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21
320
Т. С. Ратью, Нгуен Тьен Зунг
ЧЕБЫШЕВСКИЙ СБОРНИК
Том 21. Выпуск 2.
УДК 514.85+531.2+531.012 DOI 10.22405/2226-8383-2020-21-2-320-340
Интегрируемые системы в планарной робототехнике1
Т. С. Ратью, Нгуен Тьен Зунг
Ратью Тюдор Стефан — доктор наук, профессор, Школа математических наук, Шанхайский университет Цзяотун; Отдел математики, Университет Женевы; Федеральная политехническая школа Лозанны (г. Шангхай, Китай; г. Женева, Швейцария; г. Лозанна, Швейцария).
e-mail: ratiu@sjtu.edu.cn
Зунг Нгуен Тьен — доктор наук, профессор, Математический институт Тулузы (г. Тулуза, Франция).
e-mail: tienzung@math.univ-toulouse.fr
Аннотация
Основная цель данной статьи — исследовать коммутирующие потоки и интегрируемые системы на конфигурационом пространстве плоских шарнирных многоугольников. Наше исследование приводит к определению естественной формы объема на каждом конфигурационном пространтве плоских шарнирных многоугольников, понятию перекрестных произведений интегрируемых систем, а также к понятию мульти-намбу интегрируемых систем. Первые интегралы наших систем являются функциями типа Ботта-Морса, которые могут быть использованы для изучения топологии конфигурационных пространств.
Посвящается Анатолию Т. Фоменко по случаю его 75-летия
Ключевые слова: плоские шарнирные многоугольники, коммутирующие потоки, нега-мильтонова интегрируемость, форма объема, структура Намбу, перекрестное произведение интегрируемых систем
Библиография: 28 названия. Для цитирования:
Т. С. Ратью, Нгуен Тьен Зунг. Интегрируемые системы в планарной робототехнике // Чебы-шевский сборник, 2020, т. 21, вып. 2, с. 320-340.
1 Работа Т.С. Ратью частично поддержана грантом Национального фонда естественных наук Китая номер 11871334 и NCCR SwissMAP грантом Национального научного фонда Швейцарии, работа Нгуен Тьен Зун-га частично поддержана программой сотрудничества "High-End Foreign Experts"Шанхайского университета Цзяотун.
CHEBYSHEVSKII SBORNIK Vol. 21. No. 2.
UDC 514.85+531.2+531.012 DOI 10.22405/2226-8383-2020-21-2-320-340
Integrable systems in planar robotics
T. S. Ratiu, Nguyen Tien Zung
Ratiu Tudor Stefan — Doctor of Sciences, Professor, School of Mathematical Sciences, Shanghai Jiao Tong University Section de Mathématiques, Université de Geneve, Ecole Polvtechnique Fédérale de Lausanne (Shanghai, China; Geneve, Switzerland; Lausanne, Switzerland). e-mail: ratiu@sjtu.edu.cn
Zung Nguyen Tien — Doctor of Sciences, Professor, Instituí de Mathématiques de Toulouse (Toulouse, France).
e-mail: tienzung@math.univ-toulouse.fr
Abstract
The main purpose of this paper is to investigate commuting flows and integrable systems on the configuration spaces of planar linkages. Our study leads to the definition of a natural volume form on each configuration space of planar linkages, the notion of cross products of integrable systems, and also the notion of multi-Nambu integrable systems. The first integrals of our systems are functions of Bott-Morse type, which may be used to study the topology of configuration spaces.
Dedicated to Anatoly T. Fomenko on the occasion of his 75th birthday
Keywords: planar linkage, commuting flows, non-Hamiltonian integrability, volume form, Nambu structure, cross-product of integrable systems
Bibliography: 28 titles. For citation:
T. S. Ratiu, Nguyen Tien Zung, 2020, "Integrable systems in planar robotics" , Chebyshevskii sbornik, vol. 21, no. 2, pp. 320-340.
1. Introduction
This paper is a preliminary investigation of natural commuting flows and integrable systems on the configuration spaces of linkages in geometry, machinery and robotics, especially planar linkages. Our motivation is manifold:
• Firstly, linkages appear everywhere in mechanical systems (see, e. g., [18, 26]). Moreover, commuting flows may be viewed as "independent components" in a mechanical system and play an important role in robotics and motion control: the commutativitv makes it easier to compute and control the motions. In fact, one may observe many commuting flows in practice. For example, the excavator in Figure 1 (left), borrowed from Servatius et al. [23] with permission, has two sliding mechanisms, which commute with each other. This excavator may be represented by a planar linkage in Figure 1 (right), where each sliding mechanism is replaced by two connecting edges. The configuration space of this linkage admits an integrable system having two commuting vector fields. If one allows some of the lengths of the edges to be variable, then the configuration space has higher
dimension, and those adjustable lengths are first integrals of the integrable system in question.
•
(around diagonals) found by Kapovich and Millson [15], which are a special class of integrable
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Figure 1: Tho excavator and its planar linkage representation, after [2.3].
Hamiltonian systems, and the singularities of these systems have been described in a recent paper by Bouloc [3]. We want to see to what extent these results can be generalized to other situations, especially planar linkages, and also 3D linkages which are more general than just polygons. It turns out that, for 2-dimensional (2D) polygons, we have a simple construction which may be viewed as an analogue of Kapovich-Millson systems: given a 2D n-gonal linkage (n > 4), just choose arbitrary [(n — 3)/2] ([•] denotes the integer part of a real number) non-crossing diagonals which decomposes it into [n/2] — 1 quadrangles and at most one triangle (one if n is odd and zero if n is even). Then we get a non-Hamiltonian integrable system on the configuration space of type ([n/2] — 1, [(n — 3)/2]), i.e., with [n/2] — 1 commuting vector fields and [(n — 3)/2] common first integrals; ([(n — 3)/2] + [n/2] — 1 = n — 3 is the dimension of the configuration space). Each diagonal gives one first integral (its length, or rather its length squared to be smooth), each quadrangle has one degree of freedom and gives rise to one vector field, and they all commute with each other (see the rest of this paper for details). For example, when n = 7, the configuration space of a generic planar heptagonal linkage is a 4-dimensional manifold, on which we have integrable systems of type (2,2), i.e., with two commuting vector fields and two common first integrals, by this construction.
c
Figure 2: An integrable system on a the 4-dimensional configuration space of a 7-gon linkage: the two "4-bar mechanisms" ABCG and CGFD provide two commuting vector fields and the square lengths |CG|2 and IDF|2 are their smooth common first integrals.
Recall that the configuration space of 3D linkages admits a natural symplectic structure [19, 15, 10]. However, for 2D linkages, a priori, we have neither a natural symplectic structure, nor Hamiltonian systems; however, we have Nambu structures and so our vector fields can be
constructed in the spirit of Nambu mechanics [22]. Recall that a Nambu structure of order q on a manifold is a g-vector field which spans a (singular) integrable ^-dimensional distribution and which may be viewed as a leaf-wise contravariant volume element on the corresponding singular q-dimensional foliation; see, e.g., [4, 20]. Contravariant volume elements are particular cases of Nambu structures and they were used by Nambu in [22] as a way to generalize Hamiltonian mechanics,
whence the name.
•
was announced many years before by Thurston [24] but without a full proof. This theorem states that, given any closed (compact without boundary) smooth manifold, there is a planar linkage whose configuration space is diffeomorphic to a disjoint union of a finite number of copies of that manifold. Thus, by studying natural integrable systems on configuration spaces of planar linkages, we can obtain interesting integrable non-Hamiltonian systems on any closed manifold. The theory of integrable non-Hamiltonian systems (in the sense of Bogovavlenskv [2]) is relatively new compared to integrable Hamiltonian systems, especially when it comes to their topological and geometric properties (see [27] and references therein), so this huge class of integrable non-Hamiltonian systems coming from planar linkages will be very useful for the development of the theory. We summarize
below the main results of the paper.
The first result of this paper, presented in Section 2, is that every configuration space of planar linkages admits a natural volume form. This volume form, together with a choice of diagonals in the linkage, gives rise to Nambu structures on the configuration space.
In Section 3 we give a simple general method for constructing integrable systems on configuration spaces of planar linkages, by decomposing such spaces into cross products of smaller configuration spaces, and lifting integrable systems from these smaller spaces. In particular, we introduce the notion of cross products of integrable systems in this section and, as a special case of this cross product construction, we find, what we call, multi-Nambu integrable system,s.
In Section 4, the last section of this paper, we list some open questions and final remarks. In particular, we mention that our first integrals are Bott-Morse functions on many configuration spaces, which are helpful in topology. For example, in the case of hexagons, by using these Bott-Morse functions, one immediately gets the fact that the corresponding configuration spaces belong to a special class of very well-studied 3-manifolds named graph-manifolds by Waldhausen [25]; this fact also fits well with Fomenko's topological theory of integrable systems on 3-manifolds [7].
2. Configuration spaces of planar linkages 2.1. Definition of the orbit and configuration spaces
In this paper, a planar linkage C consists of:
• A finite graph Q = (V, S), where V denotes the set of vertices and S denotes the set of edges
of Q. For any two ver tices A,B e V there is at most one edge connecting t hem, denoted AB or BA. An edge is also called a bar in C.
• Each edge has an associated positive length, i.e., there is a length function
I : S ^ R+ :=]0, which specifies the length £(AB) of each edge AB e S.
• A subset B C V of vertices, which may be empty, called the set of fixed vertices or based points, which comes together with a fixing map
p : B ^ R2
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that attaches C to the Euclidean plane R2. The vertices in V \ B are called movable.
The pair (Q, B), where the lengths and the fixing are not specified, is called the linkage type of C.
A realization of a planar linkage C = (V, S, l, B, ft) is a map
r : V ^ R2
which respects the based points and the bar lengths. In other words,
Ir(A)r(B)I = l(AB),
for any A,B E V, and the restriction of r to the set B of base vertices coincides with ft. Of course,
the length condition means that r can be extended to a map from the whole graph Q to R2, that
maps each edge of Q to a straight line segment of the same length. In a planar realization of a
R2
injective. The crossing of bars may be realized in practice by laying them on parallel planes.
The space of all realizations of a planar linkage C is denoted bv Oc\ it is also called the orbit space of C. Two different realizations ri, r2 of C are said to be isometric if there is an orientation and length preserving map 0 : R2 ^ R2 (i.e., 0 E SE(2) the Euclidean group) such that 0or1 = r2.
C
by Mc.
If B = 0 is empty then the realizations of C can move freelv in R2; in this case SE(2) acts on Oc, the action is free if C has at least one bar, and we have
Mc = Oc/SE (2).
If B consists of exactly one base point then the configurations cannot be translated freely in R2 but can turn around B; in this case we have
Mc = Oc/SO(2).
BC
if and only if they coincide; in this case we have
Mc = Oc-
Let Gc be the group {e}, 5*0(2), or SE(2), depending on whether B has at least two, one, or zero fixed vertices. With this notation, in view of the considerations above, we have Mc = Oc/Gc.
Note that when studying the configuration space of a linkage, we can choose between the version with at least 2 base points and the version without any base point. Indeed, if C = (V, S,l, B,ft), where B has at least 2 points, then we can add all edges BC with vertices B,C E B, together with their lengths £(BC) = Ift(B)ft(C)|, then forget these base vertices, and we get a linkage with empty base. This new linkage, now without any base vertices, has the same configuration space as the old CC
R2
C
C
of four vertices A, B, C, D and four edges with four fixed lengths l1 = IAB|, l2 = IBC l3 = |C^|,l4 = IDAI, and no base point. Consider the same quadrangle, but now with two base points A and D, and only three free edges AB, BC, CD (the edge DA can be removed because A and D are already fixed). We get a new linkage C2, whose configuration space is exactly the same as the configuration space of C1 (see Figure 3).
Figure 3: Two versions of a quadrangle (with and without base points), also called a 4-bar mechanism, with the same configuration space.
Remark 1. When talking about a linkage one often assumes that it is connected (i.e., the corresponding graph is connected). However, for our study, it is convenient to allow linkages which are disconnected. It is clear that if £ is a disjoint union of two linkages C1 and C2, then its base is also the disjoint union of B1 and B2 and its orbit space is simply the direct product of the two corresponding orbit spaces:
0Cluc2 = 0Cl x O^.
The formula for the configuration space of C1 U C2 is more complicated. Namely, we have six cases:
(i) If both B1 and B2 have at least two vertices, then Oc = Mc = Oc1 x 0C2 = MC1 x MC2.
(ii) If B\ has at least two vertic es and B2 consists of exactly one vert ex, then Oc = Mc = Oc1xOc2 but Mc1 x Mc2 = Mc/SO(2), the SO(2)-action being on the second factor.
(iii) If Bi has at least two vertic es and B2 = 0, then Oc = Mc = GC1 x 0C2 but MCl x MC2 = Mc/SE(2), the SE(2)-action being on the second factor.
(iv) If both B1 and B2 have each exactly one vert ex, then Oc = Mc = 0Cx x 0C2 but MC1 x MC2 = Mc/(SO(2) x £0(2)), each SO(2) acting separately on its corresponding factor.
(v) If B1 consists of one vertex and B2 = 0, then Mc = Oc/SE(2) = GCl xso{2) 0C2) so we get fibrations Mc ^ 0C1) Mc ^ 0C2 whose fibers are 0C2 and 0Cx, respectively. Thus we get an SE(2)-principal fiber bundle Mc ^ MC1 x MC2.Th.e SE(2)-action is induced from the following SE(2)-action on the linkage: fix a second vertex on £1 and let SE(2) act on C2. The resulting action on Mc is free and the quotient Mc/SE(2) is MC1 x MC2.
(vi) If Bi = 0 Mid B2 = 0, we proceed as in the previous case, except that we now fix two vertices in C^ We get an SE(2)-principal fiber bundle Mc ^ MC1 x MC2.
2.2. The dimension
Consider a linkage C with s fixed vertices B1,..., Bs and n movable vertices A1,... ,An. If there are no bars, then each Ai can ta,ke any posit ion in R2 and we have
0{E,B,p) = R2 x • • • x R2 ^ R2n;
G(£ $£) denotes the set of all realizations of the planar linkage C but with variable lengths of the bars. If the linkage has q bars, then we have a joint length map
M = (m ) : 0(s ,B,P) ^ R+,
where fa ^s half the ^^^^^h square of bar number i in C. (We take the square to have smooth functions.) Clearly this map descends to the quotient M(£ ) = 0(£$$)/Gc\ we denote it by the same symbol a. because there is no danger of confusion.
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For each fixed value of a q-tuple of lengths (l1,... ,lq) we have
Oc = M-1(l2/2,.. .,l2q/2) = {L E % M I pi(L) = l2/2, V1 < i < q},
and a similar formula for Mc.
A bar in a linkage type (Q, B) is called redundant if its length is dependent on the lengths of the other bars. In other words, edge i is redundant if the function is functionally dependent on the functions ..., ,..., For example, consider a complete link of 4 vertices and 6
edges. Then one of the edges (it does not matter which one) is redundant. Eliminating a redundant edge does not change the dimension of the configuration space. So, to compute the dimension of the configuration space of a linkage C, we may assume that its linkage type (Q, B) has all bars non-redundant. In this case, we have
dim Mc = 2n — q
for any regular (generic) value of the q-tuple of edge lengths. (This formula is essentially the classical so-called Chebvchev-Griibler-Kutzbach criterion for the degree of freedom of a kinematic chain; see, e.g., [18].) In particular,
q < 2n
in a linkage all of whose edges are non-redundant, i.e., the number of edges (bars) is at most twice the number of movable vertices. A linkage all of whose bars are non-redundant is called a non-redundant linkage.
2.3. The contravariant volume element
We want to define a natural volume multivector field on Mc (a "contravariant volume also called a Nambu structure of top order). The strategy is to construct it from a volume multivector field on Oc and on Gc (recall that Mc = Oc/Gc)-
First, recall that on any orientable manifold M, any choice of volume form uniquely defines a top degree nowhere vanishing multi-vector field Am by requiring it to satisfv Am J 0m = 1- We call such a top degree multi-vector field a contravariant volume.
Second, we note that on any Lie group G there always is a left-invariant contravariant volume. Indeed, if {{1,... ,£p} is a basis of its Lie algebra g, extend these vectors to left invariant vector fields ..., and form the contravariant volume A ... A
Third, if the Lie group G acts locally freely on a dense set of the manifold M (i.e., all its isotropv subgroups at the points on this dense sent are discrete), there is always a Nambu structure of degree p = dim G on M. Indeed, if {£i,..., (p} is a basis of g, form the corresponding infinitesimal generator vector fields X\,..., Xp, then nc := Xi A ... A Xp is a Nambu structure on M tangent to the generic O-orbits.
We return now to our problem of constructing a contravariant volume on Mc. The Lie group Gc equals {e}, 5*0(2), or SE(2), depending on whether B contains more than two, one, or zero vertices. Let Ag£ be the contravariant volume on Gc which equals hence
• AG£ = 1 if Gc = {e},
• Ag£ = Z = a Jg — bthe infinitesimal generator of counterclockwise rotation in the plane R2 = {(a, b) I a,b e R}, if Gc = SO(2),
• Aqc = Z A A where (x, y) me the coordinates of R2 in SE(2) = SO(2) x R2.
This contravariant volume is given in terms of the obvious basis elements of the Lie algebra of Gc and, in view of the previous discussion, since Gc acts freely on a dense subset of Oc/, produces a Nambu multivecor field nc on Oc, tangent to the generic Oc-orbits on Oc.
Next, we construct a contravariant volume on Le t ,... : Oc ^ ]0, oo[ be the collection of the squared length maps given by ^(x) := if /2, where I is the length of the edge i and x = ((x!,yi),... (xn,yn)) G Oc- Define
/1 i \ ( & & д д д Л0с := (d^i Л ... Л )J — Л — Л • • • Л —- ^ — Л —
\ОХ1 оу1 охп ох1 оуг
.
This contravariant volume is unique up to the ordering of the bars in the linkage. Note that Aqc is invariant under the Gc-action.
Finally, let Qoc be the volume form on Oc dual to i.e., KqcJQoc = 1- Then ncJQqc '1S a basic form because of Gx-invariance of the objects in this expression. So this form projects to a form Qm^ on M^. Let Am£ be its dual multivector field, i.e., A^JQm£ = 1. This is the volume multivector field on Mc- We have proved the first part of the following statement.
Theorem 1. (i) For each non-redundant linkage C7 its orbit Oc and configuration Mc spaces admit natural canonical contravariant volumes Aqc and Ac, respectively. These canonical contravariant volumes are uniquely determined by a choice of ordering of the edges of of C: if we change the ordering by a permutation a of parity then these volume multivector fields are multiplied by (—1)|ct|.
(ii) Consider two vertices C, D of a non-redundant linkage C. Denote by = ICDl2/2 the half squared CD-length function on Oc and Mc- Assume that is not a constant function, so that the linkage C1 obtained from C by attaching to it an additional bar CD with the given length ¿(CD) = c is non-redundant. Then Oc1 (respMc 1) is the level set given by = c2/2 in Oc (respMc) and we have dim Oc1 = dim Oc — 1 dim Mc1 = dim Mc — 1, and
Aocx = d^iJ Aqc and Ac1 = d^J Ac
restricted to these level sets.
The proof of (ii) is a direct verification.
Figure 4: The snake: the canonical contravariant volume elements on its orbit space and configuration space are independent of its lengths.
Example 8. (The snake). Consider the linkage C which consists of one base vertex A0 fixed at the origin of R2 (A0 = (0,0)), k movable vertices A1 = (x1,y1),... = (xk,yu) G R2, and k fixed lengths IAi-1Ai | = Ci > 0 (i = 1,..., k). Then it is easy to check that Oc = Tk with periodic coordinates $i = arctan((^ — yi-1 )/(xi — x—)) (mod 2^), and the canonical contravariant volume element on Oc is
d d
= W, A...A ^,
which does not depend on the lengths ci,..., The configuration space Mc is isomorphic to Tk-1 with periodic coordinates $i = $i+1 — $i (mod 2^), and its canonical contravariant volume element is (up to a sign)
d d
Ac = —- A ... A
дв! двк_
1
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Figure 5: Left: An 1D0F spider, with a fixed "head" B0, fixed "feet" Bx,...,Bm ££body" O, and "legs" AiBi (i = 1,... ,n). Right: "Strandbeest", invented by the artist Theodorus Jansen in 1990s, which can be used as legs in mechanical animals.
Example 9. (Curve-drawing linkages) Linkages with one degree of freedom (1DOF for short) are those whose configuration spaces have dimension 1: generically, such configuration spaces are a finite union of closed curves. A planar 1DOF linkage is also called curve-drawing linkage because, after attaching it to a plane at the fixed vertices (if there are no fixed vertices then one can choose a bar and fix its two ends without changing the configuration space of the linkage), a movable vertex on the linkage will draw an algebraic curve when the linkage moves. By a famous en the linkage moves. By a famous 1876 theorem of Kempe [17], any algebraic curve can be obtained this way by some 1DOF planar linkage. Simple modern proofs of this theorem can be found in many places; see, for example, [5, 12]. Well-known examples of 1DOF linkages include quadrangles, spiders, the Strandbeest (Figure 5), and so on. A contravariant volume element on a manifold of dimension 1 is simply a vector field. Thus, our construction shows that on the configuration space of each curve-drawing linkage there is a unique (up to orientation) natural canonical vector field. Such vector fields will be important in our construction of integrable systems on general configuration spaces of planar linkages.
Example 10. (Zero-dimensional space). Consider a linkage C abc formed by a triangle ABC with 3 fixed lengths (i.e., 3 bars) IABI = c, IBCI = a, ICAI = b, without base points. This linkage is, of course, rigid, so its configuration space is just two points (corresponding to two possible orientations of the triangle on the plane). The contravariant volume element is just a number for each point on the configuration space. One of the numbers is twice the area of the triangle ABC and the other is its opposite (the same absolute value, but with negative sign). In particular, in the degenerate case, when the three points A, B, C are aligned, the contravariant volume element vanishes. If we fix A, B and erase the bar AB from Ca b c, then it becomes another rigid linkage C a B c same configuration space and the same contravariant volume element up to a
natural isomorphism. Now add to C a bc or CA Bc a vertex D and a bar CD with some length ICDI = d > 0. The configuration space of the new linkage is now two circles, and the corresponding
contravariant volume element is ±25—, where 0 (mod 2n) is the angle formed by the vector with ct , and S is the area of the triangle ABC.
The above simple example illustrates the following two phenomena:
• (Effect of fixing a base) If C' is obtained from C by fixing two vertices A, B of a bar AB and then removing that bar, where C is a connected linkage without base points, then not only are Mr
and Mc naturally isomorphic, but under this isomorphism we also have Ac = AC'. (This fact is not surprising, because fixing A, B amounts to contracting with the volume element of SE(2) in Ac
• (Effect of homothety) If we multiply the length of every edge of a linkage C by a positive constant c, then we get a linkage cC homothetic to C, and McC is naturally isomorphie to Mc. However, under this natural isomorphism we have the following homothety formula, whose proof is a straightforward verification:
Acc = c2(|£|-(|v|+|s|))Ac (1)
where |V| — IBI is the number of movable vertices and I£I is the number of edges, in the case when IBI > 2 (i.e., when Mc = 0c) and C is connected. In the case when B = 0 and C is connected, the above formula must be modified as follows:
Acc = c2(|£|-|v|+1)Ac (2)
2.4. Nambu structures for linkages with marked diagonals
Consider a linkage C. If two vert ices A, B of C are not connected by a bar of C, then we say that AB is a diagonaI in C. Choose a set V of diagonals of C (which may be empty) and mark the elements of V\ in this case we say that they are marked diagonals.
Given a linkage C with marked diagonals V = [A1B1, ... ,AkBk} (k > 0), we can define the following Nambu structure on the configuration space Mc:
Ac,v = (d^i A ... A d^k )JAc, (3)
where = IAíBíI2/2. (When V = 0 is empty then, of course, Ac,0 = Ac). We will use such Nambu structures for the construction of our integrable systems.
3. Integrable systems on linkage spaces 3.1. Cross products of integrable systems
In this subsection, we present a general method for constructing new integrable systems from other integrable systems. We call this technique the cross product construction.
We begin by recalling the notion of non-Hamiltonian integrable systems (fl, 27]; for an Action-Angle Theorem in this setting, see [28]). A vector field X on an ^dimensional manifold M is called integrable if there exist p vector fields X1 = X, X2,... ,XP and q functions F1,..., Fq on M, p + q = n, such that [X¿, Xj] = 0 fa all i,j = 1,... ,p, X\ A ■ ■ ■ A Xp = 0 almost everywhere, dFf (Xi) = 0 for all í = 1,... ,p and 1 = 1,..., q, and dFx A ■ ■ ■ A dFq = 0 almost everywhere. Such an n-tuple (X\,..., Xp, F\,..., Fq) is called an integrable system of type (p, q).
Consider a family of smooth manifolds Mi of dimensions m, where i belongs to some finite index set I. We assume that there is another finite index set K disjoint from I, such that for each k £ K there is a nonempty subset Sk C and for each i £ Sk there is a given smooth function fi,k : Mi ^ R on M^ Moreover, we ^to assume that for each i £ I there is an integrable non-Hamiltonian system whose functionally independent first integrals are fi,k for k £ Ki, where Ki = {k £ K | i £ Sk}, and whose linearly independent commuting vector fields are X¿,i,...,XiíPi. (By definition, we have Pi + iKil = m).
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Consider now the diagonal manifold
M = {x = (Xi) e П I flk (Xi) = fjk (xj) Vi, j e I, Vk e Si n S* с к}. (4)
íei
Denote bv ж* : Híei M* ^ ^j the projection maps. By abuse of notation, we denote the restriction of Kj to M again bv Kj, j e I. We assume that the functions -Ki*fik — ж* fjk (k e Sí n Sj) have linearly independent differentials on M, except at the points satisfying the obvious relations of the type к* fa — к*fjk = (к* fa — к* fik) + (к*fik — к* fjk); therefore, Misa smooth manifold of dimension dim M = ^ieI п* — ^keK(l^k| — 1) = J2ní — ^2keK l^k| + IK| (because each index k e К creates ISk| — 1 independent equations). Notice that YlkeK l^kI '1S the total number of functions fík and that this number equals Yliei(ni — p*), because for each г we have exactly lK.il = щ — p* functions. Therefore,
dim M = lK l + ^ Pi. (5)
iei
Denote bv Xí,j the horizontal lift of Xí,j to YlíeI Mí (for each i e I and each j < pí). In other words, Xij is the unique vector field on ^ Mí whose projections by k* (j = i) are trivial and whose projection bv ^ is Xí,j. Obviously, the vector fields X*,* commute on Mí. Moreover, they admit the pull-backs of the functions fik as common first integrals. Hence, they also admit the functions к* fa — к* fjk as first integrals and аде, therefore, tangent to M.
For each k e K, define Fk : M ^ R bv
Fk(x) = fik(k(x*)) for some i e Sk. (6)
It is immediate from the definition of M that Fk is well-defined. Moreover, these functions Fk are common first integrals of the commuting vector fields X*,* (i e I,j < pi). Notice that the number of vector fields X*, j is J^^iP^ the number of functions Fk is lKl, and lKl + eiPi '1S exactly the dimension of M (see (5)). Thus, we immediately obtain the following result.
Theorem 2. With the above notations and assumptions, on the diagonal manifold M of dimension lKl + ^eIpi we have an integrable system, of type (YheiPi, lKl), consisting of YlieiPi commuting vector fields Xitj (i e I,j < pi) and lKl first integrals Fk (k e K) given by (6).
The above constructed integrable system on M is called the cross product of the corresponding integrable systems on M*, i e I.
3.2. Multi-Nambu integrable systems
Consider now the following special case of the general cross product construction in the previous subsection.
Assume that for each i £ I, the set [fi,k I k £ Ki] (Ki = [k £ K I i £ Sk}), of functionally independent functions on Mi has exactly ni — 1 elements, where ni = dim Mi and, moreover, each Mi admits a Nambu structure Aj of top order, i.e., a «¿-vector field on M^ (In practice, Aj is often the dual of a volume form on Mi, though the construction still works if Aj has singular points.) Then, for each i £ I, define the Nambu vector field Xi on Mi by the following contraction formula:
Xi = (AkeKi dfi,k UAi. (7)
(Choose an ordering on Ki = [k £ K I i £ Sk} to fix the sign of AkeKidfi,k). Clearly, Xi admits ni — 1 first integrals fi,k (k £ Ki), since IKiI = ni — 1, and hence Xi is a completely integrable vector field on M^ In other words, on each Mi we have an integrable system (Xi, fi,k) of type (1,ni — 1). So, in the notation of the previous subsection, we have pi = 1 and ^ieiPi = III- Thus we get:
Corollary 1. With the above notations and assumptions of this subsection, on the diagonal manifold M of dimension \I\ + \K\ we have an integrable system of type (\I\, \K|); consisting of\I\ commuting vector fields Xi (i G I) and \K\ first integrals F^ (k G K).
In view of this construction, we call the systems obtained in the corollary above multi-Nambu integrable systems.
Remark 1. (Separation of variables and symplectization).
i) The construction above not only gives integrable systems, but also a kind of separation of variables for such systems: the separated variables are on the different spaces Mi (i G I). In classical mechanics, a Hamiltonian system admitting a complete separation of variables, in the sense of Hamilton-Jacobi theory, is automatically integrable. Our systems are a special case of this general idea, even though, a priori, there is no natural (pre)svmplectic or Poisson structure which turns them into Hamiltonian systems.
ii) An exception is the case when dim Mi = 2, for every i G I, and the Nambu structure Aj on Mi is non-degenerate, i.e., it is dual to a volume form w» on Mi. In dimension 2, a volume form is the same as a svmplectic form. In this situation we can pull back every Wi and take their sum to form a presymplectic form
w = ^ iei
on the diagonal manifold M. Thus we obtain a presvmplectic manifold (M, w) together with an integrable Hamiltonian system (Xi (i G I),F^ (k G K)). Each vector field Xi is Hamiltonian: XiJw = — dFfr, where k G K is the only index in Si (i.e, there is only one function on Mi). The rank of w at a generic point on M is 2\K\ in this case.
iii) If dim Mi > 3, then even if \I\ = 1, there is no natural way to turn the Nambu vector field Xi on Mi into a Hamiltonian vector field (with respect to a presymplectic, Poisson, or Dirac structure). The problem is that, in Nambu mechanics, we have n — 1 functions which play equally important roles in the definition of the Nambu vector field, while in Hamiltonian mechanics we have just one function (the energy function). In order to turn a Nambu vector field into a Hamiltonian vector field, one needs to treat these functions differently and declare that among the n — 1 function there is a special one which is more important than the other functions. Of course, this can be done. For example, a Nambu vector field X = (dF1 A... A dFn-1)jA, where A is a Nambu structure of order n on a manifold, may be viewed as a Hamiltonian vector field given by F1 with respect to the Poisson structure n = (dF2 A ... A dFn-1)jA of rank 2. Nevertheless, this is not very natural, especially in view of the diagonal construction.
iv) Moreover, there is no natural way to lift multi-vector fields from Mi to the diagonal manifold
M
is not tangent to M). Thus, even if the Mi are equipped with Poisson structures, M is still not Poisson. We cannot even lift Nambu structures Aj from Mi to M. So the Nambu structures in a multi-Nambu integrable system are not Nambu structures on the ambient manifold, only on its projected components.
3.3. Decomposition of linkages and integrable systems
We apply now the constructions in the previous subsections to the configuration spaces of planar linkages.
Consider a planar linkage C = (V, S,1) without base points. (Recall that any non-trivial linkage with base points is equivalent to some linkage without base points, configuration-wise). Choose a set V of diagonals of C such that V decomposes C into an acyclic semi-rigid connected sum of a
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T. C. PaTbK), HryeH TteH 3yHr
family of non-empty sublinkages Li = (Vi, Si,li) (i £ 1) with their corresponding sets Vi of marked diagonals. This means that the following conditions are satisfied:
• For each i £ 1, denote by Vi the set of vertices of Li. Then Li = (Vi, Si,li) is the sublinkage of L generated by Vi. This means that Si is the subset of S consisting of all edges of L such that all of their vertices belong to V^ likewise, li and Vi are also defined in a natural way. (The diagonals in the family Vi are the dements of V whose vertices belong to Vi.)
• V = UieIVh S = UieISi, and V = UieIVi.
• There is a family J^ (k £ K) of sublinkages of L, which are called joints for the sublinkages L^ in the following sense: if Li H Lj = 0 for i,j £ 1, then there is a k £ K such that Li H Lj = Jk- (It may happen that more than two sublinkges in the family Li (i £ 1) share the same joint; see Figure 6).
• Every marked diagonal is a diagonal of a joint: for any VW £ V there is a joint Jk such that V, W are vertices of J^.
• The joints are disjoint: if k\ = k2 then and Jk2 don't have any common vertex.
• Semi-rigidity: For each k £ K, J^ has at least 2 vertices. If we fix the lengths of the marked diagonals (from the family V) whose vertices lie in J^ then J^ becomes rigid, i.e., its configuration space becomes just one point (if not empty) once these diagonal lengths are fixed.
• Acyclicity: Consider the graph whose vertices are the elements of the index sets 1 and K (of course, these two index sets are assumed to be disjoint) and whose edges are the couples (i,k) £ 1 x K (connecting the vertex i to the vertex k) such that the sublinkage Li contains the joint Jk- Then this graph is a tree (i.e., it has no cycles).
B
Figure 6: Example of a connected sum with two joints and two marked diagonals AF and OP: four sublinkages share the same joint {A,F} and three sublinkages share the same joint {0,P}.
Definition 1. With the above notations and under the above conditions, we say that (L, V) is the acyclic semi-rigid connected sum of its sublinkages with marked diagonals (Li, Vi) (i £ I).
Theorem 3. (i) With the above notations, assume that (C, V) is the acyclic semi-rigid connected sum of the family (C, Ví) (i £ I). Assume that the diagonal square length functions №vw = ¿(VW)2/2 (VW £V) are functionally independent almost everywhere on the configuration space Mc, and that for every i £ I the number of elements in Ví is strictly less than the dimension dimMci of the configuration space of C. Then Mc is the cross product of the family Mci with respect to the marked square length functions:
Mc = {x = (xí) £ n M¿i l Vd(xi) = »d(xj) Vi, j £ I, yd £Vi n V^.
iei
In particular, we have
dim Mc = ^(dim Mc, — |A|) + lVl. iei
(ii) Assume, moreover, that on each configuration space Mc, we have an integrable system, of type ('Pi, qi), where qi = \Ví\ and Pi = dimMct — qi, given by Pi commuting vector fields Xi,1,..., Xi,Pi and \Dil diagonal squared length functions ^d (d £ Vi) which serve as first integrals for the system. Then by the method of cross product construction we get an integrable system of type (J2iei Pi, l^l) on Mc given by the vector fields Xi,j (which are horizontal liftings of the vector fields Xij, i £ I, j < Pi) and diagonal squared, length functions ^d (d £V).
(iii) In particular, if dimMct — \Ví\ = 1 for every i £ I, then on each dimMct we have an integrable system, of type (1, \Vi\) given by \Dil marked diagonal square length functions and the Nambu vector field Xi given by formula (7) with respect to these functions and the natural contravariant volume element on dim Mct ■ Consequently, in this case on Mc we have a multi-Nambu integrable system, of type (lll, Yol).
The proof of the above theorem is straightforward. In assertion (i), the semi-rigiditv and the acyclicitv conditions ensure that, for each point (xí) £ ni Mci satisfying the "equal common lengths" constraints, we can glue the configurations Xi together along the joints to create a configuration x £ Mc, in a unique way up to isometries. Assertions (ii) and (iii) follow immediately from of assertion (i), Theorem 2, and Corollary 1.
Example 11. The case of n-gons. Denote by (A1,..., An) the consecutive vertices of an n-gon. The configuration space Mc in this case has 2(n — 2) — (n — 1) = n — 3 dimensions.
If n is even, we can divide the n-gon (A1... An) into (n/2) — 1 quadrangles by adding (n/2) — 2 diagonals to the linkage; for example, A1A3, A1A5,..., A1A2n-1. Each quadrangle may be viewed as a curve-drawing linkage and thus produces one vector field. So we have an integrable system of type ((n/2) — 1, (n/2) — 2), i.e., with (n/2) — 1 commuting vector fields and (n/2) — 2 common first integrals.
If n is odd, divide the n-gon (A1.. .An) into (n — 3)/2 — 1 quadrangles plus one triangle by adding (n — 3)/2 diagonals to the linkage; for example, A1A3, A1A5,..., A1A2n-1. So we get an integrable system of type ((n — 3)/2, (n — 3)/2).
Remark 2. i) In assertion (iii) of Theorem 3, once the lengths of the marked diagonals are fixed, the sublinkages C become curve-drawing linkages (i.e, linkages whose configuration spaces are 1-dimensional). This shows that curve-drawing linkages play an important role in the construction of integrable systems on the configuration spaces of more general planar linkages.
ii) In a sense, the above construction is quite similar to the construction of bending flows by Kapovich and Millson [15] for obtaining integrable systems on configuration spaces of 3D polygons.
iii) The case when then dimension of Mci is equal to (instead of being greater than) the number of marked diagonals in C for some i £ / is excluded in Theorem 3 for simplicity. However, this case can, in fact, be treated similarly. In order to address this case, we have to work on the level
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T. C. PaTbio, HryeH TteH 3yHr
of orbit spaces instead of configuration spaces, so it is a bit more complicated. Some components for which the number of marked diagonals is equal to dim C still give rise to a non-trivial (and periodic) vector field in the final integrable system on Mc. For example, see the case of the snake (Example 8), which is cut into pieces. Each piece has O-dimensional configuration space, but the configuration space of the snake is a torus with natural rotational commuting vector fields.
The case when a joint Jk consists of just one point is also excluded in Theorem 3 for simplicity. However, this case can also be treated similarly. Such a Jk increases the difference Mc i dim MCi and creates additional rotational vector fields on Mc.
Likewise, the case with base points, also excluded in Theorem 3, can also be treated similarly; just put all the base points in one component.
iv) There is an obvious method to create integrable systems on configuration spaces, which works for any planar linkage. If the dimension of the configuration space Mc is n, then just mark n — 1 diagonals whose length functions are independent on Mc. Then, together with the contravariant volume element on Mc, we immediately get a Nambu integrable system of type (1,n — 1) whose first integrals are squared length functions of the marked diagonals and whose vector field is given by formula (7). However, such "obvious" systems have just one vector field each, so if we want to obtain systems with more vector fields using the cross product method, we have to decompose the linkage C into many components C, the more, the better.
A B
Figure 7: Example of an indecomposable 3DOF linkage.
v) A natural question arises: given a planar linkage L, what is the maximal number 111 of components Li {i £ 1) in a decomposition of L into acyclic semi-rigid connected sums (with any choice of marked diagonals, such that the dimension of the configuration space of each Li is strictly greater than the number of marked diagonals in it)? If this maximal number is 1, then we say
L
configuration space has dimension 3).
vi) In general, it is clear that the number of components in a "valid" acyclic semi-rigid connected sum ("valid" means that dim Mci is strictly greater than the number of marked diagonals in Li) cannot be greater than (|V| — 1)/2, where |V| is the number of vertices of the linkage L, because each time one adds a new component to such an acyclic semi-rigid connected sum, one needs to add at least 2 new vertices, and the first component has at least 3 vertices. An example where the maximal number (|V| — 1)/2 of components is achieved is shown in Figure 8. The linkage L has
L
components is valid: £1 is generated by {A, B, G}, £2 is generated by {B, C, F, G}, £3 is generated by {C, D, E, F} and the only marked diagonal is CF. So, on this 4-dimensional configuration space, we have an integrable system of type (3,1), with three commuting vector fields and one common first integral.
C D
Figure 8: A linkage which can be decomposed into 3 "elementary" components.
4. Some final remarks and open problems 4.1. Bott-Morse theory and graph-manifolds
The topology of configuration spaces of planar linkages has been extensively studied using mostly Morse theory, see, e.g., [5, 6, 9, 10, 11, 12, 13, 14] and references therein. However, in general, the square length functions that we use in this paper are not Morse functions, but rather Bott-Morse functions: their singular points are not isolated but form submanifolds and are transversally non-degenerate with respect to these submanifolds. So the idea of using Bott-Morse functions (instead of just Morse functions) for the study of the topology of linkage spaces is very natural.
At least in the case of linkages with 3-dimensional spaces which admit integrable systems of type (2,1), this idea works very well: the situation is then very similar to Fomenko's topological theory of 2-degree-of-freedom integrable Hamiltonian systems restricted to 3-dimensional isoenergy manifolds [7]. One of the main observations made by Fomenko is that, due to the Bott-Morse nature of the first integral, the 3-dimensional manifolds in question must be graph-manifolds, i.e., they can be cut into pieces which are direct products of S1 with 2-dimeiisional surfaces with boundary. The class of graph-manifolds has been introduced by Waldhausen [25] in 1967 and is a very well-understood class of 3-manifolds. (Incidentally, according to the Thurston-Perelman geometrization program, this class coincides with the class of 3-manifolds whose Gromov norm vanishes).
For example, consider the configuration space M^ of hexagons A1A2A3A4A5A6 with fixed edge lengths = ¿i (i = 1,..., 6, A7 = Ai). We assume that the lenghths are generic and that
M^ is not empty. Denote by d = A1A4 the length of the diagonal A1A4. Then d is a function on M^, which is not smooth; however, p = d2/2 is smooth and is a Bott-Morse function on Mg, whose singular points form circles. It follows immediately that Mi is a graph-manifold. To study the topology of Mi more carefully, one needs to investigate in detail the singularities of For example, assume (without loss of generality for the topology of that I1 < I2 < I3 < I4 < I5 < Iq- Then the possible critical values of d are:
• the maximal value: I1 +12 + I3,
• possible saddle values: I1 +12 —13 (if this value is positive), I3 +12 —1113 +11 —12,14 +15 — Iq (if this value is positive), Iq +15 — Iq + I4 — I5;
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T. C. PaTMO, Hrven Tten 3vnr
• the minimal value is either 0 (if both l1 + l1 — l3 and l4 + l5 — l6 are positive) or max(l3 — li — l2, l6 — l4 —15) (if this value is positive).
Then one describes the level sets at and near these critical values, the 1-cvcles on them, and so on, to recover a long list of all possibilities (see [14]). This would be an interesting exercise for graduate students.
4.2. Integrable systems on more general linkage spaces
In this paper, we used our decomposition and the cross product method to construct integrable systems on the configuration spaces of planar linkages. It it clear that our method can be applied, in a straightforward way, to other classes of linkages, including spherical linkages, 3D linkages, and higher-dimensional linkages. (See [18] for a general theory of linkage designs, including spherical linkages and spatial linkages). In particular, according to our method, the class of configuration spaces of 3D linkages admitting interesting natural integrable systems is much bigger than the class of 3D polygon spaces studied by Kapovich-Millson [15] and others. As far as we know, this bigger class has not yet been explored and it is an interesting subject of study, with potential applications in robotics.
4.3. Singularities of integrable systems on linkage spaces
Singularities of integrable Hamiltonian systems have been studied by many authors. More recently, there has been interest in formulating a theory of singularities of integrable non-Hamiltonian systems (see [27] and references therein). A detailed study of singularities of concrete integrable systems on configuration spaces of planar linkages would help the development of this theory.
4.4. Commuting flows for 2DOF components
In this paper, in the construction of integrable systems on configuration spaces, we only used components which are 1DOF (i.e., curve-drawing), once the lengths of the marked diagonals are fixed. What about 2DOF components? Can they be used effectively? In particular, is there any natural way to construct a pair of independent commuting vector fields on the configuration spaces of pentagons, for example? These configuration spaces are closed surfaces and it is known that on such surfaces there exist Reactions with nondegenerate singularities (see, e.g., [21]). The question is hence how to construct such Reactions in a natural way on 2-dimensional moduli spaces of planar linkages. What about components with more degrees of freedom (once the lengths of the marked diagonals are fixed).
Acknowledgements
We thank Marc Troyanov for some interesting discussions on the subject of this paper. We also thank B. Servatius, O. Shai, and WT. Whiteley for their permission to copy a picture from their paper [23]. Most of this work was carried out while the second author visited the first author at Shanghai Jiao Tong University for a month in 2019, under its "High-End Foreign Experts "cooperation program; he thanks SJTU and the members of the School of Mathematical Sciences, especially Xiang Zhang and Jie Hu, for the invitation and their warm hospitality.
СПИСОК ЦИТИРОВАННОЙ ЛИТЕРАТУРЫ
1. O.I. Bogovavlenskij, A concept of integrabilitv of dynamical systems, C. R. Math. Rep. Acad. Sei. Canada, 18 (1996), 163-168.
2. O.I. Bogovavlenskij, Extended integrabilitv and bi-hamiltonian systems, Comm. Math. Phys., 196(1) (1998), 19-51.
3. D. Bouloc, Singular fibers of the bending flows on the moduli space of 3D polygons, Journal of Symplectic Geometry, 16(3) (2018), 585-629.
4. J.-P. Dufour, N.T Zung, Poisson Structures and Their Normal Forms, Progress in Mathematics Vol. 242, Birkhäuser, 2015.
5. M. Farber, An Invitation to Topological Robotics, Zürich Lectures in Advanced Mathematics, European Mathematical Society Publishing House, 2008.
6. M. Farber, D. Schütz, Homology of planar polygon spaces, Geom. Dedicata, 125 (2007), 75-92.
7. А. Т. Фоменко, Симилектическая топология вполне интегрируемых гамильтоновых систем, УМН, 44:1 (1989), 145-173.
8. C.G. Gibson, J.M. Selig, Movable hinged spherical quadrilaterals, I Uber sphärische Gelenkenvierecke: the configuraion space, II singularities and reduction, Mechanism, and Machine Theory, 23(1) (1988), 13-18; 19-24.
9. C.G. Gibson, D. Marsh, On the linkage varieties of Watt 6-bar mechanisms, I Basic geometry, II The possible reductions, III Topology of the real varieties, Mechanism, and Machine Theory, 24 (1989), 106-113; 115-121; 123-126.
10. J.-C. Hausmann, A. Knutson, Polygon spaces and Grassmannians, Enseign. Math.(2), 43(1-2) (1997), 173-198.
11. J.-C. Hausmann, A. Knutson, Cohomologv rings of polygon spaces, Ann. Inst. Fourier (Grenoble), 48 (1988), 281-321.
12. D. Jordan, M. Steiner, Configuration spaces of mechanical linkages, Discrete Comput. Geom., 22 (1999), 297-315.
13. Y. Kamivama, S. Tsukuda, The Euler characteristic of the configuration space of planar spidery linkages,Algebr. Geom. Topol, 14(6) (2014), 3659-3688.
14. M. Kapovich, J. Millson, On the moduli space of polygons in the Euclidean plane, J. Diff. Geom., 42(2) (1995), 430-464.
15. M. Kapovich, J. Millson, The symplectic geometry of polygons in Euclidean space, J. Diff. Geom., 44(3) (1996), 479-513.
16. M. Kapovich, J. Millson, Universality theorems for configuration spaces of planar linkages, Topology, 41 (2002), 1051-1107.
17. A.B. Kempe, On a general method of describing plane curves of the nth degree bv linkwork, Proc. London Math,. Soc., VII, 102 (1876), 213-216.
18. J.M. McCarthy, G.S. Soh, Geometric Design of Linkages, second edition, Springer, 2011.
338
T. C. PaiMO, Hrven Tten 3vnr
19. A. Klvachko, Spatial polygons and stable configurations of points on the projective line, in Algebraic Geometry and Its Applications, Proceedings of the 8th Algebraic Geometry Conference, Yaroslavl' 1992, (A. Tikhomirov and A. Tvurin, eds.), Vieweg, Braunschweig, 1994, pp. 67-84.
20. T.H. Minh, N.T. Zung, Commuting foliations, Regular and Chaotic Dynamics, 18(6) (2013), 608-622.
21. N.V. Minh, N.T. Zung, Geometry of nondegenerate Reactions on n-manifolds, J. Math. Soc. Japan, 66(3) (2014), 839-894.
22. Y. Nambu, Generalized Hamiltonian dynamics, Phys. Rev. D., 7(8) (1973), 5405-5412.
23. B. Servatius, O. Shai, W. Whitelev, Combinatorial characterization of the Assur graphs from engineering, European J. Combin., 31(4) (2010), 1091-1104.
24. W. Thurston, Shapes of polvhedra, Preprint 1987.
25. F. Waldhausen, Eine Klasse von 3-dimensionalen Mannigfaltigkeiten, I k, II. Inventiones Mathematicae, 3 (4): 308-333 and 4 (2): 87-117 (1967).
26. J. Zhao, Z. Feng, N. Ma, F. Chu, Design of special planar linkages, Springer Tracks in Mechanical Engineering, 2014.
27. N.T. Zung, Geometry of integrable non-Hamiltonian systems, in Geometry and Dynamics of Integrable Systems, Advanced Courses in Mathematics, CRM Barcelona, pp. 85-140. Birkhauser/Springer, 2016.
28. N.T. Zung, A conceptual approach to the problem of action-angle variables, Arch. Rat. Mech. Anal, 229(2) (2018), 789-833.
REFERENCES
1. Bogovavlenskij, O.I. 1996, "A concept of integrabilitv of dynamical systems", C. R. Math. Rep. Acad. Sci. Canada, vol. 18, pp. 163-168.
2. Bogovavlenskij, O.I. 1998, "Extended integrabilitv and bi-hamiltonian systems", Comm. Math. Phys., vol. 196, no. 1, pp. 19-51.
3. Bouloc, D. 2018, "Singular fibers of the bending flows on the moduli space of 3D polygons", Journal of Symplectic Geometry, vol.16, no. 3, pp. 585-629.
4. Dufour, J.-P.& Zung, N.T. 2015, Poisson Structures and Their Normal Forms, Progress in Mathematics Vol. 242, Birkhauser.
5. Farber, M. 2008, An Invitation to Topological Robotics, Zurich Lectures in Advanced Mathematics, European Mathematical Society Publishing House.
6. Farber, M. k, Schiitz, D. 2007, "Homology of planar polygon spaces", Geom. Dedicata, vol. 125, pp. 75-92.
7. Fomenko, A.T. 1989, "Symplectic topology of completely integrable Hamiltonian systems", Russian Math. Surveys, vol. 44, no. 1, pp. 181-219.
8. Gibson, C.G. k Selig, J.M. 1988, "Movable hinged spherical quadrilaterals, I Über sphärische Gelenken vier ecke: the configuraion space, II singularities and reduction", Mechanism, and Machine Theory, vol.23, no.l, pp. 13-18; pp. 19-24.
9. Gibson,C.G. k Marsh, D. 1989, "On the linkage varieties of Watt 6-bar mechanisms, I Basic geometry, II The possible reductions, III Topology of the real varieties", Mechanism, and Machine Theory, vol. 24, pp. 106-113; pp. 115-121; pp. 123-126.
10. Hausmann, J.-C. k Knutson, A. 1997 "Polygon spaces and Grassmannians", Enseign. Math. (2), vol.43, no. 1-2, pp. 173-198.
11. Hausmann, J.-C. k Knutson, A. 1988, "Cohomologv rings of polygon spaces", Ann. Inst. Fourier (Grenoble), vol. 48, pp. 281-321.
12. Jordan, D. k Steiner, M. 1999, "Configuration spaces of mechanical linkages", Discrete Comput. Geom., vol. 22, pp. 297-315.
13. Kamivama, Y. k Tsukuda, S. 2014, "The Euler characteristic of the configuration space of planar spidery linkages", Algebr. Geom. Topol, vol. 14, no. 6, pp. 3659-3688.
14. Kapovich, M. k Millson, J. 1995, "On the moduli space of polygons in the Euclidean plane", J. Diff. Geom., vol. 42, no. 2, pp. 430-464.
15. Kapovich, M. k Millson, J. 1996, "The symplectic geometry of polygons in Euclidean space", J. Diff. Geom., vol. 44, no. 3, pp. 479-513.
16. Kapovich, M. k Millson, J. 2002, "Universality theorems for configuration spaces of planar linkages", Topology, vol. 41, pp. 1051-1107.
17. Kempe, A.B. 1876, "On a general method of describing plane curves of the nth degree by linkwork", Proc. London Math. Soc., VII, vol. 102, pp. 213-216.
18. McCarthy, J.M. k Soh, G.S. 2011, Geometric Design of Linkages, second edition, Springer.
19. Klvachko, A. 1994, "Spatial polygons and stable configurations of points on the projective line", in Algebraic Geometry and Its Applications, Proceedings of the 8th Algebraic Geometry Conference, Yaroslavl' 1992, (A. Tikhomirov and A. Tvurin, eds.), Vieweg, Braunschweig, pp. 67-84.
20. Minh, T.H. k Zung, N.T. 2013, "Commuting foliations", Regular and Chaotic Dynamics, vol. 18, no. 6, pp. 608-622.
21. Minh, N.V.& Zung, N.T. 2014, "Geometry of nondegenerate Reactions on n-manifolds", J. Math. Soc. Japan, vol. 66, no.3, pp. 839-894.
22. Nambu, Y. 1973, "Generalized Hamiltonian dynamics", Phys. Rev. D., vol. 7, no. 8, pp. 54055412.
23. Servatius, B., Shai, O. k Whitelev, W. 2010, "Combinatorial characterization of the Assur graphs from engineering", European J. Combin., vol. 31, no. 4, pp. 1091-1104.
24. Thurston, W. 1987, "Shapes of polvhedra", Preprint.
25. Waldhausen, F. 1967, "Eine Klasse von 3-dimensionalen Mannigfaltigkeiten, I k II." Inventiones Mathematicae, vol. 3, no. 4, pp. 308-333 and vol. 4, no. 2, pp. 87-117.
340
Т. С. Ратью, Нгуен Тьен Зунг
26. Zhao, J., Feng, Z., Ma, N. к Chu, F. 2014, Design of special planar linkages, Springer Tracks in Mechanical Engineering.
27. Zung, N.T. 2016, "Geometry of integrable non-Hamiltonian systems", in Geometry and Dynamics of Integrable Systems, Advanced Courses in Mathematics, CRM Barcelona, pp. 85-140. Birkháuser/Springer.
28. Zung, N.T. 2018, "A conceptual approach to the problem of action-angle variables", Arch. Rat. Mech. Anal, vol. 229, no. 2, pp. 789-833.
Получено 9.01.2019 г. Принято в печать 11.03.2020 г.
