On the free Carnot (2, 3, 5, 8) group Текст научной статьи по специальности «Математика»

ISSN 2079-3316 PROGRAM SYSTEMS: THEORY AND APPLICATIONS No.2(25), 2015, pp. 45-61

J.-P. Gauthier, Yu. L. Sachkov On the free Carnot (2, 3, 5, 8) group

ABSTRACT. We consider the free nilpotent Lie algebra L with 2 generators, of step 4, and the corresponding connected simply connected Lie group G, with the aim to study the left-invariant sub-Riemannian structure on G defined by the generators of L as an orthonormal frame.

We compute two vector field models of L by polynomial vector fields in R8, and find an infinitesimal symmetry of the sub-Riemannian structure. Further, we compute explicitly the product rule in G and the right-invariant frame on G.

Key Words and Phrases: Sub-Riemannian geometry, Carnot group.

2010 Mathematics Subject Classification: 53C17

Introduction

In this work we start to study a variational problem that can be stated equivalently in the following three ways.

(1) Geometric statement. Consider two points a0,a1 G R2 connected by a smooth curve 70 C R2. Fix arbitrary data S G R, c = (cx, cy) G R2, M = (Mxx, Mxy, Myy) G R3. The problem is to connect the points a0, a1 by the shortest smooth curve 7 C R2 such that the domain D C R2 bounded by 70 U 7 satisfy the following properties:

(1) area(^) = S,

(2) center of mass(^) = c,

(3) second order moments(^) = M.

(2) Algebraic statement. Let L be the free nilpotent Lie algebra with two generators X1, X2 of step 4:

(1) L = span(X1,..., X8),

(2) [X1,X2 ]= X3,

(3) [X1, X3] = X4 , [X2,X3] = X5,

(4) [X1, X4] = x6, [X1, X5] = [X2, X4] = , [X2, X5] = x8.

Work supported by Grant of the Russian Federation for the State Support of Researches (Agreement No 14.B25.31.0029). © J.-P. Gauthier(1, Yu. L. SAOHKOvi2, 2015

© Laboratoire des Sciences de l'Information et des Systemes, UMR, CNRS, France(1, 2015 (© Ailamazyan Program System Institute of RAS(2, 2015 (© Program systems: Theory and Applications, 2015

Let G be the connected simply connected Lie group with the Lie algebra L, we consider X\, ..., X8 as a frame of left-invariant vector fields on G. Consider the left-invariant sub-Riemannian structure (G, A, g) defined by X2 as an orthonormal frame:

Aq = span(Xi(9),X2(9)), g(Xi ,Xj ) = öy.

The problem is to find sub-Riemannian length minimizers that connect two given points q0,qi G G:

q(t) G G, q(0) = qo, q(ti) = qi,

<?C0 G Aq(t),

ft% / ■ I = \/d(ti, O) dt ^ min .

o

(3) Optimal control statement. Consider the following vector fields Xi, X2 on R8:

Xi =

d x2 d x\ + x2 d x-\_x2 d x2 d

dxi 2 dx3 2 dx5 4 dx7 6 dx8^ d xi d x\ + x2, d x3 d x\x2 d

X2 = ^ + + i : ^ + +

dx2 2 dx3 2 dx4 6 dx6 4 dx7

Given arbitrary points qo,qi € R8, it is required to find solutions of the optimal control problem

(1) </ = u1X1(q)+ u2X2(q), q € R8, (ui,u2) € R2,

(2) q(0) = qo, q(ti) = qi,

1 f

(3) J = - / (u\ + u2 ) dt ^ min .

2 o i 2

The problem stated will be called the nilpotent sub-Riemannian problem with the growth vector (2, 3, 5, 8), or just the (2, 3, 5, 8)-problem. There are several important motivations for the study of this problem:

• this problem is a nilpotent approximation of a general sub-Riemanni-an problem with the growth vector (2,3,5,8) [1-5];

• this problem is a natural continuation of the important sub-Riemanni-an (SR) problems: the nilpotent SR problem on the Heisenberg group (aka Dido's problem, growth vector (2,3)) [6, 7], and the nilpotent SR problem on the Cartan group (aka generalized Dido's problem, growth vector (2,3,5)) [8-11];

• this problem is included into a natural infinite chain of rank 2 SR problems with the free nilpotent Lie algebras of step r, r G N, and more generally into a natural 2-dimensional lattice of rank d SR problems with the free nilpotent Lie algebras of step r, (d,r) G N2;

• this problem is the simplest possible SR problem on a step 4 Carnot group, and it is the first SR problem with growth vector of length 4 that should be studied.

To the best of our knowledge, this is the first study of the (2,3,5,8)-problem (although, it was mentioned in [12] as a SR problem with smooth abnormal minimizers).

The structure of this work is as follows.

In Sec. 1 we construct two models ("asymmetric" and "symmetric") of the free nilpotent Lie algebra with 2 generators of step 4 by polynomial vector fields in R8. For these models, we use respectively an algorithm due to Grayson and Grossman [13] and an original approach. In the symmetric model, a one-parameter group of symmetries leaving the initial point fixed is found.

In Sec. 2 we describe explicitly the product rule in the Lie group G = R8, construct a right-invariant frame on G corresponding naturally to the left-invariant frame given by X1, X2 and their iterated Lie brackets, compute the corresponding left-invariant and right-invariant Hamiltonians that are linear on fibers of T*G.

In Conclusion we suggest possible questions for further study. Results of Sec. 1 of this paper appeared previously in preprint [14]. Since both results and techniques of the preprint are necessary for understanding and verification of results of Sec. 2, these materials are published completely in this work.

1. Realisation by polynomial vector fields in R8

In this section we construct two models of the free nilpotent Lie algebra L(1)-(4) by polynomial vector fields in R8.

1.1. Free nilpotent Lie algebras

Let Cd be the real free Lie algebra with d generators [15]; Cd is the Lie algebra of commutators of d variables. We have Cd = ©¿=1 Cld, where Cld is the space of commutator polynomials of degree i. Then

C« := Cd/ r+1 C\ is the free nilpotent Lie algebra with d generators of step r.

Denote ld(i) := dimCld, := dimC^r) = Y11=1 ld(i). The classical expression of ld(i) is ild(i) = d1 - Ylj\i, 1<j<i jld(j).

In this work we are interested in free nilpotent Lie algebras with 2 generators. Dimensions of such Lie algebras for small step are given in Table 1.

TABLE 1. Dimensions of free nilpotent Lie algebras £-2

i 1 2 3 4 5 6 7 8 9 10

h(i) 2 1 2 3 6 9 18 30 56 99

l(i) l2 2 3 5 8 14 23 41 71 127 226

1.2. Carnot algebras and groups

A Lie algebra L is called a Carnot algebra if it admits a decomposition L = ®\=1Li as a vector space, such that [Li, Lj] C Li+j, Ls = 0 for s > r, Li+1 = [Ll, Li].

A free nilpotent Lie algebra C^ is a Carnot algebra with the homogeneous components Li = Cld.

A Carnot group G is a connected, simply connected Lie group whose Lie algebra L is a Carnot algebra. If L is realized as the Lie algebra of left-invariant vector fields on G, then the degree 1 component L1 can be thought of as a completely nonholonomic (bracket-generating) distribution on G. If moreover L1 is endowed with a left-invariant inner product g, then (G,L1,g) becomes a nilpotent left-invariant sub-Riemannian manifold [5]. Such sub-Riemannian structures are nilpotent approximations of generic sub-Riemannian structures [1-4].

The sequence of numbers

(dimL1, dim L1 + dimL2,..., dim L1 + • • • + dim Lr = dim L)

is called the growth vector of the distribution L1 [7].

For free nilpotent Lie algebras, the growth vector is maximal compared with all Carnot algebras with the bidimension (dim L1, dim L).

1.3. Lie algebra with the growth vector (2,3,5, 8)

The Carnot algebra with the growth vector (2, 3, 5, 8) C24) = span(X1,..., X8)

is determined by the following multiplication table:

(4) [Xi,X2 ] = X3,

(5) [X!,X3 ] = X4, [X2,X3]= X5,

(6) [Xi,X4 ] = X6, [XUX5] = [X2,X4] = Xr, [^2,^5]= X8,

with all the rest brackets equal to zero. This multiplication table is depicted at Fig. 1.

X\ X2

Xe Xr X8

Figure 1. Lie algebra with the growth vector (2, 3, 5, 8)

1.4. Hall basis

Free nilpotent Lie algebras have a convenient basis introduced by M. Hall [16]. We describe it using the exposition of [13].

The Hall basis of the free Lie algebra Cd with d generators X\, ..., Xd is the subset Hall C Cd that has a decomposition into homogeneous components Hall = U?=1 Hallj defined as follows.

Each element Hj, j = 1, 2,..., of the Hall basis is a monomial in the generators Xi and is defined recursively as follows. The generators satisfy the inclusion Xi G Hall1; i = 1,... ,d, and we denote Hi = Xi, i = 1,... ,d. If we have defined basis elements Hi,..., G ®j=i Hallj,

they are simply ordered so that E < F if E G Hall^, F G Hall;, k < I: H1 < H2 < ■ ■■ < HNp-1. Also if E G Halls, F G Hallt and p = s + t, then [E, F] G Hallp if: "

(1) E > F, and

(2) if E = [G, K], then K G Hallg and t > q.

By this definition, one easily computes recursively the first components Hall,- of the Hall basis for d =2:

Hall1 = [Hu H2}, H1 = X1, H2 = X2, Hall2 = {H3}, H3 = [X2,X1],

Halls = {H4,H5}, H4 = [[X2,X1],X1], H5 = [[X2,X1],X2], Hall4 = {H6,H7,H8},

H6 = [[[X2,X1],X1],X1], H7 = [[[X2,X1],X1],X2], H8 = [[[X2,X1],X2],X2].

Consequently, c24) = span{H1,..., H8}. In the sequel we use a more convenient basis of c24) = span{X1,..., X8} with the multiplication table (4)-(6).

1.5. Asymmetric vector field model for c24)

Here we recall an algorithm for construction of a vector field model for the Lie algebra ¿¡p due to Grayson and Grossman [13]. For a given r > 1, the algorithm evaluates two polynomial vector fields H1,H2 G Vec(RN),

(r) (r)

N = dim C2 ), which generate the Lie algebra C2 ).

Consider the Hall basis elements span{ff1,..., Hn} = c2[\ Each element Hi G Hallj is a Lie bracket of length j:

Hi = [... [[H2, Hkj ], Hkj_1 ],..., Hkl ], kj = 1, kn+1 < kn for 1 < n < j - 1.

This defines a partial ordering of the basis elements. We say that Hi is a direct descendant of H2 and of each Hkl and write i y 2, i y ki, I =1,...,j.

Define monomials P2)k in x1, ..., x^ inductively by

P2,k = -Xj P2,i/(degj P2,i + 1),

whenever Hk = [H-i, Hj] is a basis Hall element, and where degj P is the highest power of Xj which divides P.

The following theorem gives the properties of the generators.

Theorem 1 (Th. 3.1 [13]). Let r > 1 and let N = dim C{2,). Then

the vector fields H1 = ——, H2 = 7;--+ / P2 — have the following

0x1 0x2^^ ' ox,,

properties:

(1) they are homogeneous of weight one with respect to the grading

Rn = Hall1 ©•••© Hallr;

(2) Lie(^1, H2) = C^.

The algorithm described before Theorem 1 produces the following

vector field basis of C

(4).

H

H2

H3

d dx\ ' d d 8x2 9x3

2 ox4 0x5

+ 6 dx6 +

d x2X2 d xix2 d

---h Xl---h X2T.-

ox3 ox4 ox5

2 Ox6 Ox7

d d d

H4 = — ---h Xi---h X2^—,

OX4 OXß Ox7

d d d

H5 = --h --h £2q-,

Ox5 ox7 ox8

H6 = —

d

dxß '

H7 = —

d dx7

H8 = —

d

3X8

2 dx7

1 JL

2 dx8 ,

+

2 dx8

with the multiplication table

(7) [H2,H! ] = H3,

(8) [H^H!] = HA, [^3,^2] = H5,

(9) [Hi, Hi] = H6, [Hi, H2] = Hr, [^5,^2] = H8.

(4)

1.6. Symmetric vector field model of C

The vector field model of the Lie algebra C2i) via the fields H\,..., H8 obtained in the previous subsection is asymmetric in the sense that there is no visible symmetry between the vector fields Hi and H2. Moreover, no continuous symmetries of the sub-Riemannian structure generated by the orthonormal frame {Hi,H2} are visible, although the Lie brackets (7)-(9) suggest that this sub-Riemannian structure should be preserved by a one-parameter group of rotations in the plane span{#i, H2}.

One can find a symmetric vector field model of C2 shortages as in the following statement.

(4)

free of such

Theorem 2. (1) The vector fields

(10) X! =

(11) X2 =

X2 d

d

dx\ 2 8x3 d

++ ^2 ^

x\x2 d x\ d

2 dx5 4 dx7 6 dx8'

+

x\ d x\ + x2 d

dx2 2 8x3

+

x3 d x2x2 d

2 dx4 6 dx6 4 dx7

+

2

3

2

d

d

X

2

d

d

X

, , ^ 8 8 8 -2 (12) X3 — -- + xi-— + X2-— +

8

(13) X4

(14) X5

(15) Xe

(16) X7

(17) X8

8 X3 9x4 8 X5 2 8 xe

8 8 8

à--+ xi7--+X2-—,

0x4 a xe 0x7

8 8 8 + x\---+ x2-

0 2 + x\x2~--+

x22 0

'8x7 2 8 x8 '

0 x5 8 0 xe 8 0 x7 8 8 x8

1 ^ 8 x7

8 x8

satisfy the multiplication table (4)-(6). Thus the fields X1, ... ,X8 G Vec(R8) model the Lie algebra Cp. (2) The vector field

(18)

d d d d d d d

Xo =X2^--X1---+ --X4---+ P---+ Q---+ —,

011 0x2 0x4 0x5 0x6 0x7 0x8

(19)

4

P = - ^r +

22 clx2

24

+ x7,

(20)

3 3

Q = xx3 + x3x2 - 2xe + 2x8,

(21) R =

12

22 l x2

12

8

- 24 -x7

satisfies the following relations:

(22) [Xo,Xi]=X2,

(23) [Xo,X4]=X5,

[Xo, X2] — —Xi,

[X0, X5] — — X4-,

[Xo, X3] — 0,

(24) [Xo, Xe] — 2X7, [Xo,X7]—X8 — Xe, [Xo,X8] — —2X7.

Thus the field Xo is an infinitesimal symmetry of the sub-Riemannian structure generated by the orthonormal frame {Xi,X2}.

4

x

Proof. In fact, both statements of the proposition are verified by direct computation, but we prefer to describe a method of construction of the vector fields X\,..., X8, and X0.

(1) In the previous work [8] we constructed a similar symmetric vector

(3)

field model for the Lie algebra £2 ), which has growth vector (2, 3, 5):

(25) £3) = spanjXi,..., X5} C Vec(R5),

X2 d ^i

(26)

(27)

(28)

(29)

(30)

X3 X4 X5

x\ + x"2 d 2 dx5' d x1 d x x\ +12 d

d

dxi 2 8x3 xi d

+--1--+ ,

dx2 2 dx3 2 dx4

d d d + xi~--+ X2-

8x3 d x4

d

OX5

d x4' d

8x5'

with the Lie brackets (4), (5). Now we aim to "continue" these relationships to vector fields X\,..., X8 G Vec(R8) that span the Lie algebra £24). So we seek for vector fields of the form

(31) d X2 d x2 + x2 d 8 + Ea> i=6

dx\ - Y - 2 dx5

(32) X2 d xi d +--L-- + 2 dx3 x2 + x"2 d 8 + Ea- i=6

9x2 2 dx4

(33) d 9x3 d d A + Xla--+ x2«--+ OX4 OX5 z—' i=6 i d a3 dXi ,

(34) X4 d 8x4 8 d a4 ^ i=6 * ,

(35) d 9x5 8 d + Y a5 — 1=6 5 dxi ,

(36) Xi 8 V^ 3 d = d^, J i=6 3 = 6, 7, 8,

• A

1 dxi

• A

2 dxi

such that span{Xi,..., X8} = ■ Compute the required Lie brackets:

d

3

3

[XUX2] = J— + X!— + -j-2 A —

ax3 3x4 3x5 \ ux\ '

34

3ai

J 3x2

[Xi,Xs ] [X2,X3 ]

[Xi,X4 ] [Xl,X5 ] [X2,X4 ] [X2,X5 ]

3 dal

3x4 3xi

3 dal

3x5 dx2

d 3a7 3x6 3xi

3 3a% 3 3x7 3xi 3x8 '

3 3al 3

3xi 3x2 3x7 3x2 3x8 '

3 a8 3

3a4 3 3xi 3xf 3a5 3 3xi 3x6 3a4 3 3x2 3x6 3a5 3

3 x2 3 x6

3 a74 3 3 a48 3

3 xi 3 x7 3 xi 3 x8

3 a57 3 3 a85 3

3 xi 3 x7 3 xi 3 x8

3 a74 3 3 a84 3

3 x2 3 x7 3 x2 3 x8

3 a57 3 3 a85 3

3 x2 3 x7 3 x2 3 x8

3

3 x6

i 3a7 3al\ 3 (30% 3a8i\ 3 3 xi 3 x2 3 x7 3 xi 3 x2 3 x8

The vector fields Xi,...,Xs should be independent, thus the determinant constructed of these vectors as columns should satisfy the inequality

D = det (Xi,...,X8)

= 0.

We will choose a? such that D = 1. It follows from the multiplication table for Xi,..., X8 that

D

d2al d2al d2a3

d2a7 dxidx2 d2 a7 d2ag

d2a8 dxidx2 d2a3 d2 2

dx\ dxidx2 dx2

6

6

6

a

a

a

6

7

8

7

7

7

a

a

a

6

7

8

8

8

8

a

a

6

7

8

In order to get D = 1, define the entries of this matrix in the following

2 2 X X

symmetric way: a3 = -1, a3 = x\x2, a| = . Then we obtain from

da3 9a? 6 x?

the multiplication table for Xi,...,X8 that —--—— = a3 = —,

oxi ox2 2

daX da[ 7 da% daS s x\

—2 - -—1 = al = xix2, -—2 - -—1 = a8 = —. We solve these ax\ ox2 0x1 ox2 2

32

(2 (2 X* I >-j X* 1 X>2

equations in the following symmetric way: aj = 0, a^ = —, ai =---—,

23

>-j X* I ft X2 Q

a2 = -, a8 =--, a? = 0. Then we substitute these coefficients

2 4 ' 1 6 ' 2 to (31), (32) and check item (1) of this theorem by direct computation.

Now we prove item (2). We proceed exactly as for item (1): we start from an infinitesimal symmetry [8]

d d d d (37) X0 = X2^--+ X5---Xi— e Vec(R5)

OXi OX2 OX4 0x5

of the sub-Riemannian structure on R5 determined by the orthonormal frame (26), (27) and "continue" symmetry (37) to the sub-Riemannian structure on R8 determined by the orthonormal frame (10), (11).

So we seek for a vector field X0 e Vec(R8) of the form (18) for the functions P,Q,R e C(R8) to be determined so that the multiplication table (22)-(24) hold.

32

The first two equalities in (22) yield X1P =--1, X2P = .

62

23

Further, X3P = [Xi,X2]P = X1X2P - X2X1P = Xi^ + X2^ =

26

x1x2. Similarly it follows that X4P = x2, X5P = x1, X6P = 0, X7P = 1, X8P = 0. Since X6P = X8P = 0, then P = P(x1,x2,x3,x4,x5, x7). Moreover, since X7P = 1, then P = x7 + a(x1,x2, x3, x4, x5). The equality X5P = x1 implies that = 0, i.e., a = a(x1,x2,x3,x4). Similarly, since X4P = x2, then a = a(x1,x2,x3). It follows from the equality d

X3P = X1X2 that —— = X1X2, i.e., a = X1X2X3 + b(x1,x2). Moreover, ox3

r, x?x^ db x2;x2

the equality X2P = - implies that —— = —x1x3--, i.e., b =

2 ax2 4

—x1x2x3--î—2 + c(x1). Finally, the equality X1P =--1 implies that

82 j 3 2 4 2 2

^— = -^ + ^ i.e., c = -+ ^. Thus equality (19) follows. ax1 6 2 24 4

Similarly we get equalities (20), (21).

Then multiplication table (22)-(24) for the vector field (18)-(21) is verified by a direct computation. □

2. Carnot group

In this section we study the Carnot group G with the Lie algebra

l = 44)-

2.1. Product rule in G

In this subsection we compute the product rule in the connected simply connected Lie group G with the Lie algebra L = £^4) on which the vector fields X,... ,X8 given by (10)-(17) are left-invariant.

Our algorithm for computation of the product rule in a Lie group G with a known left-invariant frame X\,..., Xn G Vec(G) follows from the next argument. Let gg2 G G, and let g2 = etnXn o ... o etlXl (Id), t\, ... , tn G R, where we denote by etx : G ^ G the flow of the vector field X. Then g i • g2 = g i • o ... o etlXl (Id) = etnXn o ... o etlXl (gi)

by left-invariance of Xj. So an algorithm for computation of gi • g2 is the following:

(1) Compute eUXi (g), ti G R, g G G.

(2) Compute etnXn o ... o etlXl (g), ti G R, g G G.

(3) Solve the equation etnXn o ... o e tlXl (Id) = g2 for 1i,..., tn G R (we assume that this is possible in a unique way).

(4) Compute gi • g2 = etnXn o ... o e tlXl (g2).

By this algorithm, we compute the product z = x • y in the coordinates on G (notice that as a manifold G = R8), as follows:

x = (xi,..., X8), y = (y 1,..., V8), Z = (z I,..., G G = R8, Zi=Xi + y i, Z2 = X2 +y2,

zs = X3 + y3 + 2(xi2/2 - X2yi),

Z4 = X4 + y4 + 2(xi (xi +yi) + X2(X2 +y2) +Xys), Z5 = X5 + y5 - (xi (xi +yi) + X2(X2 +^2)) +X2ys, ZQ = X6 + y6 + X2(2x2y2 + 3x\y^2 - 2y| + 6x1 y3 + 122/4),

¿7 = X7 + y7 + (3iC2;i/2(2iC2 + V2) - X2(3x2yl + 6y\y2 + 4(y2 - 6y4))

+ xi (-6x2y i+4y3 + 6yi y2 + 24x2^/3 + 24 i/5)), Z8 = X8 + i/8 + y(-2x^1 + 2y3 - 3%2y\y2 + 6x22/3 + 12^).

2.2. Right-invariant frame on G

Computation of the right-invariant frame on G corresponding to a left-invariant frame can be done via the following simple lemma. Denote the inversion on a Lie group G as « : G ^ G, i(g) = g-1.

Lemma 1. Let XUX2,X3 e Vec(G) and YUY2,Y3 e Vec(G) be respectively left-invariant and right-invariant vector fields on a Lie group G such that Yj (Id) = -Xj (Id), j = 1, 2, 3. Then

(38) i*Xj =Y, ¿ = 1, 2, 3,

(39) [Xi,X2]=X3 ^ [Y,Y2]=Y3.

Proof. Equality (38) follows by the left-invariance and right-inva-riance of the fields Xi and Y respectively. Equality (39) follows since the diffeomorphism i : G ^ G preserves Lie bracket of vector fields (see e.g. [17]). □

Thus if X,..., Xn e Vec G is a left-invariant frame on a Lie group G, then Yi,... ,Yn e Vec G, Yj = i*Xj, is the right-invariant frame such that Yj (Id) = -Xj (Id), j = 1,... ,n, and the same product rules as for X\, ..., Xn.

Immediate computation using the product rule in G given in Sub-sec. 2.1 gives the following right-invariant frame on the Lie group G = R8 :

d X2 d XX2 + 2x3 d x2 d = -7------r-T;--+

dx\ 2 dx3 2 dx4 2 dx5

x2 - 6x4 d 2X3 + 3xix2 + 12x5 d

+

6 dx6 12 dx7'

d x d x22 d x x2 - 2 x3 d Y2 = - 7--------+

d x2 2 d x3 2 d x4 2 d x5

3x2x2 + 2x3 - 12x4 d x3 + 6x5 d

+

12 dx6 6 dx8'

d

Yi = , t = 3,..., 8.

d Xi

2.3. Left-invariant and right-invariant Hamiltonians on T* G

Using the expressions for the left-invariant and right-invariant frames given in Subsec. 1.6 and Subsec. 2.2, we define the corresponding left-invariant and right-invariant Hamiltonians, linear on fibers in T* G:

h (A) = (\,Xi), 9i(X) = (\,Yi), A e T *G, i = 1,..., 8.

In the canonical coordinates (xi,..., x8, ,..., ^8) on T*G [17] we have the following:

2 2 2 3

Xo X^ + X2 ^1^2/ x2 i

hi = Wi 77^--o---- ^

2 2 4 6

2 2 3 2 Xi X2 ++ i Xi X1X2

h2 = V2 + — W3 +----y4 + — W6 +--— y7,

2 2 6 4

22 X X

h3 = ^3 + + X2^5 + y + XiX2'^7 + "2° ^8,

hA = + X\^6 + ^2^7, h5 = ^5 + X\1^7 + X2^8,

h = r^i, i = 6, 7, 8,

and

10 X\X2 + 2x3 , xl 1

91 = — Wl — Y r3--2-^4 + "2"^5

x3 — 6x4 2X3 + 3xix2 + 12x5

(40) +----^6--^-^7,

6 12

X! x2 ^X2 — 2x3 ,

92 = —W2 — Y y3 — "2" r 4 +--2-^5

3x2x2 +2x3 — 12x4 X3 + 6x5

(41) +--777-^6--7-^8,

12 6

(42) 9i = —^i, i = 3,..., 8. Conclusion

We see the following interesting questions for the (2,3,5,8)-problem:

(1) study optimality of abnormal geodesics;

(2) describe all cases where the normal Hamiltonian vector field H is Liouville intergable, integrate and study the corresponding normal geodesics;

(3) describe precisely the chaotic dynamics of the normal Hamiltonian vector field H suggested by numerical simulations.

We plan to address these questions in forthcoming works. References

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[4] A. A. Agrachev, A. A. Sarychev. "Filtration of a Lie algebra of vector fields and nilpotent approximation of control systems", Dokl. Akad. Nauk SSSR, 295 (1987), pp. 104-108 t 46, 48.

[5] R. Montgomery. A tour of sub-Riemannian geometries, their geodesies and applications: control theory from the geometric viewpoint, American Mathematical Society, 2002 t 46, 48.

[6] R. Brockett, "Control theory and singular Riemannian geometry", New directions in applied mathematics, Springer-Verlag, New York, 1982, pp. 11-27 t 46.

[7] A. M. Vershik, V. Y. Gershkovich, "Nonholonomic dynamical systems. Geometry of distributions and variational problems", Dynamical systems - 7, Itogi Nauki i Tekhniki: Sovremennye Problemy Matematiki, Fundamental'nyje Napravleniya, vol. 16, VINITI, M., 1987, pp. 5-85 \ 46, 48.

[8] Yu. L. Sachkov. "Exponential mapping in generalized Dido's problem", Mat. Sbornik, 194:9 (2003), pp. 63-90 t 46, 53, 55.

[9] Yu. L. Sachkov. "Discrete symmetries in the generalized Dido problem", Matem. Sbornik, 197:2 (2006), pp. 95-116 t 46.

[10] Yu. L. Sachkov. "The Maxwell set in the generalized Dido problem", Matem. Sbornik, 197:4 (2006), pp. 123-150 t 46.

[11] Yu. L. Sachkov. "Complete description of the Maxwell strata in the generalized Dido problem", Matem. Sbornik, 197:6 (2006), pp. 111-160 46.

[12] R. Monti, "The regularity problem for sub-Riemannian geodesics", Geometric Control Theory and sub-Riemannian Geometry, Springer INdAM Series, vol. 5, Springer International Publishing, 2014, pp. 313-332 t 47.

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[14] Yu. Sachkov. On Carnot algebra with the growth vector (2,3,5,8), http://arxiv.org/abs/1304.1035vi, 2013, 13 p t47.

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Submitted by prof. N. N. Nepeivoda

About the authors:

Jean-Paul Gauthier

Professor, extraordinary class, 2. LSIS (Laboratoire des Sciences de l'Information et des Systemes), Unite mixte de recherche CNRS 7296, Marseille. Scientific interests: geometric control theory, optimal control, control on Lie groups, observability, real applications of control theory, specially in the field of chemical engineering, image processing, quantum control, robotics. Author of 3 monographs and around 130 research papers. e-mail: [email protected]

Yuri Leonidovich Sachkov

Doctor of sciences, Chief of Research Center for Control Systems, Aylamazyan Program Systems Institute, Russian Academy of Sciences. Scientific interests: mathematical control theory, sub-Riemannian geometry. Author of 2 monographs and around 50 research papers.

e-mail: [email protected]

Sample citation of this publication:

J.-P. Gauthier, Yu. L. Sachkov. "On the free Carnot (2, 3, 5, 8) group", Program systems: theory and applications, 2015, 6:2(25), pp. 45-61. URL http://psta.psiras.ru/read/psta2015_2_45-61.pdf

УДК 517.977

Ж.-П. Готье, Ю. Л. Сачков. О свободной группе Карно с вектором роста (2, 3, 5, 8).

Аннотлция. Рассматривается свободная нильпотентная алгебра Ли Ь с двумя генераторами, ступени 4, и соответствующая связная односвязная группа Ли О, с целью исследования левоинвариантной субримановой структуры на О, заданной генераторами алгебры Ли.

Вычислены две модели алгебры Ли Ь с помощью векторных полей в К8, и найдены инфинитезимальные симметрии субримановой структуры. Явно вычислены закон умножения в группе Ли О и правоинвариантный репер на О. (Англ.)

Ключевые слова и фразы: Субриманова геометрия, группы Карно.

Пример ссылки на эту публикацию:

Ж.-П. Готье, Ю. Л. Сачков. «О свободной группе Карно с вектором роста (2, 3, 5, 8)», Программные системы: теория и приложения, 2015, 6:2(25), с. 45-61. (Англ.) http://psta.psiras.ru/read/psta2015_2_45-61.pdf

© Ж.-П. Готье(, Ю. Л. Слчков(, 2015

© Лаборатория информатики и теории систем, UMR, CNRS, Франция( , 2015

© Институт программных систем имени А. К. Айламазяна РАН(, 2015

© Программные системы: теория и приложения, 2015