---

DYNAMICAL SYSTEMS ACCEPTING THE NORMAL SHIFT

arXiv:hep-th/9405021v1 3 May 1993DYNAMICAL SYSTEMS ACCEPTING THE NORMAL SHIFTON AN ARBITRARY RIEMANNIAN MANIFOLD.Boldin A.Yu., Dmitrieva V.V., Safin S.S., Sharipov R.A.August 6, 1993Abstract. Newtonian dynamical systems which accept the normal shift on an arbitrary Riemannian manifold are considered.For them the determinating equations making the weak normality condition are derived.

The expansion for the algebra oftensor fields is constructed.1. Introduction.The concept of dynamical systems accepting the normal shift was introduced in [1] (see also [2]1).

It appears asthe result of transferring the classical Bonnet transformation from geometry to the field of dynamical systems. In [1]and [2] the Euclidean situation was studied i.e.

the dynamical systems in Rn accepting the normal shift (we refer thereader to that papers for the detailed bibliography).Apart from recent results of [1] and [2] one can find another purely geometric generalization of Bonnet transformationfrom [3] and [4] which is realized as a normal shift along the geodesics on some Riemannian manifold. The naturalquestion here is how do these two generalizations relate each other.

This question was investigated in [5]. There thespecial subclass of dynamical systems accepting the normal shift was separated for which the normal shift is equivalentto the geometrical generalization of the Bonnet transformation for some conformally-Euclidean metric in Rn.

Suchsystems are called metrizable. Comparing the explicit description of metrizable dynamical systems given in [5] withthe examples of [1] and [2] one can conclude that non-metrizable systems do exist.

Therefore the the concept ofnormal shift along the dynamical system is wider than the Bonnet transformation for conformally-Euclidean metrics.However it do not embrace the case of Bonnet transformation for arbitrary metric. In this paper below we considerthe most general situation and study the dynamical systems accepting the normal shift on an arbitrary Riemannianmanifold.2.

Newtonian dynamical systems on the Riemannian manifold.The main object considered in [1] and [2] is the second order dynamical system in Rn(2.1)¨r = F(r, ˙r)reflecting the Newton’s second law. Now let r be not radius-vector of a point in Rn but the vector of local coordinatesfor some manifold M. This case let us write the equations (2.1) in form of the system of the first order equations(2.2)˙xi = vi˙vi = Φi(r, v)Systems of differential equations of the form (2.2) are traditionally connected with vector fields on manifolds.

In thisparticular case the right hand sides of (2.2) form the components of the vector field not on M however but on thetangent bundle T M(2.3)Φ = v1 ∂∂x1 + · · · + vn ∂∂xn + Φ1 ∂∂v1 + · · · + Φn ∂∂vn1See also chao-dyn/9403003 and patt-sol/9404001Typeset by AMS-TEX1

2BOLDIN A.YU., DMITRIEVA V.V., SAFIN S.S., SHARIPOV R.A.First n components of the vector field (2.3) separately can be interpreted as components of the velocity vector(2.4)v = v1 ∂∂x1 + · · · + vn ∂∂xntangent to M. Rest part of components in (2.3) do not admit such interpretation. But if the manifold M is equippedwith Riemannian metric gij and with the metrical connection Γ kij then using the components of (2.3) one can formthe following quantities(2.5)F i = Φi + Γ ijkvkvjthat do behave like the components of some vector tangent to M when we change the local map on M(2.6)F = F 1 ∂∂x1 + · · · + F n ∂∂xnSame indices on the different levels in (2.5) and everywhere below imply summation.

Vector F from (2.6) with thecomponents of the form (2.5) is natural to be considered as a vector of force. The analogy of (2.1) and (2.2) thenbecomes complete.

The existence of the vector F lets us speak about the angle between force and velocity. It letsus also break the force F into two parts directed along the velocity and perpendicular to the velocity.

When F = 0equations (2.2) become the equations of geodesic line.3. Metric on the tangent bundle and the expansion of the algebra of tensor fields.Let us consider again the manifold M with the Riemannian metric gij.

Vector fields on the tangent bundle T Mhave the form(3.1)V = X1 ∂∂x1 + · · · + Xn ∂∂xn + W 1 ∂∂v1 + · · · + W n ∂∂vnFirst n components of V in (3.1) like in (2.4) are interpreted as the components of the vector X tangent to M. Thewhole set of components of (3.1) is transformed as follows(3.2)˜Xi = ∂˜xi∂xj Xj˜W i =∂2˜xi∂xj∂xk vkXj + ∂˜xi∂xs W swhen one change the local variables x1, . .

. , xn for ˜x1, .

. .

, ˜xn. The components of metric connection Γ kij under thesame action are transformed as follows(3.3)˜Γ ipq∂˜xp∂xj∂˜xq∂xk = ∂˜xi∂xs Γ sjk −∂2˜xi∂xj∂xkComparing (3.2) and (3.3) we conclude that the following quantities Zi produced from the components of (3.1)(3.4)Zi = W i + Γ ijkvkXjare transformed like the components of tangent vector to M under the change of local map on M. For the vector (2.3)the corresponding vector (3.4) coincides with (2.5).

So the vector V tangent to T M can be replaced by two vectorsX and Z tangent to M. This gives rise to the pair of linear maps π and ρ where π is canonical projection from thebundle T M to the base manifold M(3.5)X = π(V)Z = ρ(V)The relationship (3.4) lets us introduce the Riemannian metric on the tangent bundle T M by forming the followingquadratic form on the set of vectors (3.1)(3.6)˜g(V, V) = g(π(V ), π(V )) + g(ρ(V ), ρ(V ))

DYNAMICAL SYSTEMS ACCEPTING THE NORMAL SHIFT . .

.3In terms of differentials of local coordinates the metric (3.6) is written as follows(3.7)˜g = (gij + Γ pirvrgpqΓ qjsvs)dxidxj + 2(giqΓ qjsvs)dxidvj + gijdvidvjMetric tensor for (3.7) is determined by the initial metric tensor gij of M. Its matrix has the natural block structure(3.8)˜gij = gij + Γ pirvrgpqΓ qjsvsΓ pirvrgpjgiqΓ qjsvsgij!Upper left block corresponds to the coordinates on the base while the lower right block of (3.8) corresponds to thecoordinates on the stalk. Explicit form of metric tensor makes possible the computation of the components of metricconnection for it but we do not need them in what follows.Much more interesting objects are the images of vector fields on T M under the action of maps π and ρ from (3.5).In general they could not be treated as vector fields on M. They are the vector-valued functions whose argumentis the point of T M and whose value is the vector tangent to M at the point being the projection of the argument.Such function composes the algebra over the ring of scalar functions on T M. This algebra can easily be expandedup to the algebra of tensor-valued functions on M with the argument from T M. It is natural to call it the expandedalgebra of tensor fields on M. The first example of the element of such algebra is the force field (2.6) for Newtoniandynamical system.

This field has the real physical meaning when the manifold M is the configuration space for somereal mechanical system restricted by inner bounds.Expanded algebra of tensor fields on M is equipped with the natural operations of tensor product and contraction.Presence of metric on M adds two operations of covariant differentiation.First of them is the differentiation byvelocity or the velocity gradient. For scalar, vectorial and covector fields it is defined by the following expressions(3.9)˜∇iϕ = ∂ϕ∂vi˜∇iXm = ∂Xm∂vi˜∇iXm = ∂Xm∂viSecond is the covariant differentiation by coordinate or the space gradient.

It is the modification of the ordinarycovariant differentiation. The result of its application to the scalar, vectorial and covector fields is as follows(3.10)∇iϕ = ∂ϕ∂xi −Γ pik∂ϕ∂vp vk∇iXm = ∂Xm∂xi + Γ mip Xp −Γ pik∂Xm∂vp vk∇iXm = ∂Xm∂xi −Γ pimXp −Γ pik∂Xm∂vp vkFor other tensor fields the action of (3.9) and (3.10) is continued according to the condition of concordance withthe operations of tensor product and contraction.

Ordinary tensor fields are the part of expanded algebra. Velocitygradient for them is always zero while the space gradient coincides with ordinary covariant derivative.

Particularly(3.11)∇kgij = 0˜∇kgij = 0The relationships (3.11) express the compatibility of metric and metrical connection in terms of the above covariantderivatives.One more element of the expanded algebra is the vector field of the velocity (2.4). For the velocity and spacegradients of it we have(3.12)∇kvi = 0˜∇kvi = δikScalar field of modulus of velocity is defined by v according to the following formula(3.13)v2 = |v|2 = gijvivj

4BOLDIN A.YU., DMITRIEVA V.V., SAFIN S.S., SHARIPOV R.A.For the gradients of the scalar field defined by (3.13) we have(3.14)∇kv = 0˜∇kv = Nk = gkqN qThe quantities N q in (3.14) are the components of the unit vector field N directed along the velocity v = vN.Gradients for this vector field are the following(3.15)∇kNi = 0˜∇kNi = v−1(δik −NkN i)They are calculated on the base of (3.12) and (3.14). Components of the matrix P ik = δik −NkN i in (3.15) are thecomponents of operator valued field P of normal projectors on a hyperplane that is perpendicular to v. Covariantderivatives for P itself are(3.16)∇kP ij = 0˜∇kP ij = −v−1(gjqP qk N i + NjP ik)Along with P we define the additional projector-valued field with the components Qij = NjN i.For it we haveP + Q = 1.

Covariant derivatives of Q are easily calculated from (3.16)(3.17)∇kQij = 0˜∇kQij = v−1(gjqP qk N i + NjP ik)In addition to the properties (3.16) and (3.17) we can see that projector-valued fields P and Q are symmetric respectiveto the metric gij on M.Let the functions x1(t), . .

. , xn(t) define the parametric curve on M in local map.

The derivatives ∂txi define thetangent vector to M (it may be treated as velocity vector of a point moving along this curve). Suppose that at eachpoint of this curve we have the tangent vector U to M (but possibly not tangent to the curve) with the componentsU i(t).

In other words we have the vector-valued function on the curve. Let us produce another vector-valued functionon that curve according to the formula(3.18)∇tU i = ∂tU i + Γ ijkU j∂txkFormula (3.18) defines the covariant derivative of the vector-valued function on the curve by the parameter t of it.

Itis well known that such derivative is zero by parallel displacement of the vector along the curve.Let’s add the functions v1(t), . .

. , vn(t) to x1(t), .

. .

, xn(t) and consider all them as the curve lifted from M to T M.Such lifting is called natural if vi(t) = ∂txi(t). However here we consider arbitrary (may be not natural) liftings.

Let usdefine some vector field U of expanded algebra in some neighborhood of the lifted curve. Substituting x1(t), .

. .

, xn(t),v1(t), . .

. , vn(t) for its argument we obtain the vector-valued function on the former curve.

For this function we have(3.19)∇tU i = ∇kU i∂txk + ˜∇kU i∇tvkFormula (3.19) approves the names velocity and space gradients for the covariant derivatives (3.9) and (3.10). It iseasily modified for the case of scalar, covectorial and all other types of tensor fields of expanded algebra.Let us find the relation between the ordinary covariant derivatives on T M and the modified covariant derivatives(3.9) and (3.10) on the manifold M itself.

In order to do it consider the pair of vector fields X and Y on the tangentbundle TM. For the projections π and ρ from (3.5) applied to the commutator of these vector fields we deriveπ([X, Y]) = ∇π(X)π(Y) −∇π(Y)π(X)++ ˜∇ρ(X)π(Y) −˜∇π(Y)π(X)(3.20)ρ([X, Y]) = ∇π(X)ρ(Y) −∇π(Y)ρ(X)++ ˜∇ρ(X)ρ(Y) −˜∇ρ(Y)ρ(X) −R(π(X), π(Y))v(3.21)Here R(A, B) is the operator-valued skew-symmetric bilinear form defined by the curvature tensor of M. Formulae(3.20) and (3.21) are proved by the direct calculations in the coordinates.

DYNAMICAL SYSTEMS ACCEPTING THE NORMAL SHIFT . .

.5Let Z be one more vector field on T M and ϕ be the scalar field on T M which also may be treated as a scalar fieldof the expanded algebra on M. Then(3.22)∇Xϕ = ∂Xϕ = ∇π(X)ϕ + ˜∇ρ(X)ϕLeft hand side of (3.22) contain the ordinary covariant derivative on T M which coincides for the scalar function withthe derivative along the vector X. Covariant derivatives in the right hand sides of (3.20), (3.21) and (3.22) are thatof (3.9) and (3.10). For to calculate the covariant derivatives on T M we use the following formula from [6](3.23)2˜g(∇XY, Z) = ∂X˜g(Y, Z) + ∂Y˜g(X, Z) −∂Z˜g(X, Y)++ ˜g([Z, X], Y) + ˜g([Z, Y], X) + ˜g([X, Y], Z)Covariant derivative on T M in (3.23) is ordinary one.

Now let us use the relationship(3.24)˜g(X, Y) = g(π(X), π(Y)) + g(ρ(X), ρ(Y))The relationship (3.24) is derived directly from (3.6). As the result of substitution of (3.24) and (3.20), (3.21), (3.22)into the identity (3.23) we get two formulaeπ(∇XY) = ∇π(X)π(Y) + ˜∇ρ(X)π(Y)−−12(R(ρ(Y, v)π(X) + R(ρ(X), v)π(Y))(3.25)ρ(∇XY) = ∇π(X)ρ(Y) + ˜∇ρ(X)ρ(Y)−−12R(π(X), π(Y))v(3.26)Covariant derivatives in left hand sides of (3.25) and (3.26) are ordinary ones like in (3.23).

Covariant derivatives inright hand sides of (3.25) and (3.26) are expanded ones. They should be treated as in (3.9) and (3.10).4.

Variations of trajectories and the equations of weak normality.Let us consider the Newtonian dynamical system (2.2). In the second of the equations (2.2) we change the ordinaryderivative of the velocity by its covariant derivative according to the formula (3.18).

This gives us the equations(4.1)∂txi = vi∇tvi = F icontaining the force vector of (2.5). The equations (4.1) are more natural form for the Newton’s second law on themanifold M.For the Newtonian dynamical system (4.1) now we consider the Cauchy problem with the following initial data(4.2)xit=0 = xi(s)∂txit=0 = vi(s)depending on some parameter s. Because of (4.2) the trajectories of the dynamical system also depend on s. In localcoordinates they are given by the functions xi = xi(t, s).

Their derivatives by s are the coordinates of some vector τtangent to M. It is the vector of variations of trajectories τ i(t, s) = ∂sxi(t, s). The time derivative of τ is the vectorof variations of velocities(4.3)∇tτ i = ∂tτ i + Γ ijkτ jvkWe shall use (4.3) to obtain the equations for the vector τ from the equations (4.1) of the dynamical system itself.

Inorder to do it let us differentiate (4.1) by s and combine the result with (4.3) and the time derivative of (4.3). Thenwe obtain(4.4)∇t∇tτ i + Rijkqvjvkτ q = ∇qF iτ q + ˜∇qF i∇tτ q

6BOLDIN A.YU., DMITRIEVA V.V., SAFIN S.S., SHARIPOV R.A.Here Rijqk is the curvature tensor for M. So the variation vector τ satisfies the ordinary differential equations of thesecond order (4.4).The normal shift condition according to [1] and [2] consists in the orthogonality of τ and the vector of velocity v.It is convenient to express v via the unit vector N from (3.14) and (3.15). According to the method developed in [1]and [2] we introduce the function ϕ as a scalar product(4.5)ϕ = ⟨τ, N⟩= τ iNi = gijτ iN jFor its time derivative by differentiating (4.5) we obtain∂tϕ = gij∇tτ iN j + gijτ i∇tN jBecause of (3.11) and (3.19) the metric tensor makes no contribution by covariant differentiation.

To calculate ∇tN jwe use the relationships (3.15) and (3.19). As a result we have∂tϕ = gij∇tτ iN j + v−1gijτ iP jkF kBecause P is the symmetric operator field we may rewrite this expression in the following form(4.6)∂tϕ = gij∇tτ iN j + v−1gijF iP jq τ qDifferentiating (4.6) by t we obtain the second derivative for ϕ(4.7)∂ttϕ = ∂t(gij∇tτ iN j) + ∂t(v−1gijF iP jq τ q)Taking into account (3.15) and (3.19) we may write the first summand in (4.7) as(4.8)∂t(gij∇tτ iN j) = ∇ttτ iNi + v−1FiP iq∇tτ qThe first summand in (4.8) in turn is transformed by use of the equation (4.4) for the vector of variation of trajectory(4.9)∇ttτ iNi = Ni∇qF iτ q + Ni ˜∇qF i∇tτ q −NiRiαqβτ qvαvβLet us insert the projectors P and Q into all terms in the right hand side of (4.9).

In order to do it we use thedecomposition of identical operator as 1 = P + Q. For the first summand we have(4.10)Ni∇qF iτ q = ∇iF kNkP iqτ q + ∇qF kN qNkτ iNiFor the second summand in (4.9) by the same way we derive(4.11)Ni ˜∇qF i∇tτ q = ˜∇iF kNkP iq∇tτ q + ˜∇qF kN qNk∇tτ iNiThe third summand in (4.9) vanishes because the vectors N and v are collinear v = |v|N.

Indeed(4.12)NiRiαqβτ qvαvβ = |v|2RiαqβN iN αN βτ qCurvature tensor is skew-symmetric in i and α. Therefore the result of contraction in (4.12) is zero.Now let us transform the second summand in (4.7) using the relationships (3.14) and (3.19)(4.13)∂(v−1gijF iP jkτ k) = −v−2F kNkFiP iqτ q + v−1FiP iq∇tτ q + v−1∇t(FiP iq)τ qThen let us transform the last summand in (4.13) with the help of (3.16) and (3.19)(4.14)v−1∇t(FiP iq)τ q = ∇kF iN kP iqτ q + v−1 ˜∇kFiF kP iqτ q−−v−2F kNkFiP iqτ q −v−2FjP jkF kτ iNi

DYNAMICAL SYSTEMS ACCEPTING THE NORMAL SHIFT . .

.7As a result the second summand in (4.7) may be obtained by substituting (4.14) into (4.13) and the first one may beobtained by substitution of (4.10) and (4.11) into (4.9) followed by substitution of (4.9) into (4.8). By analyzing theobtained formulae (4.5), (4.6) and (4.7) for ϕ, ∂tϕ and ∂ttϕ we find that all they are the linear functionals with respectto vectors boldsymbolτ and ∇tτ forming the phase space for the integral trajectories of the system of differentialequations (4.4).

Using ϕ, ∂tϕ and ∂ttϕ we construct another functional L of the form(4.15)∂ttϕ −P∂tϕ −Qϕ = LThe coefficients for L in the formula (4.15) we define asP = ˜∇qF kN qNkQ = ˜∇qF kN qNk −v−2P qk FqF kThen the functional L itself may be written in the form(4.16)L = ξiP iq∇tτ q + ζiP iqτ qThe coefficients ξi and ζi in (4.16) are defined by the above calculations. They are the followingξi = ˜∇iF kNk + 2v−1Fi(4.17)ζi = (∇iFk + ∇kFi −2v−2FiFk)N k++ v−1( ˜∇kFiF k −˜∇kF qN kNqFi)(4.18)For the dynamical system (4.1) to accept the normal shift on M (see [1] and [2]) the functional L should identicallyvanish.

This condition in [1] and [2] is called the condition of weak normality. In the present situation it gives us thefollowing equations for ξi and ζi from (4.17) and (4.18)(4.20)ξiP iq = 0ζiP iq = 0The equations (4.20) may be rewritten in the following explicit form(v−1Fi + ˜∇i(F kNk))P iq = 0(4.21)(∇iFk + ∇kFi −2v−2FiFk)N kP iq+v−1( ˜∇kFiF k −˜∇kF rN kNrFi)P iq = 0(4.22)The total number of the equations (4.21) and (4.22) is 2n.

It coincides with the twiced dimension of the manifold M.However because of presence of projector matrices P iq in them they are not independent. The number of independentequations in the system of (4.21) and (4.22) is 2n −1 which is in concordance with the results of [1] and [2].The equations (4.21) and (4.22) are the covariant form of the equations of weak normality on an arbitrary Rie-mannian manifold.

In Euclidean case M = Rn they were derived in [1] and [2] by use of spherical coordinates inthe space of velocities. The question of introducing the proper spherical coordinates here in the general situation isinteresting but it is the subject for separate paper.

Analyzing the equations (4.21) one can see that if the vector offorce is decomposed into two parts first being along the velocity and second being perpendicular to it then the secondpart is defined by the first one. This fact was observed in [1] and [2].

It remains true for the general non-Euclideansituation.This paper is written under the financial support of two of authors (Boldin A.Yu. and Sharipov R.A.) by RussianFund for Fundamental Researches.References1.

Boldin A.Yu., Sharipov R.A., Dynamical systems accepting the normal shift., Theor. and Math.

Phys. 97 (1993), no.

3, 386–395.(Russian)2. Boldin A.Yu., Sharipov R.A., Dynamical systems accepting the normal shift., Dikladi Akademii Nauk.

334 (1994), no. 2, 165–167.(Russian)3.

Bianchi L., Sopra le deformazioni isogonali delle superficie a curvatura constante in geometria elliptica ed hiperbolica., Annali diMatem. 18 (1911), no.

3, 185–243.4. Tenenblat K., B¨acklund theorem for submanifolds of space forms and a generalized wave equation., Bol.

Soc. Bras.

Math. 18 (1985),no.

2, 67–92.5. Sharipov R.A., Problem of metrizability for the dynamical systems accepting the normal shift., Theor.

and Math. Phys.

(to appear).(Russain)6. Kobayashy Sh., Nomidzu K., Foundations of differential geometry., vol.

1, Intersc. Publ., New York - London, 1963.Department of Mathematics, Bashkir State University, Frunze street 32, 450074 Ufa, Russia.E-mail address: root@bgua.bashkiria.su

---

출처: arXiv:9405.021 • 원문 보기