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MULTIDIMENSIONAL DYNAMICAL SYSTEMS

arXiv:patt-sol/9404001v1 4 Apr 1993MULTIDIMENSIONAL DYNAMICAL SYSTEMSACCEPTING THE NORMAL SHIFT.Boldin A.Yu. and Sharipov R.A.April 12, 1993.Abstract.

The dynamical systems of the form ¨r = F(r, ˙r) in Rn accepting thenormal shift are considered. The concept of weak normality for them is introduced.The partial differential equations for the force field F(r, ˙r) of the dynamical systemswith weak and complete normality are derived.1.

IntroductionThe concept of dynamical system accepting the normal shift was introduced in[1] as a result of generalization of the classical geometrical Bonnet transformation(or normal shift) for the case of dynamical systems. In [1] (see also [2] and [3])the dynamical systems in R2 are studied (see [1], [2] and [3] for detailed referencelist).

In present paper we generalize the results of [1] for the multidimensional caseand considser some peculiarities absent in 2-dimensional case. These results aredeclared in [4].

They also form the section 5 in preprint [3].Let’s consider the dynamical system describing the trajectories r = r(t) in Rnparticles with unit mass in the force field F(r, ˙r)(1.1)¨r = F(r, ˙r)We shall use the trajectories of (1.1) for transforming the submanifolds of Rn. Let’sconsider the set of particles each of which is starting at t = 0 from some point Pon some hypersurface S ⊂Rn in the direction of normal vector n(P) with someinitial velocity v(P).

In the end of time interval t these particles form anotherhypersurface St ⊂Rn. This defines the one-parameter family of hypersurfaces andthe family of diffeomorphisms(1.2)ft : S −→StNow let’s recall the following two definitions from [1] and [2].Definition 1.

Each transformation f = ft of the family (2.1) is called the nor-mal shift along the dynamical system (1.1) if each trajectory of (1.1) crosses eachsubmanifold St along its normal vector n.Definition 2. Dynamical system (1.1) is called the dynamical system acceptingthe normal shift of submanifolds of codimension 1 if for any submanifold S ofcodimension 1 there is the function v = v(P) on S such that the transformation(2.1) defined by the system (1.1) and the initial velocity function |v(P)| = v(P) isthe transformation of normal shift.Typeset by AMS-TEX1

2BOLDIN A.YU. AND SHARIPOV R.A.2.

Normality conditions for the dynamical systems.Phase space for dynamical system (1.1) is defined by the pairs of vectors r andv. In all points of phase space where v ̸= 0 we introduce the spherical coordinatesin v-space.

Let u1, . .

. , un−1 be the coordinates on the unit sphere |v| = 1 andlet un = v = |v|.

We define also the unit vector N along the vector v and thederivatives of N(2.1)N = N(u1, . .

. , un−1)Mi = ∂N∂uiFor the derivatives of the introduced vectors Mi one may use the standard Wein-garten formulae with the metric connection ϑkij on the unit sphere |v| = 1(2.2)∂Mi∂uj = ϑkijMk −GijNhere Gij = Gij(u1, .

. .

, un−1) is the metric tensor on unit sphere defined by scalarproducts Gij = ⟨Mi, MjMj⟩. In (2.2) and in what follows coinciding upper andlower indices imply summation.

Force field for the dynamical system (1.1) may berepresented by the formula similar to that of [1](2.3)F = AN + BiMiThe equation (1.1) itself then is rewritten as the following system of differentialequations with respect to r, v and ui(2.4)˙r = vN˙v = A˙ui = v−1BiLet’s consider the solution of (2.4) depending on some extra parameter s and in-troduce the following notations for the derivatives of the coordinates in the phasespace(2.5)∂sr = τ∂sv = w∂sui = ziDifferentiating (2.4) and keeping in mind (2.1) and (2.5) we obtain the time deriva-tives for τ, w and zi(2.6)˙τ = wN + vMizi˙w = ∂A∂rk τ k + ∂A∂v w + ∂A∂uk zk˙zi = −Biwv2+ 1v∂Bi∂rk τ k + ∂Bi∂v w + ∂Bi∂uk zkThe equations (2.6) here are the analogs of (3.7) from [1]. In addition to (2.5) let’sintroduce the following notations(2.7)ϕ = ⟨τ, N⟩ψi = ⟨τ, Mi⟩

MULTIDIMENSIONAL DYNAMICAL SYSTEMS ACCEPTING THE NORMAL SHIFT. 3Differentiating (2.7) and taking into account (2.1), (2.2), (2.6) and (2.5) we get thefollowing equations(2.8)˙ϕ = w + Biψiv˙ψi = vGikzk + Bkv (ϑpikψp −Gikϕ)For the space gradients of A and Bi we may define the expansions(2.9)∂A∂rk = aNk + αpMpk∂Bi∂rk = biNk + βipMpkSubstituting (2.9) into (2.6) we derive the following equations(2.10)˙w = aϕ + αkψk + ∂A∂v w + ∂A∂uk zk˙zi = −Biwv2+ 1vvbiϕ + βikψk + ∂Bi∂v w + ∂Bi∂uk zkThe equations (2.8) and (2.10) form the complete system of linear differential equa-tions with respect to ϕ, ψi, w, zi.

Let’s differentiate first of the equations (2.8) byt keeping in mind all above expressions for the time derivatives of the quantitiesinvolved. As a result we obtain¨ϕ =a −BqBkv2Gqkϕ +∂A∂vw + ∂A∂ui + GikBkzi++αi + BqBkv2ϑiqk −BiAv2+ bi + 1v∂Bi∂v A + ∂Bi∂ukBkv2ψiLet’s consider the following expression denoted by L(2.11)¨ϕ −P ˙ϕ −Qϕ = Lwith P and Q being the coefficients enclosed in brackets in the above expressionfor ¨ϕP =∂A∂vQ =a −BqBkv2GqkFor L in (2.11) then we derive the following expression also being the linear com-bination of the expressions “in brackets”L = ∂A∂ui + GikBkzi +αi + BqBkv2ϑiqk −BiAv2+ bi + Av∂Bi∂v −Biv∂A∂vψiSince zi and ψi form the linearly independent set of functions L can vanish if andonly if these “brackets” vanish.

This gives us the following equations for A and BiBi = −Gik ∂A∂uk(2.12)αi + BqBkv2ϑiqk −BiAv2+ bi++ Av∂Bi∂v + ∂Bi∂ukBkv2 −Biv∂A∂v = 0(2.13)being the generalizations of (3.27) and (3.28) from [1] for the multidimensional case.

4BOLDIN A.YU. AND SHARIPOV R.A.Definition 3.

The dynamical system (1.1) with force field of the form (2.3) iscalled the system with the weak normality condition if the equations (2.12) and(2.13) hold.For the systems with the weak normality function ϕ satisfies the second orderordinary differential equation derived from (2.11)(2.14)¨ϕ −P ˙ϕ −Qϕ = 0For the dynamical system of definition 3 to be the system accepting the normalshift in the sense of definition 2 we should be able to obtain the initial conditionsϕ|t=0 = 0˙ϕ|t=0 = 0for any submanifold of codimension 1 by the choice of the modulus of initial velocityv = |v|. Let S be some arbitrary manifold of codimension 1 in Rn.

Defining theunit normal vector for each point on S we define the spherical map S −→Sn−1from S to unit sphere Sn−1 in Rn. For the submanifolds of general position thismap is the local diffeomorphism.

The latter fact let us transfer the coordinatesu1, . .

. , un−1 (see above) from the unit sphere Sn−1 to S vector N(u1, .

. .

, un−1)from (2.1) being the common unit normal vector for both. Tangent vectors to Sare defined like Mi in (2.1)(2.15)Ei = ∂r∂ui∂Ei∂uj = ΓkijEk + bijNTensor bij in (2.15) is the second quadratic form for S and Γkij are the componentsof metric connection on S while ϑkij form the metric connection on Sn−1.

From(2.1) and (2.15) we also derive(2.16)Mi = −bki EkGij = bki bqjgkqFor S of general position the matrix bki is nondegenerate. Let d = b−1 be the inversematrix.

Then from (2.16) we have(2.17)Ei = −dki Mkgij = dki dqjGkqComponents of matrix dki in (2.17) form the G-symmetric tensor(2.18)dki Gkj = Gikdkj = bij∇ibjk = ∇jbikSecond of the equations (2.18) known as the Peterson-Coddazy equation holdswith respect to both connections Γkij and ϑkij. It is known that the difference of twoconnections is the tensor(2.19)Y kij = Γkij −ϑkij = bkq∇idqjCovariant derivatives in (2.18), (2.19) and everywhere below are defined by thespherical connection ϑkij on the unit sphere Sn−1.

MULTIDIMENSIONAL DYNAMICAL SYSTEMS ACCEPTING THE NORMAL SHIFT. 5According to the definition 2 we are to determine the scalar function v =f(u1, .

. .

, un−1) such that(2.20)˙r|t=0 = f(u1, . .

. , un−1)N(u1, .

. .

, un−1)(compare with (3.23) from [1]). Variables u1 = u1(0), .

. .

, un−1 = un−1(0) here playthe same role as s in (3.23) from [1]. Denoting s = ui(0) for a while from (2.5),(2.7) and (2.20) we obtainϕ|t=0 = ⟨N, ∂sr⟩|t=0 = ⟨N, Ei⟩≡0So first initial condition for the equation (2.14) is identically zero since all particlesare starting from S along the normal vector to this submanifold.

For the secondwe have˙ϕ|t=0 = ⟨∂tN, ∂sr⟩|t=0 + ⟨N, ∂str⟩|t=0Using (2.20) and the latter expression we find the following one(2.21)˙ϕ|t=0 = −bikf Bk + ∂f∂uiTo make zero the initial condition (2.21) we need to choose the function f satisfyingthe following equations(2.22)∂f∂ui = bik(u1, . .

. , un−1)Bk(r, f, u1, .

. .

, un−1)fwhere r = r(u1, . .

. , un−1) is the vector of cartesian coordinates of a point on S.Equations (2.22) are analogs of (3.25) from [1].

Since the equations (5.22) formthe overdetermined system of differential equations one can derive from them someother equations being the compatibility conditions for (2.22). The way of obtainingthem is standard: one should differentiate (2.22) by uj and then use the equality∂ijf = ∂jif by changing the order of derivatives.

As a result of such calculationswe get(2.23)1v∂Bk∂v Bq −βkq = 1v∂Bq∂v Bk −βqk∇kBq = ∇pBpn −1 δqkFunctions Bk in (2.23) are considered as the functions of 2n independant variablesr1, . .

. , rn, v and u1, .

. .

, un−1 as in (2.3). Covariant derivatives by u1, .

. .

, un−1 arerespective to the spherical connection ϑkij.Theorem 2. The equations (2.12), (2.13) and (2.23) form the enough conditionfor the dynamical system (1.1) with force field (2.3) to be accepting the normal shiftas described by the definition 2.Equations (2.12), (5.13) and (5.23) are compatible in some sense since they havecommon solution for A and Bk (at least trivial one with A = A(v) and Bk ≡0corresponding to the geometrical situation from [5] and [6]).

Detailed analysis ofthese equations and nontrivial examples of the multidimensional dynamical systemsassociated with their solutions are the subject of separate paper.Authors are grateful to Russian Fund for Fundamental Researches for the finan-cial support (project # 93-011-273).

6BOLDIN A.YU. AND SHARIPOV R.A.References1.

Boldin A.Yu. and Sharipov R.A., Dynamical Systems Accepting the Normal Shift., Theor.and Math.

Phys. 97 (1993), no.

3, 386–395. (Russian)2.

Boldin A.Yu. and Sharipov R.A., Dynamical Systems Accepting the Normal Shift., Pbb: chao-dyn@xyz.lanl.gov, no.

9403003.3. Boldin A.Yu.

and Sharipov R.A., Dynamical Systems Accepting the Normal Shift., Preprint# 0001-M, Bashkir State University, April 1993.4. Boldin A.Yu.

and Sharipov R.A., Dynamical Systems Accepting the Normal Shift., DokladiAkademii Nauk. 334 (1994), no.

2, 165–167. (Russian)5.

Tenenblat K. and Terng C.L., B¨acklund theorem for n-dimensional submanifolds of R2n−1.,Annals of Math. 111 (1980), no.

3, 477–490.6. Terng C.L., A higher dimensional generalization of Sine-Gordon equation and its solitontheory., Annals of Math.

111 (1980), no. 3, 491–510.Department of Mathematics, Bashkir State University, Frunze str.32, 450074Ufa, RussiaE-mail address: root@bgua.bashkiria.su

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