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DYNAMICAL SYSTEMS ACCEPTING THE NORMAL SHIFT.

arXiv:chao-dyn/9403003v1 28 Mar 1993DYNAMICAL SYSTEMS ACCEPTING THE NORMAL SHIFT.Boldin A.Yu. and Sharipov R.A.March 9, 1993.Abstract.

Classical Bianchi-Lie, B¨acklund and Darboux transformations are considered.Their generalizations for thedynamical systems are discussed.For the transformation being the generalization of the normal shift the special class ofdynamical systems is defined. The effective criterion for separating such systems in form of partial differential equations isfound.1.

IntroductionB¨acklund transformations which are well known to the specialists in integrable nonlinear equations originally firstarose in the works of classics of differential geometry in last century. Let’s remind some of their results.

Consider2-dimensional surface S in R3 with the standard scalar product. Let us map each point M of the surface S onto thepoint M ′ of the other surface S′ so that the following conditions are fulfilled(1) The distance |MM ′| is constant and it is equal to R;(2) The tangent planes τ and τ′ of the surfaces S and S′ in the points M and M ′ are orthogonal: ∠(τ, τ ′) = 90◦;(3) The segment MM ′ lies in both tangents planes τ and τ′ of the surfaces S and S′ respectively.Bianchi [1] showed that if S is the surface of constant negative Gaussian curvature K = −1/R2 then in assumptions(1)–(3) the second surface S′ also is the surface of constant negative Gaussian curvature K′ = −1/R2.Lie [2]strengthened Bianchi’s result having shown that construction (1)–(3) can be realized only on the surface S of constantnegative Gaussian curvature K = −1/R2.B¨acklund [3] generalized the construction (1)–(3) having substituted condition (2) by more weak condition ofconstancy of the angle between the tangent planes τ and τ′.

Such kind of transformation also is defined only on thesurfaces of constant negative curvature and it leads to the surfaces S′ of the same curvature. Darboux [4] offeredfurther generalization of this construction having substituted condition (3) by condition of constancy of the anglesbetween line MM ′ and both tangent planes τ and τ′.

However thereby some differences there appeared: Darbouxtransformation is realized on the surfaces where some linear combination of Gaussian and mean curvatures is constant.The result of transformation is the surface where some other linear combination of Gaussian and mean curvatures isconstant.Differential equations which are obtained in considering the transformations (1)–(3) as well as their generalizationsbecame the objects of the numerous investigations.For the modern state of such investigations one can enquirethe monographs [5] and [6]. The generalization of the above transformations for the multidimensional spaces andsubmanifolds in them is made in papers [7], [8] and [9].

The generalizations for the submanifolds in the arbitraryRiemannian manifold of constant sectional curvature is considered in [10] and [11].The normal shift (or the Bonnet transformation) is some particular case in the general Darboux construction. Inthis case tangent planes τ and τ ′ are parallel while the segment MM ′ is orthogonal to them both.

This is degenerateparticular case since it can be realized on any surface imposing no limitations for its curvatures.In this paper we discuss the generalization of the Bianchi-Lie, B¨acklund and Darboux transformation for the spaciallyanisotropic case substituting the straight line segment or geodesic segment MM ′ by the segment of trajectory for somedynamical system. Let’s consider the following dynamical system in Rn(1.1)¨r = F(r, ˙r)Typeset by AMS-TEX1

2BOLDIN A.YU. AND SHARIPOV R.A.This is the natural second order dynamical system defined by the force F. For each point M on some submanifoldS ⊂Rn let us consider the particle starting from it with some initial velocity v. In the end of some time interval(same for all particles) these particles form some other submanifold S′ ⊂Rn.

In most general form the problem maybe stated as follows: what kind of limitations for the dynamical system (1.1) itself and for the choice of submanifoldS and initial velocities of particles on it arise if one require the angles defining the mutual arrangement of tangentspaces τ, τ ′ and particle trajectories to be constant. Such problem has a lot of possible specializations one of whichleading to the meaningful results is considered below.2.

The normal shift along the dynamical system.Let’s consider the dynamical system (1.1) in Euclidean space Rn with the standard scalar product. Let S be thesubmanifold of codimension 1 in Rn and let n be the vector field of unit normal vectors on S. We direct the initialvelocity of particles along n defining the modulus of velocity as some smooth function v = v(M) on the submanifoldS.

Then the dynamical system (1.1) produces the family of submanifolds St(2.1)ft : S −→Sttogether with the diffeomorphisms ft binding S with St.Definition 1. Each transformation f = ft of the family (2.1) is called the normal shift along the dynamical system(1.1) if each trajectory of (1.1) crosses each submanifold St along its normal vector n.Definition 2.

Dynamical system (1.1) is called the dynamical system accepting the normal shift of submanifoldsof codimension 1 if for any submanifold S of codimension 1 there is the function v = v(M) on S such that thetransformation (2.1) defined by the system (1.1) and the initial velocity function |v(M)| = v(M) is the transformationof normal shift.Note that the transformation (2.1) may implement the normal shift for some particular submanifolds even whenthey do not satisfy the definition 2. Dynamical systems accepting the normal shift for arbitrary submanifold form thespecial class of dynamical systems narrow enough to be described in much details.

In the following two sections weconsider such dynamical systems in R2 and derive the partial differential equation for the force function F of them.3. Dynamical systems in R2 accepting the normal shift.Let’s consider the second order dynamical system (1.1) in Euclidean space R2 with the standard scalar product.Phase space of the system (1.1) is four-dimensional in this case.

The locus of points where v = ˙r = 0 is the two-dimensional plane in it. Everywhere out of this locus we define the unit vector(3.1)N = N(v) = v|v|and the unit vector M(v) perpendicular to N(v).

Using (3.1) the right hand side of (1.1) can be rewritten as follows(3.2)F(r, v) = A(r, v)N(v) + B(r, v)M(v)Cartesian components of N(v) and M(v) satisfy the following differential equations(3.3)∂N k∂vi = MiM k|v|∂M k∂vi= −MiN k|v|Gradients of the functions A(r, v) and B(r, v) in (3.2) also can be expressed in terms of components of N(v) andM(v)∂A∂ri = α1Ni + α2Mi∂A∂vi = α3Ni + α4Mi(3.4)∂B∂ri = β1Ni + β2Mi∂B∂vi = β3Ni + β4Mi(3.5)

DYNAMICAL SYSTEMS ACCEPTING THE NORMAL SHIFT.3For the dynamical system (1.1) with the right hand side (3.2) we consider the Cauchy problem with the initial datadepending on some scalar parameter s(3.6)r(t, s)|t=0 = r(s)∂r(t, s)|t=0 = v(s)Let τ(t, s) be the derivative of r(t, s) by the parameter s. Differentiating (1.1) we derive the following equation forthe components of the vector τ(t, s)(3.7)¨τ k = ∂F k∂ri τ i + ∂F k∂vi ˙τ iUsing the expression (3.2), the differential equations (3.3) and the formulae (3.4) and (3.5) we my write the partialderivatives in (3.7) in the following form∂F k∂ri = (α1Ni + α2Mi) N k + (β1Ni + β2Mi) M k(3.8)∂F k∂vi = α3NiN k +α4 −B|v|MiN k++ β3NiM k +β4 + A|v|MiM k(3.9)Time derivative for the vector v is defined by the equation (1.1). For the length of this vector then we have(3.10)∂t|v| = A∂t|v|−1= −A|v|2For the derivatives of A and B in (3.4) and (3.5) we obtain(3.11)∂tA = α1|v| + α3A + α4B∂tB = β1|v| + β3A + β4BTaking into account (3.4) and (3.5) from (3.3) and (3.10) we derive the formulae for the time derivatives of N and M(3.12)∂tN = BN|v|∂tM = −BN|v|Time dynamics of the vector τ due to (3.7) and the relationships (3.12) determine the dynamics of the scalarproducts ⟨τ, N⟩and ⟨τ, M⟩.

Let’s denote them as follows(3.13)⟨τ, N⟩= ϕ⟨τ, M⟩= φDifferentiating (3.13) and taking into account (3.12) we obtain(3.14)⟨∂tτ, N⟩= ∂tϕ −B|v|ψ⟨∂tτ, M⟩= ∂ψ + B|v|ϕDifferentiating the left hand sides in (3.14) we get∂t ⟨∂tτ, N⟩= ⟨∂ttτ, N⟩+ ⟨∂tτ, ∂tN⟩== ⟨∂ttτ, N⟩+ B|v| · ∂tψ + B2|v|2 ϕ(3.15)∂t ⟨∂tτ, M⟩= ⟨∂ttτ, M⟩+ ⟨∂tτ, ∂tM⟩== ⟨∂ttτ, M⟩−B|v|∂tϕ + B2|v|2 ψ(3.16)

4BOLDIN A.YU. AND SHARIPOV R.A.Differentiating the right hand sides in the same equations (3.14) and taking into account (3.10), (3.11) and (3.12) wehave∂t ⟨∂tτ, N⟩= ∂ttϕ −B|v|∂tψ + BA|v|2 ψ−−β1ψ −β3A|v|ψ −β4B|v|ψ(3.17)∂t ⟨∂tτ, M⟩= ∂ttψ + B|v|∂tϕ −BA|v|2 ϕ++ β1ϕ + β3A|v|ϕ + β4B|v|ϕ(3.18)The second derivative of the vector τ in (3.15) and (3.16) can be calculated on the base of (3.7), (3.8) and (3.9).

Thenwe get⟨∂ttτ, N⟩= α1ϕ + α2ψ + α3∂tϕ −B|v|ϕ++α4 −B|v|·∂tψ + B|v|ϕ(3.19)⟨∂ttτ, M⟩= β1ϕ + β2ψ + β3∂tϕ −B|v|ψ++β4 + A|v|·∂tψ + B|v|ϕ(3.20)As a result of equating (3.15) with (3.17) and (3.16) with (3.18) after substituting (3.19) and (3.20) for ϕ and ψ weobtain∂ttϕ −B|v|∂tψ + BA|v|2 ψ −β1ψ −β3Aψ|v| −β4Bψ|v| == α1ϕ + α2ψ + α3∂tϕ −B|v|ψ+ α4∂tψ + B|v|ϕ(3.21)∂ttψ + B|v|∂tϕ −BA|v|2 ψ + β1ϕ + β3Aϕ|v| + β4Bϕ|v| == β1ϕ + β2ψ +β3 −B|v|·∂tϕ −B|v|ψ++ α4β4 + A|v|·∂tψ + B|v|ϕ(3.22)Let the dynamical system (1.1) with the force field (3.2) be accepting the normal shift in R2. Submanifolds ofcodimension 1 in R2 are the plane curves.

It is convenient to use the natural parameter on them measuring the arclength referenced to some fixed point on the curve. Let the parameter s in the Cauchy problem initial data (3.6) docoincide with the natural parameter on the curve.

Then for the transformations of the normal shift these initial datamay be rewritten as follows(3.23)r(t, s)|t=0 = r(s)∂tr(t, s)|t=0 = v(s)n(s)The derivative ∂sr(s) = τ(s) is the unit tangent vector for the curve while n(s) is the unit normal vector for it.From (3.23), (3.14) and from the orthogonality of the vectors τ(s) and n(s) we derive the following initial data forthe function ϕ introduced in (3.13)(3.24)ϕ|t=0 = 0∂tϕ|t=0 = ∂sv(s) +Bv(s)

DYNAMICAL SYSTEMS ACCEPTING THE NORMAL SHIFT.5By the proper choice of the function v(s) in (3.23) and (3.24) (see also the definition 2) one can vanish the right handside of the second initial data statement in (3.24). Such proper choice is defined by the following differential equation(3.25)v′s = −B(r(s), v(s)n(s))v(s)The right hand side of (3.25) as well as the function v(s) itself then depend on the form of the curve.

Once the choice(3.25) for v(s) is made the initial data (3.14) take the form(3.26)ϕ|t=0 = 0∂tϕ|t=0 = 0Now let’s come back to the differential equations (3.21) and (3.22). They are linear with respect to ϕ and ψ andaltogether they form the complete system of the differential equations.

As for the initial data (3.26) they are notenough to determine the Cauchy problem for this system. However one can make the special choice of the functionsA and B in (3.2) such that the coefficients of ψ and ∂tψ in (3.21) vanish making (3.21) the separate equation of thesecond order with respect to the function ϕ.

Because of the linearity the Cauchy problem (3.26) for this equation hasthe unique solution being identically zero. In this case vectors τ and N are perpendicular for all s and t which is justthe condition of normal shift.

The proper choice of A and B leading to this case is described by the following theorem.Theorem 1. Dynamical system (1.1) with the right hand side of the form (3.2) is the system accepting the normalshift in R2 if and only if the following relationshipsB = −|v|α4(3.27)BA|v|2 −β1 −β3A|v| −β4B|v| = α2 −α3B|v|(3.28)are fulfilled.

Here α1, α2, α3, α4 and β1, β2, β3, β4 are the coefficients in (3.4) and (3.5).4. Some examples in R2.Because of the derivatives in (3.4) and (3.5) the equations (3.27) and (3.28) are the system of the partial differentialequations with respect to A and B defining in turn the force field F according to (3.2).

In order to make theseequations more explicit one should use the explicit form of the vectors N and MN = 1|v|v1v2M = 1|v|−v2v1For the function B then from (3.27) we may obtain its expression via the function A(4.1)B = v2 ∂A∂v1 −v1 ∂A∂v2Further substitution of (4.1) into (3.5) and (3.28) let us bring the system of equation (3.27) and (3.28) to the formof one equation with respect to A. We do not write this equation here because it is very huge.

But in place of it weconsider some examples when this equation can be simplified to rather observable size.Case 1. Spacially homogeneous force field directed along the vector v of velocity.

Function A in this case doesnot depend on r and B is equal to zero identically. From (4.1) then we obtain that A depends only on the modulusof velocity A = A(|v|).

Therefore α2 = 0 and the equation (3.28) become the identity. Note that this case is trivialfrom geometrical point of view since trajectories of particles are straight lines and associated transformation in (2.1)coincides with the classical normal shift.Case 2.

Spacially homogeneous but anisotropic force field. Both functions A and B do not depend on r. Let’sdenote v = |v| the modulus of velocity and denote via θ the angle between v and some fixed direction in space.

ThenB = −∂θA = −Aθ and the equation (3.28) is written as follows(4.2)AAθ −vAAθv + AθAθθ = −vAθAv

6BOLDIN A.YU. AND SHARIPOV R.A.Equation (4.2) has the particular solution with the separated variables A = A(v) cos(θ).

Force field then is of the form(4.3)F = A(v)N cos(θ) + A(v)M sin(θ)The modulus of force here |F| = A(v) is some arbitrary function of v. The direction of force F form with the fixeddirection in space the angle 2θ twice as greater than the angle between v and that direction (see fig. 1).Placeforfig.1.Placeforfig.2.Case 3.

Spacially unhomogeneous force field with the central point. Here it is convenient to introduce new variablesρ, γ, v and θ according to the fig.

2(4.4)r1 = ρ cos(γ)r2 = ρ sin(γ)v1 = v cos(γ + θ)v2 = v sin(γ + θ)From (4.4) it is not difficult to derive the following formulae for different functions in the equations (3.27) and (3.28)α2 = −sin(θ)Aρ + cos(θ)ρ(Aγ −Aθ)α4 = 1v Aθα3 = Avβ1 = cos(θ)Bρ + sin(θ)ρ(Bγ −Bθ)β4 = 1v Bθβ3 = BvLet’s substitute these expressions into (3.27) and (3.28) and impose one additional condition Aγ = 0 (the condition ofisotropy for all rays coming out from the central point). As a result we obtain the following equations(4.5)B = −Aθ−AAθv2+ cos(θ)Aθρ −sin(θ)ρAθθ + AAθvv−AθAθθv2== −sin(θ)Aρ −cos(θ)ρAθ + AθAvvThe solution of the equations (4.5) with separated variables here has the following formA = A(v)ρcos(θ)B = A(v)ρsin(θ)where A(v) is some arbitrary function of modulus of velocity v. Corresponding dynamical system has the followingforce field(4.6)F = A(v)N cos(θ) + A(v)M sin(θ)ρ(see fig.2).Note, that both dynamical systems with force fields (4.3) and (4.6) are integrable via quadratures.Moreover for some special choice of A(v) they are explicitly integrable.

For instance when A(v) = const the trajectoriesof the system (4.3) are the cycloids.References1. Bianchi L., Ricerche sulle superficie a curvatura constante e sulle elicoid., Ann.

Scuola Norm. Pisa 2 (1879), 285.2.

Lie S., Zur Theorie der Fl¨achen konstanter Kr¨ummung, III, IV., Arch. Math.

og Naturvidenskab 5 (1880), no. 3, 282–306, 328–358and 398–446.3.

B¨acklund A.V., Om ytor med konstant negativ kr¨okning., Lunds Universitets ˚Ars-skrift, 19 (1883).4. Darboux G., Le¸cons sur la th´eorie g´en´erale des surfaces, III., Gauthier-Villars et Fils, Paris, 1894.5.

Ibragimov N.Kh., Groups of transformations in Mathematical Physics., Nauka, Moscow, 1983.6. Olver P.J., Application of Lie Groups to Differential Equation., Springer-Verlag.

DYNAMICAL SYSTEMS ACCEPTING THE NORMAL SHIFT.77. Teneblat K. and Terng C.L., B¨acklund theorem for n-dimensional submanifolds of R2n−1., Annals of Math.

111 (1980), no. 3, 477-490.8.

Terng C.L., A higher dimensional generalization of Sine-Gordon equation and its soliton theory., Annals of Math. 111 (1980), no.

3,491–510.9. Chern S.S. and Terng C.L., An analogue of B¨acklund theorem in affine geometry., Rocky Mount Journ.

of Math. 10 (1980), no.

1, 105.10. Bianchi L., Sopra le deformazioni isogonali delle superficie a curvatura constante in geometria elliptica ed hiperbolica., Annali diMatem.

18 (1911), no. 3, 185–243.11.

Teneblat K., B¨acklund theorem for submanifolds of space forms and a generalized wave equation., Bol. Soc.

Bras. Math.

18 (1985),no. 2, 67–92.Department of Mathematics, Bashkir State University, Frunze str.

32, 450074 Ufa, RussiaE-mail address: root@bgua.bashkiria.su

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