฝ).
(Notice that the formulas for all the ฮฉ๐ผ require the inverse of the full matrix ๐บ).We now construct a family of metrics for which the projections become very
simple. Consider an even dimensional vector space, of dimension ๐ = 2๐ , where halfof the basis is denoted ๐ด๐ and the other half ๐ต๐ผ. Let
๐บ = โ ๐ผ ๐๐โ ๐ผ
} .
where ๐พ is a symmetric matrix so that ๐บ is still invertible. Actually, we only need๐พ to commute with its transpose. It is immediate to verify that in this case
๐บโ1 = โ (๐ผ โ ๐๐โ )โ1 โ๐(๐ผ โ ๐๐โ )โ1โ๐โ (๐ผ โ ๐๐โ )โ1 (๐ผ โ ๐๐โ )โ1 } .
๐๐บ = (๐ผ โ ๐โ.
Taking into account that
๐๐ฝ๐ =)โโโโโ]
๐ฟ๐ฝ๐, 1 โฉฝ ๐ฝ โฉฝ ๐
๐๐ผ๐, ๐ + 1 โฉฝ ๐ฝ = ๐ + ๐ผ โฉฝ ๐
the resulting equations for the nh dynamics are
ฮฉ๐ = โ1โฉฝ๐,โโฉฝ๐
(๐๐โ๐ +๐
โ๐ผ=1
๐๐ +๐ผโ๐ ๐๐ผ๐)ฮฉ๐ฮฉโ, ๐ = 1 . . . ๐ .
We reintroduced the summation symbols for better clarity. We obtain a resultsimilar to the โmaster equationsโ.
Proposition 4.5. The nonholonomic dynamics is independent of the matrix๐ if and only if the expression
(4.13) ๐ด๐โ,๐ =
๐
โ๐ผ=1
๐๐ +๐ผโ๐ ๐๐ผ๐
is skew symmetric in the indices โ,๐, for all ๐ = 1, . . . ๐ .
38 EHLERS AND KOILLER
We can now re-check Terra and Olivaโs example. We shuffle the basis for ๐ ๐(4).For ๐4, ๐5, ๐6 we take the standard 3ร3 skew symmetric matrices bordered by zerosin the fourth row and column,
(๐๐)๐๐ = (๐ด๐)๐๐ = โ๐๐๐๐, ๐, ๐, ๐ = 1,2,3
and for ๐1 = ๐ต1, ๐2 = ๐ต2, ๐3 = ๐ต3 the only nonzero entries will be in the last rowand colum, as shown below:
๐ด1 =
โจโโโโโโโโช
0 0 0 00 0 โ1 00 1 0 00 0 0 0
โฌโ โ โ โ โ โ โ โฎ
, ๐ด2 =
โจโโโโโโโโช
0 0 1 00 0 0 0โ1 0 0 00 0 0 0
โฌโ โ โ โ โ โ โ โฎ
, ๐ด3 =
โจโโโโโโโโช
0 โ1 0 01 0 0 00 0 0 00 0 0 0
โฌโ โ โ โ โ โ โ โฎ
,
๐ต1 =
โจโโโโโโโโช
0 0 0 โ10 0 0 00 0 0 01 0 0 0
โฌโ โ โ โ โ โ โ โฎ
, ๐ต2 =
โจโโโโโโโโช
0 0 0 00 0 0 โ10 0 0 00 1 0 0
โฌโ โ โ โ โ โ โ โฎ
, ๐ต3 =
โจโโโโโโโโช
0 0 0 00 0 0 00 0 0 โ10 0 1 0
โฌโ โ โ โ โ โ โ โฎ
.
The structure constants are
(๐ด๐,๐ด๐โ = ๐๐๐๐๐ด๐, (๐ต๐,๐ต๐โ = ๐๐๐๐๐ด๐, (๐ด๐,๐ต๐โ = ๐๐๐๐๐ต๐.
Relative to the basis ๐1 = ๐ต1, ๐2 = ๐ต2, ๐3 = ๐ต3, ๐4 = ๐ด1, ๐5 = ๐ด2, ๐6 = ๐ด3, asdone in the previous subsection, the family of metrics is given by the 6 ร 6 matrix(symmetric, positive definite) ๐บ with
๐ = ๐๐ผ3ร3.
For ๐ = 0 the metric is bi-invariant. The unconstrained dynamics system is trivial:ฮฉ๐ผ = 0, ๐ผ = 1, . . . ,6. In fact, since ๐๐ฝ๐ = ๐ฟ๐ฝ๐ , we get ฮฉ๐พ = ๐๐ฝ๐ฟ๐พฮฉ๐ฝฮฉ๐ฟ. Observethat all the non-vanishing structure constants satisfy ๐๐ฝ๐ฟ๐พ = โ๐๐ฟ๐ฝ๐พ so the terms inthe right hand side cancel in pairs6.
We test the condition (4.13) with 0 < ๐ < 1.Due to the symmetries between the indices 1,2,3 in ๐บ and in the Lie brackets,
it is enough to test for ๐ = 1 and โ,๐ = 2,3, with ๐๐๐ = ๐๐ฟ๐๐.
โ =๐ = 2 โถ ๐521๐22 but ๐521 = ๐212 = 0.
โ =๐ = 3 โถ ๐631๐22 but ๐631 = ๐313 = 0.
โ = 2,๐ = 3 โถ ๐621๐33 but ๐621 = ๐213 = โ1.โ = 3,๐ = 2 โถ ๐531๐22 but ๐531 = ๐312 = +1.
The skew symmetry requirement of Proposition 4.5 is fulfilled. Although theunconstrained EulerโArnold system does not vanish, it projects orthogonally over๐ธ, so the nonholonomic dynamics is trivial.
6This is true for ๐๐(๐). The standard bi-invariant metric in ๐ ๐(๐) is given by the Killingform, โฮฉ1,ฮฉ2 = โ(1โ2) tr(ฮฉ1ฮฉ2) which defines an isomorphim ๐ ๐(๐) โก ๐ ๐(๐)โ. Eulerโs equationsthen write as ๏ฟฝ๏ฟฝ = (๐,ฮฉโ, ๐ = ๐บ(ฮฉ) โ ๐ ๐(๐). The positive definite bilinear operator giving thetotal energy of the ๐-dimensional rigid body is defined by ๐ป = โ๐บ(ฮฉ1),ฮฉ2. Obviously for ๐บ = ๐๐,
๏ฟฝ๏ฟฝ = (ฮฉ,ฮฉโ โก 0. ๐๐(4) is special: it has a two parameter family of bi-invariant metrics.
CARTAN MEETS CHAPLYGIN 39
5. Cartan meets Chaplygin
We outline a research proposal: equivalence problems for Chaplygin systems.There is a new twist. One can focus on the dynamics on the base. The horizontallift to the total space will be considered โjustโ a quadrature (this makes no justiceto linear ODEs with time varying coefficients).
Given two purely inertial ๐บ-Chaplygin systems (symmetry group ๐บ),
(5.1) (๐บ๐ โช ๐๐ โ ๐๐ , ๐, ๐ โถ ๐๐โ ๐ข), (๐บโช ๏ฟฝ๏ฟฝ๐ โ ๐๐ , ๐,๏ฟฝ๏ฟฝ โถ ๐๏ฟฝ๏ฟฝโ ๐ข),
with constraint distributions ๐ธ = ker๐, ๏ฟฝ๏ฟฝ = ker ๏ฟฝ๏ฟฝ, respectively, find what are theinvariants that imply the existence of a local diffeomorphism ๐ โถ ๐ โ ๏ฟฝ๏ฟฝ satisfyinga required criterion, which could be, if ๐ธ = ๏ฟฝ๏ฟฝ,
i) Preserving the nh connections, as discussed in sections 3 and 4:
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๐ = ๐โ(๐ท๐๐ ), ๏ฟฝ๏ฟฝ = ๐โ(๐), ๐ = ๐โ๐.
ii) Sending the straightest paths in ๐ to the straightest paths in ๏ฟฝ๏ฟฝ.
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ = ๐โ(๐ท๐๐), ๏ฟฝ๏ฟฝ = ๐โ(๐).
If ๐ธ โ ๏ฟฝ๏ฟฝ (we change the connection in addition to the metric):iii) Preserving the projected connections in ๐.
iv) Sending the reduced paths in ๐ to the reduced paths in ๐.
In equivalence problems it is traditional to work locally, on neighborhoods ofpoints ๐ โ ๐ and ๐ โ ๏ฟฝ๏ฟฝ. So there is no loss in generality to assume
๐ = ๐ ร๐บ, ๏ฟฝ๏ฟฝ = ๏ฟฝ๏ฟฝ ร๐บ,
with ๐ = ๏ฟฝ๏ฟฝ an open set in R๐ . To start, one needs to describe the initial subbundlesinside the full coframe bundle ๐น โ(๐ ร๐บ) and the corresponding matrix group.
5.1. The reduced affine connection. Let ๐ be a ๐บ-invariant riemannianmetric in ๐. Given ๐,๐ vectorfields in ๐, one defines an affine connection in ๐as follows. Lift them to horizontal vectorfields โ(๐), โ(๐ ) in ๐. Going back todefinition 1.1, compute
๐ทโ(๐)โ(๐ ) = Proj๐ธ โโ(๐)โ(๐ )
where โ is the Levi-Civita connection in ๐ associated to the metric ๐ . It is easyto see that it is also ๐บ-invariant.
Definition 5.1. The reduced affine connection is
(5.2) ๐ท๐๐๐ = ๐โ(๐ทโ(๐)โ(๐ )).
Query 5.1. ๐ท๐ is an affine connection in ๐ in the usual sense. One should beable to compute its structure equations, plus the torsion and curvature tensors interms of moving frames in ๐. For this task, one can use results by A. Lewis in [69].
40 EHLERS AND KOILLER
Definition 5.2. The reduced Chaplygin dynamics is governed by
๐ท๐๐๐ = 0
the geodesic (i.e. straightest path) equation of the affine connection ๐ท๐ . It is ametric connection with respect to the projected metric ๐๐ .
Problem. In (5.1), fix a principal bundle ๐บ๐ โช ๐๐ โ ๐๐ , with vertical spaces๐ . We allow ๐ and ๐ to change. When do the reduced connections ๐ท = ๐ท๐ and๏ฟฝ๏ฟฝ = ๏ฟฝ๏ฟฝ๐ given by the definition (5.2) have the same geodesics (criterion iv)?
One considers the difference tensor
๐ต(๐,๐ ) = ๏ฟฝ๏ฟฝ๐๐๐ โ๐ท๐
๐๐
which is easily seen to be bilinear. As it is well known (see [52, section 5.4.]), ๐ท and
๏ฟฝ๏ฟฝ have the same geodesics if and only if ๐ต is skew-symmetric, which is equivalentto the (obvious) condition
๐ท๐๐๐ = ๏ฟฝ๏ฟฝ๐
๐๐.
At the ๐ level, this amounts to the requirement that
(5.3) Proj๐ธ(โโ(๐)โ(๐)) โProj๏ฟฝ๏ฟฝ(โโ(๐)โ(๐)) โ ๐
for all vectorfields ๐ in ๐. ๐ is the (same) vertical distribution of the two principalconnections, the first orthogonal projection is relative to metric ๐ and the secondto ๐ , and likewise one takes the two distinct Levi-Civita covariant derivatives.
6. Final comments and other research directions
6.1. Cartan equivalence of Chaplygin systems. The basic task is how touse (5.3) in section 5 to obtain the initial structure for Cartan equivalence in termsof frames in ๐ and the Lie algebra of the group ๐บ. For integrable distributions thesolution is very simple. Both reduced dynamics are hamiltonian, with ๐ป = ๏ฟฝ๏ฟฝ beingthe respective kinetic energies in ๐ restricted to the corresponding distributionsand then projected to ๐. Since for metrics the unique invariant is the Riemanniancurvature, the requirement is that ๐ โ๐ธ and ๐ โ๏ฟฝ๏ฟฝ be the lifts (to their distributions)of a same metric ๐ in ๐, the complements being arbitrary. The reduced connectionis simply the Levi-Civita connection of the metric ๐๐.
For nonintegrable distributions there is a complication: the projected system in๐ non-hamiltonian. The reduced equations are of the form ๐ท๐
๐๐ = ๐น (๐), where ๐นis a โgyroscopic typeโ force, i.e. ๐(๐น (๐),๐) โก 0. This force comes from a โ(๐ฝ,๐พ)โterm combining the momentum map of the ๐บ-action in ๐ and the curvature of theconnection (see eg [62]). The problem becomes more interesting and is related tothe query 5.1. We also thank the referee for pointing out that a duality existsbetween a Levi-Civita type connection and an almost-Poisson bracket, in the moreabstract context of mechanics in Lie algebroids [9,49,67].
6.2. Non-equivalent connections with the same straightest paths.The example of Terra and Olive generates interest in finding in what circumstancesthe weak equivalence does not imply the strong. We saw that there is a need forhaving โroomโ for at least three dimensions between ๐ธ and ๐ธ + (๐ธ,๐ธโ. In suchcases, are there consequences for the equivalence problem? For hamiltonization?
CARTAN MEETS CHAPLYGIN 41
6.3. Perspectives on Hamiltonization. Chaplyginโs reducing multipliersmethod was geometrized in [21]. We mentioned in the introduction that it istied intimately to hamiltonization and integrability. Currently, the almost Poissondescription of nh systems is how one starts the quest for hamiltonization [19].Hamiltonization is done usually after reduction of symmetries and a change of timescale depending on reduced variables (in Chaplygin systems the base manifold).Time change is related to the existence of a smooth invariant measure.
Unfortunately, our impression is that Cartanโs equivalence approach (changingthe metric) is of no avail for these purposes (we hope to be wrong). At any rate,we are preparing a sequel paper, in which Cartan moving frames in ๐ are extendedto moving frames in ๐ โ๐ and then used to produce a simple way to obtain theMaschke/Schaft bracket [24,72,79]7.
Necessary and sufficient conditions for the existence of invariant measures canbe found in eg. [25,43]. Compiling a list of the known nh systems with a smoothinvariant measure is in order, together with information about which ones have beenhamiltonized and their integrability. For a rolling rigid body (several versions ofthe problem) results on the existence of an invariant measure and Hamiltonizationhave been gathered in the form of tables [15,16,18]. In these papers, a conformallyHamiltonian representation is found for a reduced system on ๐ โ๐2. See [46] fora negative case. Chaplygin sphere was shown to be Hamiltonizable in [13] andthe methodology explained in [17]. A glimpse of the subject (we apologize for themany omissions since the theme is booming) can be found in references such as (inrandom order) [6,7,34,39โ42,44,48,53,56โ61,85,86]. There are many more andmore will be coming! Some recent information follows, kindly given to us by LuisNaranjo and Ivan Mamaev.
6.3.1. Possibility of hamiltonization in the first level (๐ฟ๐๐(๐ธ) = โณ โ ๐ โ๐).Sufficient conditions in order for a Chaplygin system to be hamiltonizable at thefirst level ๐ โ(๐โ๐บ), similar to Stanchenkoโs [83], were found by Naranjo and Mar-rero (personal communication). For Chaplyginโs sphere, such conditions are notsatisfied. It is believed that, in general, one cannot achieve hamiltonization at thefirst level. However, this question should be formulated in a clearer manner8.
6.3.2. Obstructions to Hamiltonization. The conformal Hamiltonization of abivector ๐ต seems a very interesting problem. Given a function ๐ป, a mandatoryrequirement is the existence of an invariant measure for ๐๐ป = ๐ต โ ๐๐ป. In the caseof nh systems, some obstructions to Hamiltonization are discussed in [14,19,20].
6.3.3. Turning integrals into Casimirs. The underlining idea is to modify theMashke van der Schaft bracket [72,79] to make first integrals into Casimirs of thereduced system [47].
7Adding semi-basic bivectors that do not affect the dynamics (a la Stanchenko [83]) willbecome evident in this approach. The experts will agree that such preliminary constructions,
before reduction, can help hamiltonization.8 For instance, in the case of Chaplyginโs sphere, if it is homogeneous, then the system
on โณ can be understood as an invariant subsystem of a Hamiltonian system, namely, that of a
homogeneous sphere moving in ๐ 2 without constraints. In this case ๐ฟ๐๐(๐ธ) is a level set of thefirst integrals of the Hamiltonian system on ๐ โ๐.
42 EHLERS AND KOILLER
6.4. Themes from Cartanโs ICM1928 paper. One wonders why Cartandid not discuss the weak notion of equivalence. We believe that he was concernedmainly with ๐ท๐ธ
๐๐ in the strongly nonholonomic case (1 step, ๐ธ + (๐ธ,๐ธโ = ๐๐).Recall, we saw that in this case, for the preservation of the partial nh-connectionon ๐ธ, the metric can be changed outside ๐ธ as long as the orthogonal space ๐น = ๐ธโฅ
with respect to the original metric in ๐ stays the same. Cartan was interested ina complementary partial connection on ๐น . For this he shows in ยง 6 to 8 that thereis an intrinsically defined metric in ๐น such that the partial connection ๐ท๐น (nowprojecting to ๐น ) plus a suitable 2-tensor has zero torsion.
Moreover, it seems to us that at their time Cartan and contemporaries antici-pated a construction that is nowadays popular in robotics, control theory, computervision and statistics on manifolds (see eg. [74,81]). Namely, obtaining the devel-opment of a curve in ๐ over a fixed tangent space using the connection. The โtrueโmechanics at a point ๐ โ ๐ is transported to a mirror Euclidian setting. The in-ertial motion in ๐ becomes a motion with Euclidian metric but with an appliedforce (using parallel transport of an object at ๐). We think it will be rewardingto revisit those aspects of Cartanโs paper, not discussed in this review, and justtouched upon in [63].
Acknowledgments. We thank Profs. Waldyr Oliva and Glaucio Terra forsending us their example prior to publication; thanks to Profs. Luis Naranjo andIvan Mamaev for up-to-date information about the status of the Hamiltonizationproblem. We thank Profs. Vladimir Dragovic and Bozidar Jovanovic for the invi-tation to submit a contribution to the special volume in memory of Sergey Alex-eyevich Chaplygin (1869โ1942) on the occasion of his 150th anniversary. J. Koillerwas supported by a UFJF visiting fellowship.
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46 EHLERS AND KOILLER
ะะะ ะขะะ ะกะ ะะะ ะงะะะะะะะะ
ะ ะตะทะธะผะต. ะะฐ ะะตััะฝะฐัะพะดะฝะพะผ ะบะพะฝะณัะตัั ะผะฐัะตะผะฐัะธัะฐัะฐ 1928, ะะฐััะฐะฝ jะต ะฝะฐะฟะพ-ะผะตะฝัะพ ะดะฐ ัะตะณะพะฒ โะผะตัะพะด ะตะบะฒะธะฒะฐะปะตะฝัะธjะตโ ะดะพะฒะพะดะธ ะดะพ ะธะฝะฒะฐัะธjะฐะฝัะธ ะฝะตั ะพะปะพะฝะพะผะฝะพะณ ัะธ-ััะตะผะฐ ะฝะฐ ะบะพะฝัะธะณััะฐัะธะพะฝะพะผ ะฟัะพััะพัั ๐ ัะฐ ะะฐะณัะฐะฝะถะธjะฐะฝะพะผ ะพะดัะตัะตะฝะธะผ ะบะธะฝะตัะธั-ะบะพะผ ะตะฝะตัะณะธjะพะผ [29]. ะะฐััะฐะฝ jะต ะดะฐะพ ะบะฐัะฐะบัะตัะธััะธะบั ะฟัะพะผะตะฝะต ะผะตััะธะบะต ะฒะฐะฝ ะดะธัััะธ-ะฑััะธjะต ะฒะตะทะฐ ๐ธ โ ๐๐ ะบะพjะฐ ะพััะฒะฐะฒะฐ ะฝะตั ะพะปะพะฝะพะผะฝั ะฟะพะฒะตะทะฐะฝะพัั ๐ท๐๐ = Proj๐ธ โ๐๐ ,๐,๐ โ ๐ธ, ะณะดะต jะต โ๐๐ ะะตะฒะธ-ะงะธะฒะธัะฐ ะฟะพะฒะตะทะฐะฝะพัั ะฝะฐ ๐ ะธ Proj๐ธ ะพััะพะณะพะฝะฐะปะฝะฐ ะฟัะพ-jะตะบัะธjะฐ ะฝะฐ ๐ธ. ะะฒะดะต ัะฐะทะผะฐััะฐะผะพ ะฝะฐะฒะตะดะตะฝะธ ะฟัะพะฑะปะตะผ ะตะบะฒะธะฒะฐะปะตะฝัะธjะต ะฝะตั ะพะปะพะฝะพะผะฝะธั ะฟะพะฒะตะทะฐะฝะพััะธ ะทะฐ ะงะฐะฟะปะธะณะธะฝะพะฒะต ัะธััะตะผะต [30,31,62]. ะขะฐะบะพัะต ัะฐะทะผะฐััะฐะผะพ ะธ ะฟัะธะผะตั- ะผะฐัะตะผะฐัะธัะบะธ ะดัะฐะณัั! - ะบะพะณะฐ ัั ะฟัะพะฝะฐัะปะธ ะะปะธะฒะฐ ะธ ะขะตัะฐ [76]. ะขั ะธะผะฐะผะพ ะฒะธัะตัะปะพะฑะพะดะต (ะดะฐะบะปะต ะธ ะฒะธัะต ะผะพะณััะฝะพััะธ) ะบะพัะธััะตัะธ ัะปะฐะฑะธjั ะตะบะฒะธะฒะฐะปะตะฝัะฝะพัั - ััะฒะฐjััะต ัะฐะผะพ ะฝะฐjะฟัะฐะฒัะธ ะฟััะตะฒะธ: ๐ท๐๐ = 0. ะะฟะฐะบ, ััะฐะถะตัะต ะฟัะธะผะตัะฐ ะบะพjะธ ัั ัะปะฐะฑะพ,ะฐะปะธ ะฝะต ัััะพะณะพ ะตะบะฒะธะฒะฐะปะตะฝัะฝะธ ะดะพะฒะพะดะธ ะดะพ ะฟัะตะพะดัะตัะตะฝะพะณ ัะธััะตะผะฐ jะตะดะฝะฐัะธะฝะฐ, ััะพะดะฐjะต ะธะฝะดะธัะธjั ะดะฐ ัั ัะฐะบะฒะธ ัะธััะตะผะธ ัะตัะบะธ. ะะธ ะฟะพะบะฐะทัjะตะผะพ ะดะฐ ัะต ะดะฒะฐ ะฟัะธัััะฟะฐะฟะพะบะปะฐะฟะฐjั ั ัะปะตะดะตัะธะผ ัะปััะฐjะตะฒะธะผะฐ: i) ะะธัััะธะฑััะธjะฐะผะฐ ัะฐะฝะบะฐ 2. ะะฐ ะฟัะธะผะตั, ัะะฐััะฐะฝะพะฒะพะผ ะฟัะธะผะตัั ััะตัะต ะบะพjะฐ ัะต ะบะพัััะฐ ะฟะพ ัะฐะฒะฝะธ ะฑะตะท ะบะปะธะทะฐัะฐ ะธ ัะฒะธjะฐัะฐะบะฐะดะฐ ะธะผะฐะผะพ (2,3,5) ะดะธัััะธะฑััะธjั; ii) ะะฐะดะฐ jะต ะดะธัััะธะฑััะธjะฐ ัะฐะฝะบะฐ 3 ะธะปะธ ะฒะธัะต,ะฝะตะพะฟั ะพะดะฐะฝ ััะปะพะฒ ะดะฐ ัะต ะฟัะธัััะฟะธ ัะฐะทะปะธะบัjั jะต ะดะฐ jะต ะบะพัะฐะฝะบ ะพะด D ั D+[D,D]ะฑะฐัะตะผ 3. ะะฒะพ ะธัะบััััjะต ะผะพะณััะฝะพัั ะฝะฐะปะฐะถะตัะฐ ะฟัะธะผะตัะฐ ั ัะปััะฐjั (3,5) ะดะธัััะธ-ะฑััะธjะฐ, ะบะฐะพ ะบะพะด ะงะฐะฟะปะธะณะธะฝะพะฒะต ััะตัะต. ะะฐะบะปะต ะฟัะตะดะธะฒะฐะฝ ะฟัะธะผะตั (3,6) ะดะธัััะธะฑััะธjะตะบะพjะธ ัั ะดะพะฑะธะปะธ ะะปะธะฒะฐ ะธ ะขะตัะฐ jะต ะผะธะฝะธะผะฐะปะฐะฝ.
Mathematics Department (Received 16.01.2019.)
Truckee Meadows Community College (Revised 22.05.2019.)Reno (Available online 25.06.2019.)
Departamento de Matematica
Universidade Federal de Juiz de ForaJuiz de Fora
Brazil