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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 21. Выпуск 2.
УДК 514.752.8+514.763+514.765+514.764227 DOI 10.22405/2226-8383-2020-21-2-43-64
ПМП, (ко)присоединённое представление и нормальные геодезические левоинвариантных (суб)финслеровых метрик на группах Ли1
В. Н. Берестовский, И. А. Зубарева
Берестовский Валерий Николаевич — доктор физико-математических наук, профессор, Институт математики им. С. Л. Соболева СО РАН, Новосибирский государственный университет (г. Новосибирск). e-mail: [email protected]
Зубарева Ирина Александровна — кандидат физико-математических наук, старший научный сотрудник, Институт математики им. С. Л. Соболева СО РАН (г. Омск). e-mail: [email protected]
Аннотация
С помощью основ теории групп и алгебр Ли, их (ко)присоединеппых представлений и принципа максимума Понтрягина для задачи оптимального быстродействия даны независимое обоснование методов геодезического векторного поля поиска геодезических левоинвариантных (суб)финслеровых метрик на группах Ли и поиска соответствующих локально оптимальных управлений в (суб)римановом случае, а также несколько их применений.
Ключевые слова: алгебра Ли, группа Ли, (ко)присоединенное представление, левоинва-риантная (суб)риманова метрика, левоинвариантная (суб)финслерова метрика, математический маятник, нормальная геодезическая, оптимальное управление.
Библиография: 27 названий. Для цитирования:
В. И. Берестовский, И. А. Зубарева. ПМП, (ко)присоединённое представление и нормальные геодезические левоинвариантных (суб)финслеровых метрик на группах Ли // Чебышевский сборник, 2020, т. 21, вып. 2, с. 43-64.
1 Работа выполнена при поддержке Математического Центра в Академгородке, соглашение с Министерством науки и высшего образования Российской Федерации номер 075-15-2019-1613.
CHEBYSHEVSKII SBORNIK Vol. 21. No. 2.
UDC 514.752.8+514.763+514.765+514.764227 DOI 10.22405/2226-8383-2020-21-2-43-64
PMP, (co)adjoint representation, and normal geodesies, of left-invariant (sub-)Finsler metric on Lie groups
V. N. Berestovskii, I. A. Zubareva
Berestovskii Valerii Nikolaevich — doctor of physical and mathematical Sciences, Professor, Sobolev Institute of Mathematics, Novosibirsk State University (Novosibirsk). e-mail: [email protected]
Zubareva Irina Aleksandrovna — candidate of physical and mathematical Sciences, Senior Researcher, Sobolev Institute of Mathematics (Omsk). e-mail: [email protected]
Abstract
On the ground of origins of the theory of Lie groups and Lie algebras, their (co)adjoint representations, and the Pontryagin maximum principle for the time-optimal problem are given an independent foundation for methods of geodesic vector field to search for normal geodesies of left-invariant (sub-)Finsler metrics on Lie groups and to look for the corresponding locally optimal controls in (sub-)Riemannian case, as well as some their applications.
Keywords: (co)adjoint representation, left-invariant (sub-)Finsler metric, left-invariant (sub-)Riemannian metric, Lie algebra, Lie group, mathematical pendulum, normal geodesic, optimal control.
Bibliography: 27 titles. For citation:
V. N. Berestovskii, I. A. Zubareva, 2020, "PMP, (co)adjoint representation, and normal geodesies, of left-invariant (sub-)Finsler metric on Lie groups" , Chebyshevskii sbornik, vol. 21, no. 2, pp. 43-64.
1. Introduction
After Gromov's 1980s papers, homogeneous sub-Finsler manifolds, in particular, sub-Rieman-nian manifolds were actively studied fl], [15], [22], [26]. Their investigation is based on the Rashevsky-Chow theorem which states that any two points of a connected manifold can be joined by a piecewise smooth curve tangent to a given totally nonholonomic distribution [14], [20].
1) Every homogeneous manifold with intrinsic metric is the quotient space G/H of some connected Lie group G by its compact subgroup H, equipped with G-invariant Finsler or sub-Finsler metric d\ in particular, it may be Riemannian or sub-Riemannian metric [3], [4], [5];
2) moreover, according to a form of metric d, there exists a left-invariant Finsler, sub-Finsler, Riemannian or sub-Riemannian metric p on G such that the canonical projection (G, p) ^ (G/H, d) is a submetrv [5], [2], [18].
The search for geodesies of homogeneous (sub-)Finsler manifolds are reduced to the case of Lie groups with left-invariant (sub-)Finsler metrics.
The shortest arcs on Lie groups with left-invariant (sub)-Finsler metrics are optimal trajectories of the corresponding left-invariant time-optimal problem on Lie groups [3]. This permits to apply the Pontryagin maximum principle (PMP) for their search [13]. By this method, in [7] are found all
geodesies and shortest arcs of an arbitrary sub-Finsler metric on the three-dimensional Heisenberg group.
In [8] is proposed a search method of normal geodesies on Lie groups with left-invariant sub-Riemannian metrics. The method is applicable to Lie groups with left-invariant Riemannian metrics, since all their geodesies are normal.
In this paper, to find geodesies of left-invariant (sub-)Finsler metrics on Lie groups and corresponding locally optimal controls in (sub-)Riemannian case we use the geodesic vector field method (Theorems 7, 8) and an improved version of method from [8], applying (co)adjoint representations. The version is based on differential equations from Theorem 9 for controls, using only the structure constants of Lie algebras of Lie groups.
An interesting feature of these two methods in (sub-)Riemannian case is that locally optimal controls on Lie algebras of Lie groups for geodesies and corresponding geodesic vector fields on Lie groups (their integral curves are geodesies, i.e., locally optimal trajectories) can be determined independently of each other. Moreover, controls on different Lie algebras could be solutions of the same mathematical pendulum equation (see sections 6-8).
Analogues of Theorems 4 and 7 (but for the last theorem is only along one geodesic) are proved in the book [22] on the basis of more complicated concepts and apparatus. Apparently, other researchers did not apply PMP for the time-optimal problem to find geodesies of left-invariant metrics on Lie groups.
2. Preliminaries
The left and the right shifts lg : h £ G ^ g ■ h, rg : h £ G ^ h ■ g, g,h £ G, of a Lie group (G, ■) by an element g are diffeomorphisms with the inverse shifts lg-i, rg-i, and their differentials (dlg)h : ThG ^ TghG Wlyd (drg)h : T^G ^ ThgG are linear isomorphisms of tangent vector spaces to G at corresponding points.
There exist an open neighborhoods U of zero in the Lie algebra g = TeG of the Lie group G and W of unit e in G such that exp : U ^ W is a diffeomorphism. If dim G = n then after introduction of arbitrary Cartesian coordinates (xi,..., xn) with zero origin 0 in g, it is naturally identified with Rra. Then exp-1 : W ^ U c Rra is a local chart (a coordinate system) on G in the neighborhood W of the point e £ G. This coordinate system in W is called a coordinate system of the first kind.
The group GL(n) = GL(n, R) of all nondegenerate real squared (n x n)-matrices is a Lie group relative to the global map that associates to each matrix g £ GL(n) its elements gij, i,j = 1,... ,n. Obviously, for every g £ G the mapping I(g) : G ^ G such that
I(g)(h) = g ■ h ■ g~l = (lg o rg-i)(h) = (rg-i o lg)(h)
is an automorphism of the Lie group (G, ■), I(g)(e) = e, and the differential
(dl(g))e := dlg o drg-i : TeG ^ TeG
is a nondegenerate linear map (i.e. an element of the Lie group GL(n) relative to some vector basis in TeG,ii dim G = n), denoted with Ad(g). The calculation rule for the differential of composition gives
Ad(gi ■ g2) = (dl(gi ■ gi))e = (d(I(gi) o I(g2)))e =
(dl(gi))e o (dl(g2))e = Ad(gi) o Ad(g2),
i.e., Ad : G ^ GL(n) is a homomorphism of Lie groups, called the adjoint representation of the Lie group G.
3. Theoretical results
Definition 1. Let (l, [■, ■]) be a Lie algebra; p, q c l are nonzero vector subspaces. By definition,
[p, q] = {[v,w] : v G p,w g q}.
//dimp > 2 then by definition,
m
P1 = P, Pfc+1 = [P, Pk], Pm = ^ Pfc.
k= 1
The vector subspace p c l generates the Lie algebra (l, [■, ■]), if l = pm for some natural number m;
the smallest number m := s with such property is called the generation degree (of the algebra (l, [■, ■]) p
It is clear that subsets from Definition 1 are vector subspaces of l.
Let {e1,..., er} be any basis of the vector subspace p c g, generating the Lie algebra (g, [■, ■]) of a Lie group (G, ■). One can prove the following special case of the Rashevskv-Chow theorem.
Theorem 1. Let (G, ■) be a connected Lie group and a vector subspace p C g generates Lie algebra (g, [■, ■]). Then the control system,
g = dig(u), u g p, (1)
is controllable (attainable) by means of piecewise constant controls
u = u(t) g p, 0 < t < T, (2)
where u(t) = ±ej, j = 1,... ,r, in the constancy segments of the control. In other words, for any elements go,g1 G G there exists a piecewise constant control (2) of this type such that g(T) = g1 for solution of the Cauchy problem
9(t) = dlg(t) (u(t)), g(0) = g0.
Every left-invariant (sub-)Finsler metric d = dp on a connected Lie group G with Lie algebra (g, [■, ■]) is defined by a sub space p c g, generating g, and some no rm F on p. A distance d(g,h) for g,h G G is defined as the infimum of lengths JQT lg(t)ldt of piecewise smooth paths g = g(t), 0 < t < T, such that dlg(t)-i(g(t)) G p and g(0) = g, g(T) = h; T is not fixed, |<jf(i)| = F(dlg(t)-i (g(t))). The existence of such paths and, consequently, the finiteness of d are guaranteed by Theorem 1. Obviously, all three metric properties for d ^re fulfilled. If p = g then d is a left-invariant Finsler metric on G; if F(v) = ^J(v, v), v G p, where {■, ■) is some scalar product on g, then d is a left-invariant sub-Riemannian metric on G, and d is a left-invariant Riemannian metric, if additionally p = g.
The following statements were proved in [4]. The space (G, d) is a locally compact and complete. Then in consequence of S.E. Cohn-Vossen theorem [12] the space (G, d) is a geodesic space, i.e. for any elements g,h G G there exists a shortest arc c = c(t), 0 < t < T, in (G, d), which joins them. This means that c is a continuous curve in G, whose length in the metric space (G, d) is equal to d(g, h). Therefore we can assume that c is parameterized by arc length, i.e. T = d(g,h) and d(c(t1), c(t2)) = t2 — ^ if 0 < t1 < t2 < d(g, h). Then c = c(t), 0 < t < d(g, h), is a Lipschitz curve
relative to the smooth structure of the Lie group G. Therefore this curve is absolutely continuous.
a
almost everywhere defined derivative function c(t), 0 < t < d(g, h), and c(t) = c(0) + c(t)dr, 0 < t<T.
Theorem 2. [3] Every shortest arc g = g(t), 0 < t < T = d(g0,g\), in (G,d) with g(0) = g0, g(T) = g\, is a solution of the time-optimal problem for the control system, (1) with compact control region
U = {u e p : F(u) < 1}
and indicated endpoints.
In consequence of Theorem 2, one can apply the Pontrvagin maximum principle [13] for the time-optimal problem from Theorem 2 and a covector function tp = ip(t) e T*(t) to find shortest arcs on the Lie group G with left-invariant sub-Finsler metric d. The function ip can be considered as a left-invariant 1-form on (G, ■) and therefore it is natural to identify it with a covector function ip(t) e g* = T*G. Then every optimal trajectory g(t), 0 < t < T, is determined by some mesurable optimal control u = u(t) e U, 0 < t < T. Moreover, for some non-vanishing absolutely continuous function tp = ip(t), 0 < t < T, we have
H = H (g,tp,u) = ^(dlg (u))= ip(u), (3)
• 9H 1 9H tA\
g = w * = - n¿, (4)
H(r) := H(g(r),^(T),m(r))= ^(t)(u(t)) = max<P(t)(u) (5)
uEU
for almost all t e [0,T].
Definition 2. Later on, an extremal for the problem from Theorem 2 is called a parameterized curve g = g(t), t e R, satisfying PMP for the time-optimal problem.
Remark 1. For every extremal, H(t) = const := Mo > 0, t e R, [1], [13].
Definition 3. An extremal is called normal (abnormal), if M0 > 0 (M0 = 0). Every normal extremal is parameterized by arc length; proportionally changing ip = ip(t), t e R, if it is necessary, one can assume that M0 = 1. Every normal extremal for a left-invariant (sub-)Riemannian metric on a Lie group is a geodesic, i.e. a locally shortest curve [23].
Theorem 3. ¡8] The Hamiltonian system, for the function H on the Lie group G = GL(n) with the Lie algebra g = g[(n) has a form
g' = g ■ u, g e G, u e g, (6)
^(v)' = ^([u, v]), g eG, u,v e g. (7)
Proof. Each element g e G = GL(n) c Rra is defined by its standard matrix coordinates gij, i,j = 1,... ,n, and ^ is defined by its components ^ij = eij), i, j = 1,... ,n, where eíj e g is a 1
In consequence of (3),
n / n \ n
H(g= ^ ^ij I ^guUlA = ^ (gT^)ijuij. (8)
i,j=1 \l=1 J l,j = 1
The variables gij, Wij must satisfy the Hamiltonian system of equations
9'ij = oH(9, w, u) = ^ 9nUU = (9u)ij, (9)
own l=l
^'ij = -= - = -(^uT)ij. (10)
m=1
dg,
The formula (9) is a special case of the formula (6). It is clear that
n / n \
S E 3il Vln .
,3=1 \l=1 )
= ^(gv) = Aj
i,j=l
On the ground of formulae (9) and (10) we get from here that
n / n \ n / n \
WW = ^ ^ ^güviA + ^ ^ij ^g'uViA = i,j=1 \l=1 / i,j=1 \l=1 /
n / n n \ n In \
y] y] gu Vlj\ + ^ ^ij I ^ gimUmlVlj I
i,j=1 \m=1 1=1 / i,j=1 \l,m=1 I
n / n \ n / n \
- S E9h(vu)IA + ^ ^9Ü(uv)IA = i,j=1 \l=1 J i,j=1 \l=1 J
n i,j=1
which proves the formula (7). □
Theorem 4. [8] The Hamiltonian system, for the function H on a Lie group G with Lie algebra g has a form
g = dig(u), g £ G, u £ g, (11)
^(v)' = ^([u,v]), g £ G, u,v £ g. (12)
PROOF. In consequence of Theorem 3, Theorem 4 holds for every matrix Lie group and for every Lie group (G, ■), because it is known that (G, ■) is locally isomorphic to some connected Lie subgroup (mav be, virtual) of the Lie group GL(n) C Mra . □
It follows from Theorem 4, especially from (12), and Remark 1 that
Theorem 5. //dimG = 3, dimp > 2 in Theorem, 2 then every extremal of the problem from Theorem 2 is normal.
The following lemma holds.
Lemma 1. [16] Let g = g(t), t £ (a, b), be a smooth path in the Lie group G. Then
(9(t)-1)' = -g(t)-lg'(t)g(t)-1. (13)
proof. Differentiating the identity g(t)g(t)-1 = e by t, we get
0 = (g(t)g(t)-1)' = g'(t)g(t)-1 + g(t)(g(t)-1)', □
Theorem 6. [16] Let W e g* = T*G be a covector,
Ad*W(9):=(Adg)*(W)=W ° Ad(g), geG, an action of the coadjoint representation of the Lie group G on W- Then
(d(Ad* W)(w))(v) = ((Adgo)*(ip))([u, v]),
if
u,v e g, w = dl90 (u) eTg0G, g0 eG. G
Ad(g)(v) = gvg-1, dlg (u) = gu, u,v e g, geG.
We choose a smooth path g = g(t), t e (s, e), in the Lie group G such that g(0) = go, g'(0) = w. Then by Lemma 1,
(d(Ad*W)(w))(v) = (W(9(t)vg(t)-1))'(0)=W((9(t)vg(t)-1)'(0)) =
w(9Xfyvg-1 + gov(g(t)-1),(0)) = ^gowg-1 - gov(g-1 g'(0)g-1^ = W(9ouvg-1 - gov(g^goug-1)) = Wigouvg-1 - govug-1) = W(9o[u, v\g-l) = ((AdgoYWWu v}),
as required. □
It follows from Theorems 4 and 6 that
Theorem 7. 1. Any normal extremal g = g(t) : R ^ G (parameterized by arc length and with origin e e G), of left-invariant (sub-)Finsler metric d on a Lie group G, defined by a norm F on the subspace p C g with closed unit ball U, is a Lipschitz integral curve of the following vector field
v(9) = dlg(u(g)), u,(g) = Wo(Ad(g)(w(g)))w(g), w(g) e U,
Wo(Ad(g)(w(g))) = maxWo(Ad(g)(w)),
wGU
where W0 e g* is some fixed covector with max.veu Wo(v) = 1.
2. (Conservation law) In addition, W(t)(9(t)~19*(t)) = 1 for aM t e R where W(t) :=
(Adg(t))*(Wo).
Remark 2. Every extremal with origin g0 is obtained by the left shift lga from some extremal .
Remark 3. In (sub-)Riemannian case, the vector u(g) is characterized by condition {u(g), v) = = Wo(Ad(g)(v)) for all v e p. In Riemannian case, every extremal is a normal geodesic, and we can assume that W0 is unit vector in (p = g, {■, ■)), setting W0(v) = {W0, v), v e g. Moreover, 9(0) = Wo.
Theorem 8. If v(go) = 0, go e G, then an integral curve of the vector field v(g),g e G, with o
factor \dlfl_i (v(go))|.
Proof. Let g(t), t e R, be an integral curve under consideration and set 7 = ^(t) = g0 1g(t), t e R. Then 7 is an integral curve of vector field dlg-i (v(g)), g e G, with origin e. Hence
7 (t) = dlg-i (g(t)) = dlg-i (dlg{t) (u(g(t)))) = dll{t)(u(g(t))). (14)
Ad(g(t))* = Ad(g0 ■ 7(t))* = Ad(7(t))* o Ad(g0)*. (15)
In addition, By definition
u(g(t)) = Ad(g(t))*(^ )(w(g(t)))w(g(t))
(g(t
that by (15) can be rewriten as
Ad(g(t))*(^ )(w(g(t))) = max Ad(g(t))* (^ )(w),
wGU
u(g(t))=Ad(7 (t))*(^0 )(w(g(t))), Ad(7 (t))*(^0 )(w(g(t))) = maxAd(1(t))*(^0 )(w),
where = Ad(go)*C0o). As a result of this and (14), we see that u(g(t)) plays a role of u(7(t)) for constant covector (instead of ^o). Due to point 2 of Theorem 7 the curve 7(t) is a normal extremal parameterized proportionally to arc length with the proportionality factor |d/ -i(v(go))|. Then its left shift g(t) = go ■ j(t) also has this propertv. □
Remark 4. Theorem 8 holds for left-invariant Riemannian metrics on (connected) Lie groups. In this case, v(go) = 0 fa all % e G.
Let us choose a basis {e1,..., en} in g, assuming that {e1,..., er} is an orthonormal basis for the scalar product (■, ■) on p in case of left-invariant (sub-)Riemannian metric. Define a scalar product (■, ■) on g, considering {e1,..., en} as its orthonormal basis. Then each covector ^ e g* can be considered as a vector in g, setting ^(v) = (^, v) for every v e g. If ^ = Y1 v = Y1 n=i vkek,
then ^(v) = ^ ■ v, where ^ and v are corresponding vector-row and vector-column, ■ is the matrix multiplication. If Z : g ^ g is a linear map, then we denote by (I) its matrix in the basis {e1,..., en}.
If g(t), t e R, is a normal geodesic of a left-invariant (sub-)Riemannian metric dona Lie group G, then u(g(t)) is the orthogonal projection onto p of the vector (Adg(t))*(^o) in the notation of
(■, ■) g.
Theorem 9. Every normal parameterized by arc length geodesic of left-invariant (sub-) Riemannian metric on a Lie group G issued from the unit is a solution of the following system, of differential equations
n
g(t) = dlg{t)(u(t)), u(t) = £i>l(t)el, K0)| = 1, fa(t) = ^^ck(t), (16)
i=1 k=1 i=1
where j = 1,... ,n, ck are structure constants of Lie algebra gin its basis {e1,..., en}. In Riemannian case, r = n.
Corollary 1.
mi = |u(i)l = 1, t e R. (17)
proof. The first equality in (17) is a consequence of the first equality in (16) and left in variance of the scalar product. Therefore, due to the equality |^^(0)| = 1, it suffices to prove that |(u(t),u(t)) = 0. Now by (16),
^ ^ r n r
| (u(t),u(t)) = | £ 2(t) | =2^2 ^ (t) = 4 v, (
,3=1 ) 3=1 k=1 i,3=1
which is zero by the skew symmetry of ck with respect to subscripts. □
Remark 5. In fact, the same equations for tpj(t) from (16) in a different interpretation were obtained in [21] as "normal equations". Their derivation there uses more complicated concepts and techniques.
4. Lie groups all of whose left-invariant Riemannian metrics have constant negative curvature
The only Lie groups which do not admit left-invariant sub-Finsler metrics are commutative Lie groups and Lie groups Gn, n > 2, consisting of parallel translations and homotheties (without rotations) of Euclidean space En_i [5], [17]. Up to isomorphisms, Lie groups Gn can be described as connected Lie groups every whose left-invariant Riemannian metric has constant negative sectional curvature [24].
The group Gn, n > 2, is isomorphic to the group of real block matrices
9 = (yxE'n~X V!) , (18)
where En_i is unit matrix of order n — 1, yT is a transposed (n — 1) —vector-row y, 0 is a zero (n — 1)—vector-row, x > 0.
It is clear that in vector notation the group operations have a form
(yi,xi) • (y2,x2)=xi(y2,x2) + (yi, 0), (y,x)-1 = x-1 (—y, 1). (19)
Let Eij, i,j = 1,...,n,be a (n x n)-matrix having 1 in the ith row and the jth column and 0 in all other places. Matrices
n_ i
CA = Ein, i = 1,...,n — 1, en = ^2,Ekk (20)
k=i
constitute a basis of Lie algebra gn of the Lie group Gn. In addition,
[ei, ej ] = 0, i,j = 1,...,n — 1; [e,n, ei] = a, i = 1,...,n — 1,
so all nonzero structure constants in the basis {ei,..., en} are equal to
4 = — ¿in = 1, i = 1,...,n — 1. (21)
Let (•, •) be a scalar product on gn with the orthonormal basis ei,..., en. Then we get left-invariant Riemannian metric d on the Lie group Gn of constant sectional curvature —1 [24].
On the ground of Theorem 9 and (21), ^j, = t), i = 1,...,n, are solutions of the Cauchv problem
{n_i
i>i(t) = Wi(t)tPn(t), i = 1,...,n — 1, ipn(t) = — £ i>1 (t);
n i=i (22)
Wi(0) = <Pi, i = 1,...,n, =1.
i=i
It follows from (22) that
n_ i
Mt) = —2ipn(t) ^rf(t) = 2ipn(t)ipn(t) = №)'(t), =i
whence on the ground of initial data of the Cauchv problem (22), it follows that
Mt) = ^n(t) — 1, Wn(0) = ^n-
Solving this Cauchv problem, we find that
, , . fn coshi — sinhi
Wn(t) = -—-r-T-7 .
coshi — fn smhi
= 1, . . . , n — 1
t
, . , , ,. f fn cosh t — sinh t , ,,, . ,i,ii
In \ipi(t)\ = -:-——dr + ln\ipi\ = — ln | cosht — fn sinht\ + ln\(fi\,
J cosh t — fn sinh t 0
if fi = 0, so
Mt)= ,, fi ■ , ,, i = 1,...,n — 1, cosh t — (fn sinh t
and these formulae are true also when fi = 0. Consequently, on the ground of (16),
U(t) = cosht -fn sinht (£ fi6i + ((n cosh * — sinh f) . (23)
n
If g £ Gn is defined by formula (18), u = £ uiei £ gn, then
=i
gu = ( (xUn)0En_i , v = (xui,..., xUn_i)T . (24)
Therefore on the base of Theorem 9 and (23) in the notation (18), the corresponding parameterized by arc length normal geodesic g = g(t), t £ R, of the space (Gn, d) with g(0) = e is a solution of the Cauchv problem
{xc(f) = x(t), m = x(l), i = 1,..., n — 1, /25)
x(0) = 1, yi(0) = 0, i = 1,...,n — 1. K '
Solving the problem, we find
t
x(t) =_1_, yi(t) = f__=_( sinht_. (26)
coshi — fn sinhi' i J (coshi — fn sinhi)2 coshi — fn sinhi
0
This implies that
x(t) = e±t, yi(t) = 0, i = 1,...,n — 1, if (n = ±1. (27)
Let (f n < 1- Let us show that for any t £ R, the equality
n_ i n_ i
Y,(Vi(t) — ai)2 +x2(t) = + 1 (28)
=i =i
holds, where a^ i = 1,...,n — 1, are real numbers such that
n— 1
am = <Pn. (29)
i=1
n-1 n-1
We introduce a function f (t) = ^(yi(t)—ai)2 +x2(t). Due to initial data (25),/(0) = ^ a2 + 1.
i=l i=1 On the ground of (25), (26) and last equation in (22), we get
2 f w = fsm- <n)vi w+norn = E (cosh1;^ t - «) w+
=l =l
s'nh t[Y1^'i — 1)+<£n cosht n-i <pn cosh t — s'nh t \i=1 ) sr^
cosht — fn s'nhi cosht — fn s'nhi
n-1
Pn — ai^i = 0.
i=1
Consequently, f (t) = f(0) and the equality (28) is proved. It is easy to check that the equality (29) holds for
n- 1
• 1 „ 1. .........
=1
ai = <pi<fin/(1 — Pn), i = 1,...,n — 1; moreover ^ a2 + 1 = --2. (30)
^ 1 — Wn
=1
These numbers a-i are obtained as halves of sums of limits ^(i) when t ^ t ^ —«, which
are equal to <$i/(1 — <pn) and —<^i/(\ + <pn) respectively.
Formulae (19) show that the group Gn is a simply transitive isometrv group of the famous Poincare's model of the Lobachevskv space Ln in the half space R++ with metric ds2 = = (mldyl + dx2)/x2.
Ln
this model, passing through the point (0,..., 0,1), are semi-straights or semi-circles (with centers (a1,... ,an-1, 0) and radii 1/^J1 — <^1) (30)), orthogonal to the hvperplane Rn-1 x {0}. Since all other geodesies are obtained by left shifts on the group, in other words, by indicated parallel translations and homotheties of this model, then also all straights and semi-circles, orthogonal to the hvperplane Rn-1 x {0}, are geodesies of the space Ln.
We got a well-known description of geodesies in this Poincare's model.
Now let us look what the vector field method gives us for the problem.
Every vector p £ gn can be considered as a covector g*, setting p(v) = (tp, v) for v £ gn. Then any (co)vector po from Theorem 7 has a form
n n
po = ^2^^ = 1.
=1 =1
n
Let w = wiei £ fln) 9 £ Gn is defined by formula (18). It is easy to see that =1
n- 1
Ad(g)(w) = gwg-1 = ^(wiX — w,nyi) ei + w,ne.n,
n_ i n_ i n_ i
{tpo, Ad(g)(w)) = Y,(^x — WnVi)fi + Wnfn = x ^ fiWi + (fn — ^ fiUi) Wn. =i =i =i
It is clear that
n_ i n_ i
u(g) = x fiei + \(fn — fiVi) en, =i =i n n_ i n_ i
v(g) = gu(g) = x ^ Uiei = x2 ^ (eA + x i(,n — ^ fyA e,n.
i=i i=i \ i=i J
= ( ) £ R (0) =
x(t) = (fn — fiVi(t)^ x(t), yi(t) = fix2(t), i = 1,...,n — 1, x=0) = 1, yi(0) = 0, i = 1,..., n — 1.
x( ),
lnx(i) := z(t). Differentiating both sides of the resulting equation and using the second equation
n
in (31) and the equality ^f2 = 1, we get
i
n_ i
z(t) = — £f2x2(t) = —(1 — fn)e2z(t\ z(0) = 0, ¿(0) = fn. i
If fn = ±1 then z(t) = 0 and due to the initial data and the second equation in (31), we get z(t) = ±t, x(t) = e±t, yi(t) = 0, i = 1,...,n — 1.
Let 0 < fn < 1. Let us multiply both sides of the resulting equation by 2z. Then
(31)
2zz = —(1 — fn)e 2z, d(z)2 = —(1 — fzn)e2zd(2z), z2 = —(1 — f2n)e2z + C.
Taking into account the initial conditions for z(t),we get C = 1 and z(t)2 = 1 — (1 — f2n)e2z(t). The expression on the right is positive for t sufficiently close to zero. Therefore, with these t, we get
z(t) = ,
where the sign coincides with the sign of fn, if fn = 0. Separating variables, we get
d d
dt =
ez y/T-% /(1 - &)) - i
= ^d cosh
y/(e-2z/(1 - &)) - 1 \ \y/i=
^cosh 1
<P-
n ,
± cosh i | —6 | = c — t, c = cosh i | —, 1 |
V1 — (p2J ' W1-^,
cosh
e z(t') . , . , cosh t - юп sinh t
= cosh с cosh t — sinh с sinh t =
\/1 — fn ^J1 — f
x(t) = ez(t) =-1-
cosht - ipn sinhi
Since the right sides of the system of differential equations (31) are real analytic, this equality is true for all t e R. We obtain from this and the second system in (31) the same solutions yi(t), te R, i = 1,...,n — 1, as in (26).
Using formulae (19) and (26) for x = x(t), yi = yi(t), we shall find a formula for distances d between group elements, or, which is the same, between points of the Lobachevskv space in Poincare's model under consideration. We obtain from (26)
1 , coshi + wn sinhi coshi + wn sinhi
- = cosh t — ifn sinh t, x =-2---^ =---—-,
x cosh21 — sinh21 1 + (1 — Vn) sinh2 t
n— 1
i=1
n— 1
yi/x)2 =sinh2 = (1 - vn)sinh2
i=1
[x2 + £ y^ = Xc[x2 + £ ^ ,
[l + x2 + ]—1 y^ ,
cosh t + Vn sinh t = XX ( x2 + Vi ) = X ( x2 + Vi
COSht = 2x \1 + x +' X
d((0,1), (y, x)) = cosh
1
2x
(
n— 1
1+x2 + £ Vi
=1
d
(Vl,xi) (y2,x2)=x— (-yi, 1)(y2,x2) = (x— (y2 - yi),x— x2), d((yi,xi), (V2,x2)) = d((0,1), (x——1(y2 - yi),x—1x2)) =
cosh
-1
cosh
2x1x2
1
n 1
xi (1 + x2 + (y2>* -
x x
2x2
n 1
=1
xi +x2 ^ y2,i - yi,i)2
=1
= d((yi,xi), (V2,x2)).
5. The three-dimensional Heisenberg group
This Heisenberg group is a nilpotent Lie group of upper-triangular matrices
1 x
H = < h = i 0 1 y | } , x,y,z e R. 001
It is easy to compute that
1 — x xy - z h—1 = i 0 1 -y 0 0 1
(32)
(33)
(34)
Clearly, H is naturally diffeomorphic to R3 and H is a connected Lie group with respect to this differential structure. Matrices
010 1 = i 0 0 0 000
2=
000 001 000
3=
001 000 000
(35)
1
1
constitute a basis of Lie algebra If) of Heisenberg group H. In addition,
[ e 1, e2] = eie2 — e2ei =
Hence the vector subspace p c ) with basis {e1, e2} generates ).
Thus the triple (H, ), p) satisfies all conditions of Theorems 1 and 2.
Let us search for all geodesies of the problem from Theorem 2. They are all normal by Theorem 5, and we can use Theorem 7.
Let us define a scalar product (■, •} on ) with orthonormal basis {e 1, e2, e3}. Then each vector p £ I) can be considered as a covector from )*, if we set p(v) = (tp, v} for v £ ). Then any (co)vector p0 from Theorem 7 has a form
p0 = cos£>e1 +sin{e2 + fie3, £ R.
(36)
Let
v £ p, Vk £ R, k = 1, 2.
Using formulae (33), (34), we get
0 v1 —y v1 + xv 2 0 0 V2
0 0 0
v(h) = hu(h) = (cost; — fiy) e1 + (sin{ + 0x) e2 + x(sin{ + 0x) e3.
Therefore h(t) is a solution of the Cauchv problem
(37)
x(0) = (0) = (0) = 0 Let us turn to the coordinate system, X, y, z of the first kind on the Lie group H :
Hence X = x, y = y, z = z — (xy)/2. It is easy to see that for 0 = 0 we get
x(t)=tcos y(t)=tsin z(t)
2
cos {sin z(t) = 0, t£ R,
and geodesic is a 1-parameter subgroup
g(t) = exp(i(cos{e1 + sin{e2)), t <E R. If 3 = 0, the calculations are more difficult:
x = —3y = —3(sin{ + fix) = —32x — 3 sin
sin
x(t) = Ci cos ¡t + C2 sin ¡t----1.
3
Since x(0) = 0, x(0) = cos£, then Ci = (sin 0/3, C2 = (cos 0/3,
m = ^i«*3- wo = - **<■>-, o«)
y = 3x = 3 (cos £ — 3y) = —32y + 3 cos 0
cos
y(t) = C1 cos 3t + C2 sin 3t + —--1.
3
Since y(0) = 0, y(0) = sin£, then Ci = —(cosO/3, C2 = (sin^/3,
y(t) = \(— cosí; cos3t + sin£sin3t + cosO = \(— cos(lI + 3t) + cos£), (39)
3 3
( x ) 1 1
2 = 2--=xy — 2¿(xy + xy) = 2¿(xy — xy) =
[(sin(Z + 3t) — sinO sin(Z + 3t) — cos(i + 3t)(— cos(C + 3t) + cosC)] =
[1 — (sin£sin(£ + 3t )+cos(C + 3t )cosO] = (1 — cos3t ) = z'. Since ¿(0) = 0 then
1 sin 3
'(t) = w{t — -J3-)' te R (4">
It follows from equalities (38), (39), (40) that the projection of geodesic g = g(t) onto the plane x,y is a circle with radius 1/\3\ and center (1/3)(— sin0 cost;), T = 2tt/\3\ is a circulation period, while z(t), t <E R, does not depend on the parameter Therefore, if we fix 3 = 0 then for different £ all geodesic segments g(3, t), 0 <t< 2tt/\3\, start at e and finish at the same point. It
3 = 0
(respectively, 3 = 0) then every segment (respectively, of the length less or equal to T = 2n/\3\) of these geodesies is a shortest arc. There is no other geodesic or shortest arc except indicated above and their left shifts.
6. Controls for left-invariant sub-Riemannian metrics on SO(3)
It is well known that every two-dimensional vector subspace p of Lie algebra (so(3), [■, ■]) of the Lie group SO(3) generates so(3). Moreover, there exists a basis {e1, e2} of the space p such that [e2, e3] = e1, [e3, e1] = e2 for the vector e3 = [e 1, e2]. Let (■, ■) be a scalar product on so(3) with orthonormal basis {e1, e2, e3}. Then if a scalar product (■, ■) on p defines a left-invariant sub-Riemannian metric d on the Lie group G = SO(3), then there exists a basis {v,w} in p that is orthonormal relative to (■, ■), orthogonal relative to (■, ■), and such that (v, v) = a2 < b2 = (w,w), [v,w] = (ab)e3, where 0 < a < b. Let v,w be new vectors e1, e2. Then
[e 1, e.2] = (ab)e3, [e3, ei] = (b/a)e2, [e.2, e*] = (a/b)ei, 0 <a < b. (41)
It follows from (41) that all nonzero structure constants are
c12 = — 41 = c31 = — c13 = b/a, c23 = — c^2 = a/b.
Let g(t), t £ R, be a geodesic of the space ( SO(3), d), parameterized by arc length, and g(0) = e. On the ground of Theorem 9,
g'(t) = g(t)u(t), u(t) = p1(t) d + p2 (t)&2,
where
a2 2
p[(t) = —abp2(t)p3(t), p2(t) = abp1(t)p3 (t), P3(t) = ———p1(t)p2(t). (42) Since lu(t)l = 1 then p1(t) = cos((t), p2(t) = sin((t) and (42) is written as
— sin((t)((t) = — ab sin£(t)pp3(t), cos£(t)£(t) = ab cos £(t)pp3(t),
a2 2
m = ^^ cosat)sinat).
Then p3(t) = 1 (t) and ( = £(t) is a solution of the differential equation
a2 2
e(t) = —¿—sin^t). (43)
If a = b then (''(t) = 0, ('(t) = const = 0. Then geodesies are obtained from geodesies in the a = = 1 = / a.
d, a = = 1
found in papers [9] and [10].
The case 0 < a < b is reduced to the case a2 — b2 = — 1 by proportional change of the metric d. Then the variable w(t) := 2£(t) allows us to rewrite the equation as the mathematical pendulum equation
u''(t) = — sinu(t). (44)
In [11], I.Yu. Beschastnvi and Yu.L. Sachkov studied geodesies of left-invariant sub-Riemannian S O(3)
replacement b2 — a2 by a2 and £ by p, the equation (43) coincides with the equation (2.4) from their paper, obtained by another method.
7. To search for geodesies of a sub-Riemannian metric on SH(2)
S H(2)
/ A v \ , ( coshw sinhw \ ( x\ ^2 . ,„s
°={o 1 ); A ={s°n4 coshl), v={ y) £ R. <45>
It is not difficult to see that
^ (A v\ 1 (A-1 —A-1v \
V 0 1) \ 0 1 )'
(46)
Clearlv, matrices
0 1 0 0 0 1 0 0 0 e\ = | 100 I , e2 = I 0 0 0 I , e3 = I 00 1 | (47)
0 0 0 0 0 0 0 0 0
constitute a basis of Lie algebra sh (2). In addition,
[ ei, = ез, [e2, ез]=0, [e 1, e3] = e2. (48)
Let us define a scalar product {■, •} on sh (2) with orthonormal basis {e1, e2, e3} and the subspace p with orthonormal basis {e1, e2} generating Lie algebra sh(2). Thus a left-invariant sub-Riemannian metric d is defined от the Lie group SH(2).
Let us take а (со)vector = cosae1 + sinae2 + /e3 £ sh (2). We calculate
фд(w) = {ipg, w} = {ipo,gwg-1} g £ SH(2), w = wiei +W2e2 £ p.
cosh p sinh p x \ / 0 w1 w2 \ / cosh p — sinh p —x cosh p + y sinh p gwg-1 = I sinhp coshp у I I w1 0 0 II — sinhp coshp x sinhp — у cosh p 0 0 1/ \0 0 0/ \0 0 1
= w1e1 + (—w+ w2 coshp)e2 + (—w1x + w2 sinhp)e3, фд (v) = w1 cos a + (—w+ w2 coshp) sin a + (—w1 x + w2 sinhp) /3 = w1(cosa — у sin a — /x) + w2(coshp sin a + / sinhp).
Therefore,
u(g) = (cos a — у sin a — /x)e 1 + (sin a cosh p + / sinh p)e2, v(g) = gu(g) =
cosh p sinh p x \ / 0 cos a — y sin a — /x sin a cosh p + / sinh p
sinh p cosh p I I cos a — sin a — / x 0 0
0 0 1 J \ 0 0 0
sinh p(cos a — у sin a — /x) cosh p(cos a — у sin a — /x) cosh p(sin a cosh p + / sinh p) cosh p(cos a — y sin a — /x) sinh p(cos a — у sin a — /x) sinh p(sin a cosh p + / sinh p) 0 0 0
Hence integral curves of vector field v(g), g £ SH(2), satisfy the system of differential equations
p = cos a — y sin a — /x,
x = coshp(sina coshp + / sinhp), (49)
у = sinh p(sin a cosh p + / sinh p).
The geodesic g(t), t £ R, with g(0) = e is a solution of this system with initial data p(0) = x(0) = y(0) = 0. In this case, lu(g(t))l = 1, i.e.
g(t) £ M1 = {(sina coshp + / sinhp)2 + (cosa — ysina — /x)2 = 1} С SH (2). (50)
Therefore there exists a differentiable function 7 = ^(t) such that
7 7
cos—= sin a coshp + / sinhp, sin— =cosa — ysina —/x. (51)
Since p(0) = x(0) = y(0) = 0^ ^^^^ ^^^^ ^ад assume that 7(0) = ж — 2a. On the ground of (51) the sistem (49) is written in the form
p = sin 2,
x = cos 2 coshp, (52)
у = cos 2 sinhp.
Differentiating the first and the second equalities in (51) and using (52), we get
7 7 7
— — sin- = (sinasinhp + fcoshp)p = sin — (sinasinhp + fcoshp),
7 ' '
— cos — = — y sin a — fee = — cos — (sin a sinh p + f cosh p),
whence
7 = —2(sin a sinh p + f cosh p), 7(0) = —2f3. Consequently, on the ground of the first equality in (51) and (52)
7 7
7 = —2(sin a cosh p + f sinh p)p = —2 cos — sin — = — sin 7.
We got the mathematical pendulum equation. In paper [19] this equation together with equations
p .
8. To search for geodesies of a sub-Riemannian metric on SE(2)
The Lie group SE(2) is isomorphic to the group of matrices of a form
(A M ; A = ( cosp — sinp ) , ,= ( M € R2. (53)
\ 0 1 J \ sinp cosp J \y J
The same formula (46) is true. It is clear that matrices
/0 -1 0 \ / 0 0 1 \ /000
ei = I 1 0 0 I , = I 000 I , ез = I 001 | (54)
\ 0 0 0 J \ 0 0 0 J \000
constitute a basis of Lie algebra se(2). In addition,
[ ei, e2] = ез, [ei, ез] = -e2, [e2, ез] = 0. (55)
Let us define a scalar product (-, •) on se(2) with orthonormal basis [e 1, e2, e3} and the subspace p with orthonormal basis [e 1, e2} generating Lie algebra se(2). Thus a left-invariant sub-Riemannian metric d is defined от the Lie group SE(2) (see [6], [25], [27] and other papers). Let us take a (co)vector = cosae 1 + sinae2 + 3e3 £ se(2). We calculate
фд(w) = {ipg, w) = {ipo,gwg-1), g £ SH(2), w = wiei + W2e2 £ p.
. 4 / 0 -w1 w2 \ / cosp sin^ —x cosp — у sinp
_1 . cosp sin p x \ I n n I I
gwg 1 0 0 л I \ w1 0 ^ — s.np cosp x s.np — у cosp
0 0 'V000/V0 0 1
1=(
w\e\ + (w\y + w2 cos p)e2 + (—w\x + w2 sinp)e3, (w) = w\ cos a + (w\y + w2 cos p) sin a + (—w\x + w2 sin p)f = w\(cos a + y sin a — fx) + w2 (sin a cos p + f sin p).
Consequently,
u(g) = (cos a + y sin a — fx) e\ + (sin a cosp + f sinp) e2, v(g) = gu(g) =
cosp — sinp x \ / 0 — cos а — у sin а + (x sin а cosp + ( sinp
sin p cos p у I I cos а + у sin а — fix 0 0
0 0 1 / \ 0 0 0
sin p((3x — cos а — у sin а) cos p((3x — cos а — у sin а) cosp(sinа cosp + ( sinp) \ cosp(cosа + y sin а — (x) sin p((3x — cos а — у sin а) sinp(sinа cosp + ( sinp) I. 000
Hence integral curves of vector field v(g), g £ SE(2), satisfy the system of differential equations
p = cos а + у sin а — fix,
X = cosp(sinа cosp + ( sinp), (56)
у = sin p(sin а cos p + ( sin p).
The geodesic g(t), t £ R, with g(0) = e is a solution of this system with initial data p(0) = x(0) = y(0) = 0. In this case, lu(g(t))l = 1, i.e.
g(t) £ Mi = {(sin а cosp + ( sinp)2 + (^а + ysinа — fix)2 = 1} с SE (2). (57)
Therefore there exist differentiable functions ш = w(t) = 2£(i) such that
sin Ш^ = sinа cosp + ( sinp, cos ш(^=cosа + y sin а — fix. (58)
Given the equality p(0) = x(0) = y(0) = 0 we can assume that ш(0) = 2£(0) = 2а. On the ground of formula (58) the system (56) is written in a form
p = cos 2,
x = sin j cosp, (59)
у = sin j sinp.
Differentiating the first and the second equalities in (58) and using (59), we get Ш cos Ш = — (sin а sin p — ( cos p) p = — cos Ш (sin а sin p — ( cos p),
—Ш sin Ш = y sin а — fix = sin Ш (sin а sin p — ( cos p),
whence
ш = 2(( cosp — sin £ sin p), ш(0) = 2£(0) = 2(3. (60)
Differentiating the last equality, we get in view of formulae (58) and (59)
ш = —2(3 sin p + sin а cos p)p = —2 sin Ш cos Ш = — sin ш. (61)
We get again the mathematical pendulum equation.
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22. Jurdjevich V. Geometric control theory. — Cambridge: Cambridge Univ. Press, 1997.
23. Liu W., Sussman H. J. Shortest paths for sub-Riemannian metrics on rank-two-distributions // Memoirs of the Amer. Math. Soc. — Vol. 118, no. 564. ^(Amer. Mth. Soc., Providence, 1995).
24. Milnor J. Curvatures of left invariant metrics on Lie groups // Adv. Math. —1976. —Vol. 21. —P. 293329.
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26. Montgomery R. A tour of subriemannian geometries, their geodesies and applications. — AMS, 2002.
27. Sachkov Yu. L. Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane // ESAIM: ('OCX". 2010. Y. 16, no. 4. — P. 1018-1039.
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Получено 14.09.2019 г.
Принято в печать 11.03.2020 г.
