Geometrical structures associated to implicit first order dynamical systems with some holonomic constraints Virgil Ob˘adeanu and Monica Ciobanu Dedicated to the 70-th anniversary of Professor Constantin Udriste Abstract. In [5], [6], [7], [8], it was shown that to any implicit first (or higher) order dynamical system without constraints, it corresponds, on the configuration space, a geometrical structure given by a (nondegenerate and in general not symmetric) generalized d-metric and a nonlinear d-connection. Thus, there exist a covariant differential operator, a parallel transport of the vector fields and autoparallel curves. The following property holds: if the equations which describe the dynamics of the system do not depend explicitly on time, the set of autoparallel curves associated to the structure build in this way, coincide with the set of solutions of the given dynamical system. In this paper we generalize these results to the case of implicit first order dynamical systems with holonomic constraints. M.S.C. 2000: 34C40, 53C07, 70G45, 70H45. Key words: Dynamical systems; nonlinear connections; holonomic constraints.
1
Geometrical structures defined on a differentiable manifold
Let M = Mm be a differentiable manifold of dimension m. On it we define the following geometrical structures ([6]).
1.1
Natural frames ([1], [2])
On a change of local chart, the coordinates on T M change by the rules: (1.0)
x ¯h = x ¯h (xi ), x ¯˙ h =
¯˙ h ∂x ¯h ∂x ¯h i ∂ x x ˙ , = . ∂xi ∂ x˙ i ∂xi
∗ Proceedings 17. The International Conference ”Differential Geometry-Dynamical Systems 2009” BSG (DGDS-2009), October 8-11, 2009, Bucharest-Romania, pp. 151-162. c Balkan Society of Geometers, Geometry Balkan Press 2010. °
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Virgil Ob˘adeanu and Monica Ciobanu
The corresponding natural frames change respectively by the rules: ∂ ∂xi ∂ ∂xi ∂ ∂ x˙ i ∂ ∂ = + , = , ∂x ¯h ∂x ¯h ∂xi ∂x ¯h ∂ x˙ i ∂ x ∂x ¯h ∂ x˙ i ¯˙ h (1.00 ) ∂x ¯˙ h i ∂ x ¯h i ∂x ¯h i ˙h = dx , d x ¯ dx + dx˙ . d¯ xh = ∂xi ∂xi ∂xi It follows, by the formulas (1.00 ), that in each point x of the domain of the con∂ generate a subspace of the space T Mx , which does not sidered chart, the vectors ∂ x˙ i depend on the chosen chart, called vertical subspace.
1.2
Generalized Lagrange d-metric
It is called generalized Lagrange d-metric defined on M , a d-tensor of the second rank, two times covariant and nondegenerated (in general not symmetric). In a local chart (U, χ), with the coordinated (xi ) of a current point ¡ ¢ ¡ x, the ¢components of the d-tensor are given by the functions a1ij = a1ij t, xh , x˙ h , t, xh , x˙ h ∈ J 1 M , with the property that, on a change of local chart, they change by the rules: ¡ ¢ ∂xi ∂xj 1 aij , det a1ij 6= 0, ∀x ∈ U. h k ∂x ¯ ∂x ¯ We make no additional hypothesis related to the symmetry or signature. ¡ ¢ ij h By the definition, it follows the existence of the functions aij ˙h , 1 = a1 t, x , x components of the reciprocal matrix of (a1ij ). On a change of local chart, we have: (1.1)
(1.10 )
1.3
a ¯1hk =
a ¯hk 1 =
∂x ¯h ∂ x ¯k ij a . i ∂x ∂xj 1
Nonlinear d-connection
We say that on the manifold M it is given¡ a nonlinear d-connection, if in each ¢ local chart we have a set of functions a0ij = a0ij t, xh , x˙ h , which change, on a change of chart, by the rules: · ¸ ∂xi ∂xj 0 ∂ x˙ j 1 (1.2) a ¯0hk = a + a . ∂x ¯h ∂ x ¯k ij ∂ x ¯k ij The functions a0ij are called Christoffel symbols of first type of the d-connection.
1.4
Mixed coefficients of connection
By contracting the above coefficients with the contravariant components of the d-metric, we obtain the functions: (1.20 )
0 Mji = aih 1 ahj ,
called the Christoffel symbols of second type or coefficients of the nonlinear connection. They change, on a change of local chart, by the rules ([3], [4]): · ¸ ∂x ¯i ∂xk h ∂ x˙ k h i M + δ . (1.3) Mj = ∂xh ∂ x ¯j k ∂x ¯j k
Geometrical structures
1.5
153
Adapted frames
Given a d-connection, we can build an adapted dual frame and, thus, an adapted frame by ([4]): δ ∂ ∂ ( i = − Mij j , δx = dxi (= δji dxj ), i i δx ∂x ∂ x˙ and (1.4) δ ∂ δ x˙ i = dx˙ i + Mji dxj = (Mji dxj + δji dx˙ j ) = , δ x˙ i ∂ x˙ i respectively. On a change of local chart, the rules of change: (1.5)
δx ¯i =
∂xh δ ∂x ¯i h ∂x ¯i h δ ∂xh δ δ ˙i = δx , δ x ¯ δ x ˙ , and = , = ∂xh ∂xh δx ¯i ∂x ¯i δxh δ x ∂x ¯i δ x˙ h ¯˙ i
hold respectively.
δ generate a second subspace, δxi called horizontal subspace, and the tangent space to T M in a point (x, x) ˙ is the direct sum of the vertical and horizontal spaces in that point. δ δ The Lie brackets of the vector fields and are expressed by the formulas: δxi δxj ¸ · δMjh δMih δ δ h δ h = , = R , where: R − . ij ij δxi δxj δ x˙ h δxj δxi By the formulas (1.5) it follows that the vectors
This tensor is called tensor of curvature. On a change of local chart, its components ∂x ¯q ∂xi ∂xj p q change by the rules: Rhk = R . ∂xp ∂ x ¯h ∂ x ¯k ij
1.6
Differential operator, covariant derivative on M associated to a nonlinear d-connection
We call d-vector field on M a special vector field on J 1 M , locally written as ∂ X = X i (t, x, x) ˙ , which change, on a change of local chart, by the rules: ∂xi (1.6)
i
X =
∂x ¯i h X ∂xh
and we denote there set by Xd (M ). It is obviously that X(M ) ⊂ Xd (M ). Let given the spaces (rings) Λo (J 1 M ), Λo (J 2 M ), the modulus Xd1 (M ), where X1 ∈ ∂ , and a nonlinear connection N of local components Xd1 (M ) if X1 = X1i (t, x, x) ˙ ∂xi j Mi . A differential operator D : Λo (J 1 M ) → Λo (J 2 M ), is defined, along a curve df ∂f ∂f ∂f xi = ci (t), by D : f → Df = = + x˙ i i + x ¨i i . To the connection N it dt ∂t ∂x ∂ x˙ d is associated a linear differential operator DN , defined on the µ set X¶ 1 (M ) of the d∂ ∂ d vector fields and with values in X2 (M ), given by its values: DN = Mij j on ∂xi ∂x
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the bases vectors and satisfying the relation DN (f X) = f DN (X) + D(f )X, for any f ∈ Λo (J 1 M ), X ∈ Xd1 (M ). On an arbitrary field · d-vector ¸ X, the local expression of dX i ∂ the operator DN is given by DN (X) = + Mji X j and it is called the N dt ∂xi δX i dX i covariant derivative of X. The expressions = + Mji X j are called covariant δt dt derivatives of the components of the d-vector field X with respect to the nonlinear δX i connection N . On a change of local chart, the covariant derivatives change by δt i δX ∂x ¯i δX h the rules: = . δt ∂xh δt ∂ We say that a d-vector field X = X i i is transported by parallelism, along a curve ∂x dX i i i x = c (t), if, along this curve, its covariant derivative vanish: + Mji X j = 0. dt j d2 ci i dc + M = 0, that means its A curve xi = ci (t), is called autoparallel if j dt2 dt i dc tangent vector is transported by parallelism. dt
2
Submanifolds Vp of a structured differentiable manifold (Mm , a1ij , a0ij )
Let V = Vp be a submanifold of the differentiable manifold M , locally defined by the functions (2.0)
xi = xi (ua ), i = 1, m, a = 1, p, 1 ≤ p < m,
which give us a parametrical representation of V [9]. By the definition of the submanifold, it is assumed that the condition: ° i° ° ∂x ° 0 ° (2.0 ) rank ° ° ∂ua ° = p is fulfilled. The atlas of the submanifold V is not necessarily the restriction to V of the atlas of M .
2.1
Induced frames on V
Given a canonical frame by the relations (1.00 ), it induces on V the frame µ ¶ ∂ ∂ and on T V the frame , by the formulas: ∂ua ∂ u˙ a
µ
∂ ∂ua
¶
∂xi ∂ ∂ ∂xi ∂ ∂ x˙ i ∂ ∂ = , = + . ∂ u˙ a ∂ua ∂ x˙ i ∂ua ∂ua ∂xi ∂ua ∂ x˙ i It is necessary that, on a change of local chart µ on M and ¶ a change of parameters on ∂ ∂ V , respectively, the elements of the frame , to change by the rules: ∂ua ∂ u˙ a
(2.1)
(2.2)
∂ua ∂ ∂ u˙ a ∂ ∂ ∂ ∂ua ∂ = + , . = ∂u ¯b ∂u ¯b ∂ua ∂u ¯b ∂ u˙ a ∂ u ∂u ¯b ∂ u˙ a ¯˙ b
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We have: Lemma 1. The following diagram is commutative: ¾ ½ ¾ ½ ∂ ∂ h1 −→ ∂ x˙ i ∂x ¯˙ h ½
v1 ↓ ∂ ∂ u˙ a
¾
&&
½
h2
−→
↓ v2 ∂ ∂u ¯˙ c
¾
∂ ∂ ∂xi ∂ → = . ∂ x˙ i ∂u ¯c ∂ x˙ i ∂u ¯˙ c Lemma 2. The following diagram is commutative: ½ ¾ ½ ¾ ∂ ∂ h1 −→ ∂xi ∂x ¯h
Indeed, v2 h1 = h2 v1 :
½
v1 ↓ ∂ ∂ua
¾
&& h
2 −→
½
↓ v2 ∂ ∂u ¯c
Indeed, by (1.00 ) and (2.2) it follows v2 h1 = h2 v1 : Thus, we have:
¾
∂ ∂ ∂xi ∂ ∂ x˙ i ∂ → = + c i. i c c i ∂x ∂u ¯ ∂u ¯ ∂x ∂u ¯ ∂ x˙
Theorem 1. The frame (1.00 ) given on T M induces a frame on T V , defined by (2.1), which change, on a change of local chart on M and of parameters on V , by the rules (2.2). The frame on T M induces a dual frameµon T V , but ¶ it is difficult to define it ∂ ∂ directly. Let us consider the induced frame , on T V , we associate to it ∂ua ∂ u˙ a its dual frame (dua , du˙ a ). We define the last one by the implicit formulas: dxi =
∂xi a du , ∂ua
dx˙ i =
∂ x˙ i a ∂xi a du + du˙ . ∂ua ∂ua
It is necessary that, on a change of local chart and a change of parameters, the forms of the frame to change (by hypothesis) by the rules: d¯ ua =
∂u ¯a b du , ∂ub
du ¯˙ a =
∂u ¯˙ a b ∂ u ¯a b du + du˙ . ∂ub ∂ub
This is proved by: Lemma 3. The following diagram is commutative: h
dxi
1 ←−
v1 ↑
-- ↑ v2
dua
2 ←−
h
d¯ xh
d¯ uc
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Virgil Ob˘adeanu and Monica Ciobanu ∂xi c d¯ u . ∂u ¯c Lemma 4. The following diagram is commutative:
Indeed, h1 v2 = v1 h2 : d¯ uc → dxi =
h
dx ¯˙ h
dx˙ i
1 ←−
v1 ↑
-- ↑ v2
du˙ a
2 ←−
h
du ¯˙ c
∂ x˙ i c ∂xi c d¯ u + du ¯˙ . ∂u ¯c ∂u ¯c Let us consider the equations (2.0), which define the submanifold V . The formula:
Indeed, h1 v2 = v1 h2 : du ¯˙ c → dx˙ i =
∂xh i ∂xi a δ = δ , h ∂ub ∂ua b
(2.3)
where δhi and δba are the Kronecker’s symbols on M and V respectively, holds.
2.2
Induced d-metric on V
1 The d-metric, locally given by a1ij , of the manifold M , induces a d-metric αab on the submanifold V , by the formula:
(2.4)
1 = αab
∂xi ∂xj 1 a . ∂ua ∂ub ij
¡ 1 ¢ By the definition, it follows det αab 6= 0, ∀x ∈ U . On a change of local chart on M and on V , respectively, we obtain: (2.5)
1 α ¯ cd =
∂ua ∂ub 1 α . ∂u ¯c ∂ u ¯d ab
This property is proved by: Lemma 5. The following diagram is commutative: a1ij
h
1 −→
a ¯1hk
v1 ↓
&& ↓ v2
1 αab
2 −→
h
1 α ¯ cd
∂xi ∂xj 1 a . ∂u ¯c ∂ u ¯d ij of the induced d-metrical tensor can be impli-
1 = Indeed, we have v2 h1 = h2 v1 : a1ij → α ¯ cd
The contravariant components α1ab citly defined by the formulas: (2.6)
aij 1 =
∂xi ∂xj ab α . ∂ua ∂ub 1
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157
Lemma 6. The following diagram is commutative: aij 1
h
1 ←−
a ¯hk 1
v1 ↑
-- ↑ v2
α1ab
2 ←−
h
α ¯ 1cd
∂xi ∂xj cd α ¯ . ∂u ¯c ∂ u ¯d 1 ab ac 1 The components α1 satisfy the relations: α1 αcb = δba . 1 Indeed, let aij 1 be the contravariant components corresponding to ahk , we have j ∂x i 1 aih , it follows: 1 ahj = δj . By contracting each term with the functions ∂ub Indeed, we have h1 v2 = v1 h2 : α ¯ 1cd → aij 1 =
1 aih 1 ahj
2.3
∂xi ∂xh ac 1 ∂xj ∂xj ∂xj ∂xi a ∂xi ac 1 1 = δji b = δb = α1 ahj b = α α ⇒ α1ac αcb = δba . b a a c ∂u ∂u ∂u ∂u ∂u ∂u ∂ua 1 cb
Induced d-connection
Let us consider on the manifold M the d-metric a1ij and the connection a0ij and 1 on its submanifold V the induced metric αab . The connection a0ij induces on V a 0 connection αab by the formulas: · ¸ ∂ x˙ j 1 ∂xi ∂xj 0 0 a + a , (2.7) αab = ∂ua ∂ub ij ∂ub ij such that, on a change of local chart on M and on V respectively, the rules of change: · ¸ ∂ua ∂ub 0 ∂ u˙ b 1 0 (2.8) α ¯ cd = α + α ∂u ¯c ∂ u ¯d ab ∂ u ¯d ab hold. We have: Lemma 7. The following diagram is commutative:
(2.9)
a0ij
1 −→
h
a ¯0hk
v1 ↓
&&
↓ v2
0 αab
2 −→
h
0 α ¯ cd
0 Indeed, d1 = v2 h1 and d2 = h2 v1 , lead us to the relations: d1 = d2 : a0ij → α ¯ cd = · j ¸ i j ∂x ∂x 0 ∂ x˙ 1 a + a . ∂u ¯c ∂ u ¯d ij ∂ u ¯d ij By the Lemmas 5 and 7, it follows: Theorem 2. The following diagram is commutative: ¢ ¡ 1 0¢ ¡ 1 h1 ¯0hk aij , aij −→ a ¯hk , a
(2.10)
v1 ↓ ¡
0 1 , αab αab
&& ¢
h2
−→
↓ v2 ¢ ¡ 1 0 ¯ cd α ¯ cd , α
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Thus, we can associate to the submanifold V an induced geometry by the geometry of M . 1 0 Given the functions αab and αab , they define the coefficients (Christoffel symbols of second type) on V : 0 µab = α1ac αcb ,
(2.11)
which change, on a change of local chart, by the rules: · ¸ ∂u ¯c ∂ub a ∂ u˙ b a c cs 0 (2.12) µ ¯d = α ¯1 α ¯ sd = µ + δ . ∂ua ∂ u ¯d b ∂ u ¯d b By the relations (2.6) and (2.7), between the metric and the connection on M and on V , respectively, we will express the relations between the coefficients Mji and µab . These relations will determine µab implicitly, as function of Mji . Theorem 3. Given the coefficients of connection Mji on M , the induced coeffi∂xi a ∂xj i ∂ x˙ j i cients µab on V are obtained, implicitly, by the formulas: µ = M + δ . b ∂ua ∂ub j ∂ub j Indeed, we have: · ¸ ∂xi a ∂xi ac 0 ∂xi ac ∂xh ∂xj 0 ∂ x˙ j 1 µ = α α = α a + a = ∂ua b ∂ua 1 cb ∂ua 1 ∂uc ∂ub hj ∂ub hj · j ¸ ∂ x˙ j 1 ∂xj i ∂ x˙ j i 0 ih ∂x + = a1 a a = M + δ . hj hj ∂ub ∂ub ∂ub j ∂ub j We have: Lemma 8. The following diagram is commutative: h
Mii
1 −→
h
Mk
v1 ↓
&&
↓ v2
µab
2 −→
h
µ ¯cd
∂xi c ∂xj i ∂ x˙ j i µ ¯ = M + δ , it follows v2 h1 = h2 v1 : d ∂u ¯c ∂u ¯d j ∂ u ¯d j i c Mj → µ ¯d , which proves the above statement (formula (2.12)). 1 0 With help of the metric αab and the connection αab , we can build a geometrical structure on V , called the induced structure of the structure on M . Indeed, by the implicit formula:
2.4
Induced adapted frames
On the submanifold V we define an induced adapted dual frame by the formulas: δua = dua , δ u˙ a = du˙ a + µab dub . On a change of parameters, they change by the rules: δu ¯b =
∂u ¯b a ∂u ¯b a ˙b = δu , δ u ¯ δ u˙ . ∂ua ∂ua
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The relation between the adapted dual frame on M and the induced adapted frame on V is given by the formulas: δxi =
∂xi a ∂xi a i δu , δ x ˙ = δ u˙ . ∂ua ∂ua
The induced adapted frame is defined by: δ ∂ ∂ δ ∂ = − µba b , = . δua ∂ua ∂ u˙ δ u˙ a ∂ u˙ a It is obtained by the adapted frame on M by the formulas: δ ∂xi δ δ ∂xi δ = , = . a a i a δu ∂u δx δ u˙ ∂ua δ x˙ i On a change of parameters, the vectors of this frame change by the rules: δ ∂ua δ δ ∂ua δ = , = . b b a b δu ¯ ∂u ¯ δu ∂u ¯b δ u˙ a δu ¯˙ The tensor of curvature is defined by the brackets: · ¸ δ δ δ δµc δµc , = ρcab c , where: ρcab = ab − ab . a b δu δu δ u˙ δu δu ∂xh c ∂xi ∂xj h ρab = R . c ∂u ∂ua ∂ub ij of the tensor of curvature change
h The relations between Rij and ρcab are given by the relations:
On a change of parameters, the components ρcab ∂u ¯s ∂ua ∂ub r by the rules: ρ¯scd = ρ . ∂ur ∂ u ¯c ∂ u ¯d ab
2.5
The covariant differential operator on V
Given a function f on M , we associate to it its restriction to V , and by the df invariance of the total derivative operator it follows that D : f |V → Df = = dt ∂f ∂f ∂f ∂ + u˙ a a + u ¨a a . Any vector field X = X i i induces on V a vector field ∂t ∂u ∂ u˙ ∂x ∂ X1 = X1a a , where the functions X1a , X1a = X1a (t, x(u), x(u, ˙ u)), ˙ are defined on V ∂u µ ¶ ∂xi a ∂ ∂ implicitly, by the relations X i = X . We define: DN = µba b , which ∂ua 1 ∂ua ∂u has satisfy the f DN (X1 ) + D(f )X1 . Thus, DN (X1 ) = µ to ¶ property: DN (f X1 ) = dX1a ∂ δX1a dX1a a b + µb X1 . The functions = + µab X1b are the covariant derivaa dt ∂u δt dt tives of the components X1a . A vector is transported by parallelism if the covariant dX1a derivatives of its components vanish: + µab X1b = 0. A curve ua = ca (t) is dt autoparallel if its tangent vector is transported by parallelism.
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3
Implicit first order differential dynamical systems with holonomic, scleronomic constraints given by parametric equations ([9])
An implicit first order differential dynamical system is defined by a function F : (t, x, x) ˙ ∈ R ×M T M → T ∗ M , which is written, in a local chart, as: F = Fi (t, x, x) ˙ dxi ∈ Tx∗ M. Its kernel is locally expressed by ¯ © ª KerF = (t, x, x) ˙ ∈ R × T M ¯Fi (t, x, x) ˙ dxi = 0 . Thus, the system of equations: (3.1)
Fi (t, x, x) ˙ =0
represents the dynamical system. By the definition, it follows that these functions change, on a change of local chart, by the rules: Fh =
(3.2)
∂xi Fi . ∂x ¯h
To the dynamical system given by the equations (3.1) we associate the functions: (3.3)
a1ij =
∂Fi , ∂ x˙ j
a0ij =
∂Fi , ∂xj
which change, on a change of local chart of the form (1.0), by the rules (1.1) and (1.2) respectively. It follows that the dynamical system (3.1) defines on the space M a geometrical structure (a d-metric and a d-connection) and thus, we have a geometry. A solution of the dynamical system (3.1) is built by a set of functions xi = ci (t) dci such that they, together with there derivatives c˙i = , transform the equations of dt the system in identities: Fi (t, c(t), c(t)) ˙ ≡ 0. Let us now consider a dynamical system (3.1), whose equations ¡ do not depend ¢ explicitly on time and let xi = ci (t) be a solution of it such that: Fi ch (t), c˙h (t) ≡ 0. ∂Fi dc˙j ∂Fi dcj dc˙j d2 cj By derivation, these relations lead us to: + = 0. By = , ∂ x˙ j dt ∂xj dt dt dt2 2 i j dc d c + Mji = 0. Thus, we have: (3.3) and (1.20 ), we obtain: dt2 dt Theorem 4. Any solution of the given system is an autoparallel curve with respect to the geometrical structure imposed by the system. dcj d2 ci =0 Conversely, let xi = ci (t) be an autoparallel curve. The equation 2 +Mji dt dt i j dc˙ 0 dc can be written in the form: +aih = 0 and, by contraction with a1ki , it follows 1 ahj dt dt ∂Fi dc˙j ∂Fi dcj dFi that: + = = 0. If we integrate, we obtain: Fi = Ci (constant, ∂ c˙j dt ∂cj dt dt holonomic manifold). For the constants Ci = 0 it follows that the autoparallel curves are solutions of the given system.
Geometrical structures
4
161
Implicit first order differential dynamical systems with holonomic constraints given by parametric equations
Let us consider a dynamical system (3.1) on a differentiable manifold M and the submanifold V of the holonomic constraints imposed to the system, assumed as given by the parametric equations (2.0) with the condition (2.00 ). Thus, the equations of the system with these constraints, lead us to the restriction of the system to the submanifold V . The restrictions to V of the equations of the system given by the kernel of the function F = Fi dxi , defined on R × T M with values in T ∗ M , are: ¡ ¡ ¢ ¡ ¢¢ ∂xi a F |V = Fi t, xh ub , x˙ h ub , u˙ b du = Φa dua = 0. ∂ua ∂xi We have: Φa = Fi |V = 0. To the system Φa = 0 we associate the functions ∂ua ∂Φ ∂Φ a a 1 0 αab = and αab = . On a change of local chart on M and a change of param∂ u˙ b ∂ub ¡ 1 ¢ ∂ua ∂ub 1 1 eters on V , these functions change by the rules: α ¯ cd = α , det αab 6= 0, c ∂u d ab ∂ u ¯ ¯ · b ¸ a b ∂u ∂u 0 ∂ u˙ 1 0 and α ¯ cd = α + α respectively. These functions represent nothing ∂u ¯c ∂ u ¯d ab ∂ u ¯d ab ∂Fi 0 ∂Fi else but the restrictions of the functions a1ij = , a = of the geometrical ∂ x˙ j ij ∂xj structure, to the submanifold V . Lemma 9. The following diagram is commutative: h
Fi
1 −→
v1 ↓
&&
Φa
2 −→
h
∂xi Fi ∂x ¯h ↓ v2
Fh =
Φb =
∂ua Φa ∂u ¯b
Theorem 5. The constraints manifold (on which we find the admissible solutions of the given system) has as structure the structure of the system reduced to it. This follows from Lemma 9 and Theorem 4.
References [1] Gh. Atanasiu, M. Neagu, Canonical nonlinear connections in the multi-time Hamilton Geometry, Balkan J. Geom. Appl. 14, 2 (2009), 1-12. [2] V. Balan, M. Neagu, Jet geometrical extension of the KCC-invariants, Balkan J. Geom. Appl. 15, 1 (2010), 8-16. [3] I. Buc˘ataru, R. Miron, Finsler Lagrange Geometry, Romanian Academy Eds., Bucharest 2007. [4] R. Miron, M. Anastasiei, Vector Bundles, Lagrange Spaces, Applications in Relativity Theory, Romanian Academy Eds., Bucharest 1987.
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[5] M. Neamt¸u, The study of Differential Dynamical Systems using Geometric Structures (in Romanian), Ph.D. Thesis, U.V.T. Eds., Timi¸soara, 2001. [6] V. Ob˘adeanu, Structures g`eom´etriques associ`ees ´ a certaines syst`emes dynamiques, Sem. Mec. 67, U.V.T. Eds., Timi¸soara, 2000. [7] V. Ob˘adeanu, M. Ciobanu, The evolution of some deformable continuous media, Tensor N.S. 69 (2008), 127-132. [8] V. Ob˘adeanu, M. Ciobanu, M. Neamt¸u, Dynamical Systems and Associated Geometrical Structures, Sem. Mec. 69, U.V.T. Eds., Timi¸soara, 2000. [9] Sh. Kh. Soltakhanov, M. P. Yushkov, S. A. Zegzhda, Mechanics of Nonholonomic Systems. A New Class of Control Systems, Foundations of Engineering Mechanics, XXXII, Springer, 2009. Authors’ addresses: Virgil Ob˘adeanu West University of Timi¸soara, Department of Mathematics, Faculty of Mathematics and Informatics, 4 V. Pˆarvan Blvd., 300223 Timi¸soara, Romania. E-mail:
[email protected] Monica Ciobanu “Vasile Goldi¸s” Western University of Arad, Faculty of Informatics, 85-87 Revolut¸iei Blvd., 310130 Arad, Romania. E-mail: cm
[email protected]
