The authors use the procedure developed in [9] to develop a Hamiltonian structure into the variational problem given by the integral of the squared curvature on the spatial curves.

The solutions of that problem are the elasticae or nonlinear splines. The symmetry of the problem under rigid motions is then used to reduce the Euler–Lagrange equations to a firstorder dynamical system

Muñoz Masqué, Jaime

Pozo Coronado, Luis Miguel

English

Proceedings of the third international conference on symmetry in nonlinear mathematical physics, Kyiv, Ukraine, July 12-18, 1999. Part 1. Transl. from the Ukrainian. Kyiv:
Institute of Mathematics of NAS of UkraineThe authors use the procedure developed in [9] to develop a Hamiltonian structure into the variational problem given by the integral of the squared curvature on the spatial curves.
The solutions of that problem are the elasticae or nonlinear splines. The symmetry of the problem under rigid motions is then used to reduce the Euler–Lagrange equations to a firstorder dynamical system.Depto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasTRUEpu

Docta Complutense

Proceedings of Institute of Mathematics of NAS of Ukraine 2000, Vol. 30, Part 1, 170–176.Hamiltonian Formulation and Order Reductionfor Nonlinear Splines in the Euclidean 3-SpaceJ. MUÑOZ MASQUÉ † and L.M. POZO CORONADO ‡† IFA-CSIC, C/ Serrano 144, 28006-Madrid, SpainE-mail: jaime@iec.csic.es‡ Departamento de Geometŕıa y Topoloǵıa,Universidad Complutense de Madrid, 28040-Madrid, SpainE-mail: luispozo@eucmos.sim.ucm.esThe authors use the procedure developed in [9] to develop a Hamiltonian structure into thevariational problem given by the integral of the squared curvature on the spatial curves.The solutions of that problem are the elasticae or nonlinear splines. The symmetry of theproblem under rigid motions is then used to reduce the Euler–Lagrange equations to a first-order dynamical system.1 IntroductionElasticae, or nonlinear spline curves, are the extremal curves of the second order variationalproblem given by the functional∫κ2ds (cf. [2, 4, 5]), where κ is the geodesic curvature of thepath, and ds is the arc-length element. This is one of the simplest examples of a general typeof variational problems called parameter-invariant problems (see [6]), as the action integral isinvariant under arbitrary changes of parameter. As all the problems in this class, our problemis singular, that is, its Hessian vanishes. As a consequence, the standard Hamiltonian formalism(momenta, Hamilton equations) cannot be directly applied. Hence it is not clear how to reducethe order of the equations by using Noether invariants attached to the symmetries of the problem.In [9], the authors devised a general procedure called parameter elimination, in order to passfrom parameter-invariant Lagrangian to a nonparametric version of it. Roughly speaking, theparameter elimination consists in taking the parameter “time” apart and using one of the “spa-tial” coordinates as the new parameter. So, the Lagrangian projects onto a “deparametrized”Lagrangian, whose extremals, parametrized arbitrarily, are the extremals of the original prob-lem. Furthermore, the projected Lagrangian gives rise in some cases (which can be suitablycharacterized) to a regular problem, and hence Hamiltonian formalism can be applied.In this paper this general procedure is applied to the particular case of the nonlinear splines inthe 3-dimensional space, thus introducing a natural Hamiltonian formulation to the problem.Within this setting, the Noether invariants associated to the rigid motions of the space are found(the rigid motions – translations and rotations – are symmetries of the variational problem, asboth the curvature and the arc-length are invariant under isometries). This invariants are thenused to reduce the order of the equations of extremals, from a fourth order system to a nonlinearsystem of the formdy′dx= F (y′, z′),dz′dx= G(y′, z′),where y′ = dy/dx and z′ = dz/dx.Hamiltonian Formulation and Order Reduction for Nonlinear Splines 171The term nonlinear splines is used to distinguish the extremal curves of the squared curvaturefunctional of the “usual” linear splines (or just splines), or piecewise cubic polynomials. The lat-ter are the extremals of the linear approximation to the variational problem under consideration.Of course, we are just interested in the exact extremals, not in the approximations.2 Hamiltonian formulation for elasticaeIn the standard coordinates of R3, the expression of the nonlinear splines Lagrangian is (see [12]):L = (ẏz̈ − ÿż)2 + (ẋz̈ − ẍż)2 + (ẋÿ − ẍẏ)2(ẋ2 + ẏ2 + ż2)5/2.The variational problem defined by this Lagrangian is not regular. In fact, as a simple compu-tation shows, we have det(∂2L/∂ẍi∂ẍj) ≡ 0. Hence, Hamiltonian formalism cannot be applied.Nevertheless, this variational problem is parameter-invariant (see [6]); i.e.,∫ baL (j2t (σ ◦ u))dt =∫ βαL (j2u(σ)) du,for every orientation-preserving diffeomorphism u: [a, b] → [α, β]. This fact can easily be checked,either geometrically, or making use of the Zermelo conditions (see [6]).Using the procedure described in [9], we pass from the Lagrangian L given above to thedeparametrized version L̄. Roughly speaking, this is done by eliminating the variable t, using xas the new independent variable, and passing from the “dots” (which represent derivatives withrespect to t), to the “primes” (standing for derivatives with respect to x). The following relationsare used:y′ =ẏẋ, z′ =żẋ, y′′ =ẋÿ − ẏẍẋ3, z′′ =ẋz̈ − żẍẋ3. (1)It is important to notice that the process is applied to the Lagrangian density Ldt to convert itinto a density L̄dx modulo a contact form dx− ẋdt.The following are some of the main results given in [9] (here L stands for a generic second-order Lagrangian):(i) L̄ is regular if and only if the Hessian matrix of L has rank 2.(ii) The extremals of Ldt are the extremals of L̄dx, endowed with an arbitrary parametri-zation.In the case of the squared curvature functional, the rank of the Hessian matrix is 2, so thenon-parametric Lagrangian, which can be easily computed by using the formulas (1),L̄ = (y′z′′ − y′′z′)2 + y′′2 + z′′2(1 + y′2 + z′2)5/2, (2)is regular, and we can apply the Hamiltonian formalism to it.2.1 Hamilton equations for elasticaeUsing the standard Hamiltonian formalism for second order problems (see, [1, 3, 4, 7, 11]) wecan write the Jacobi–Ostrogradski momenta associated to the non-parametric Lagrangian L̄,p = v−5[−2(z′2 + 1)y′′′ + 2y′z′z′′′]+ v−7[5y′(z′2 + 1)y′′2+ 2z′(−2y′2 + 3z′2 + 3)y′′z′′ − y′(y′2 + 6z′2 + 1)z′′2],172 J. Muñoz Masqué and L.M. Pozo Coronadoq = v−5[2y′z′y′′′ − 2(y′2 + 1)z′′′]+ v−7[−z′(6y′2 + z′2 + 1)y′′2+ 2y′(3y′2 − 2z′2 + 3)y′′z′′ + 5z′(y′2 + 1)z′′2],p′ = v−5[2(z′2 + 1)y′′ − 2y′z′z′′],q′ = v−5[−2y′z′y′′ + 2(y′2 + 1)z′′],v =(1 + y′2 + z′2)1/2,and also the Hamiltonian H = py′ + qz′ +(v3/4) [(p′y′ + q′z′)2 + (p′)2 + (q′)2], the Poincaré–Cartan formΘ̄ = L̄dx+ p(dy − y′ dx) + q(dz − z′ dx) + p′(dy′ − y′′ dx) + q′(dz′ − z′′ dx)= −H dx+ p dy + q dz + p′ dy′ + q′ dz′,(3)and, finally, the Hamilton equations:dydx= y′,dzdx= z′,dpdx= 0,dqdx= 0,dy′dx=12v3[(y′2 + 1)p′ + y′z′q′],dz′dx=12v3[y′z′p′ +(z′2 + 1)q′],dp′dx= −p− 14v[y′(5y′2 + 2z′2 + 5)(p′)2 + 2z′(4y′2 + z′2 + 1)p′q′ + 3y′(z′2 + 1)(q′)2],dq′dx= −q − 14v[3z′(y′2 + 1)(p′)2 + 2y′(y′2 + 4z′2 + 1)p′q′ + z′(2y′2 + 5z′2 + 5)(q′)2].(4)Our aim shall be to reduce this system to a first-order ordinary differential system in thevariables y′, z′. Also note that p′ = q′ = 0 if and only if the spline curve is a geodesic; i.e.,y = α1x+ β1, z = α2x+ β2, αi, βi ∈ R, i = 1, 2.3 Generalized symmetries and reduction to first orderIt is straightforward to see that the variational problem defined by the squared curvature func-tional is invariant under isometries, as the curvature and arc-length element are themselvesinvariant under isometries. Thus, the infinitesimal generators of the rigid motions of R3 areinfinitesimal symmetries of Ldt, that is, if X ∈ X(R3) is the infinitesimal generator of a rigidmotion, LX(2)(Ldt) = 0, where X(r) is the prolongation of the vector field (0, X) ∈ X(R×R3) toJr(R,R3) by means of infinitesimal contact transformations (cf. [8, 9]). For infinitesimal symme-tries, Noether’s Theorem (see [7]) states that the function fX = iX(3)Θ(Ldt): J2r−1(R,M) → Ris constant along each extremal of the variational problem defined by Ldt. The function fX iscalled the Noether invariant associated to X.Nevertheless, as it was stated in [9], if (0, X) ∈ X(R×R3) is an infinitesimal symmetry of Ldt,it does not need to be X ∈ X(R×R2) an infinitesimal symmetry of L̄dt (in fact, it can even benot projectable onto R2). But it is a generalized infinitesimal symmetry of L̄dt, i.e., LX[2](L̄dt)Hamiltonian Formulation and Order Reduction for Nonlinear Splines 173is a contact form, that is, it vanishes on every 2-jet of curve on M (cf. [10, Definition 5.25]).Here, X[r] denotes the prolongation of X ∈ X(R3) to X(Jr(R,R2)) by means of infinitesimalcontact transformations.It can be shown that if X ∈ X(R × M) is a generalized infinitesimal symmetry of L̄dx,then fX is constant on the extremals of L̄dx; i.e., generalized symmetries also produce Noetherinvariants.Now we shall use the (generalized) symmetries of the deparametrized squared curvatureLagrangian to reduce (by means of the associated Noether invariants) the order of the Hamiltonequations for the nonlinear splines on R3. More precisely, in the general case we shall reducethe equations (4) to a system of the form:dz′dy′= F(y′, z′),dy′dx= G(y′, z′).Hence the problem is reduced to two ordinary differential equations in the plane as we can firstsolve z′ in terms of y′ from the first equation above and then substitute the obtained expressionfor z′ into the second equation thus leading us to an equation of the form dy′/dx = Ḡ(µ, x, y′),where µ is a constant.As we have already said, the infinitesimal generators of the isometries of R3 are infinitesimalsymmetries of the Lagrangian density Ldt, and hence infinitesimal generalized symmetries ofL̄dx. The isometries of R3 are generated by the translations along the three axes, and therotations around these axes. The infinitesimal generators of the translations are ∂/∂x, ∂/∂yand ∂/∂z, which are their own prolongations by infinitesimal contact transformations. As thePoincaré–Cartan form has the expression given in (3), the Noether invariants associated to thetranslations are H = −Θ̄(∂/∂x), p = Θ̄(∂/∂y) and q = Θ̄(∂/∂z). The infinitesimal generatorsof the three rotations areX = z∂∂y− y ∂∂z, Y = −z ∂∂x+ x∂∂z, Z = −y ∂∂x+ x∂∂y.We obtain their prolongations X[3], Y[3], Z[3] ∈ X(J3(R,R2)) by imposing that X[3], Y[3] and Z[3]project onto X, Y and Z respectively, and leave invariant the differential system spanned bythe contact forms dy − y′ dx, dz − z′ dx, dy′ − y′′ dx, dz′ − z′′ dx, dy′′ − y′′′ dx and dz′′ − z′′′ dx:X[3] = X + z′ ∂∂y′− y′ ∂∂z′+ z′′∂∂y′′− y′′ ∂∂z′′+ z′′′∂∂y′′′− y′′′ ∂∂z′′′,Y[3] = Y + y′z′∂∂y′+(z′2 + 1) ∂∂z′+(y′z′′ + 2y′′z′) ∂∂y′′+ 3z′z′′∂∂z′′+(y′z′′′ + 3y′′z′′ + 3y′′′z′) ∂∂y′′′+(4z′z′′′ + 3z′′2) ∂∂z′′′,Z[3] = Z + (y′2 + 1)∂∂y′+ y′z′∂∂z′+ 3y′y′′∂∂y′′+(2y′z′′ + y′′z′) ∂∂z′′+(4y′y′′′ + 3y′′2) ∂∂y′′′+(y′′′z′ + 3y′′z′′′ + 3y′z′′′) ∂∂z′′′.The associated Noether invariants are:Cx = pz − qy + p′z′ − q′y′,Cy = Hz + qx+ y′z′p′ +(z′2 + 1)q′,Cz = Hy + px+(y′2 + 1)p′ + y′z′q′.174 J. Muñoz Masqué and L.M. Pozo Coronado3.1 The level H = 0The Hamiltonian H has a unique critical point at the point y′ = z′ = p = q = p′ = q′ = 0,which is non-degenerate of signature (4, 2) as follows from Morse’s lemma taking into accountthat H = py′ + qz′ + P 2 +Q2, whereP =12v3/2((y′2 + 1)1/2p′ + y′z′(y′2 + 1)−1/2q′),Q =12v5/2(y′2 + 1)−1/2q′.If the Hamiltonian vanishes, then we can substitute the expressions of Cy and Cz into thethird and fourth Hamilton equations (4) to obtain the following system of first-order ordinarydifferential equations:dy′dx=12v3(Cz − px), dz′dx=12v3(Cy − qx).3.2 H = 0If H is not zero, we use the formulas of Cy and Cz to eliminate y and z, and the third andfourth Hamilton equations (4) to eliminate p′ and q′. Substitution in the expressions of H andCx then yields, after defining C as2C = Cx − pCy − qCzH,the following system of ordinary differential equations of the first order:Cv3 =( qH+ z′) dy′dx−( pH+ y′) dz′dx, (5)(H − py′ − qz′)v5 =(z′2 + 1) (dy′dx)2− 2y′z′dy′dxdz′dx+(y′2 + 1) (dz′dx)2. (6)3.2.1 The case C = 0If C vanishes, then the equation (5) is just(Hz′ + q)dy′dx− (Hy′ + p)dz′dx= 0,or, equivalently,(dy′/dxdz′/dx)⊥=(dz′/dx−dy′/dx)⊥(H(y′z′)+(pq)).Hence, for every x there exists a λ(x) ∈ R such thatdy′dx= λ(Hy′ + p),dz′dx= λ(Hz′ + q).Solving this pair of differential equations, we obtainy′(x) = KyeHΛ − pH, z′(x) = KzeHΛ − qH,Hamiltonian Formulation and Order Reduction for Nonlinear Splines 175where Λ(x) =∫λ(x)dx and Ky,Kz ∈ R. Substitution in equation (6) then yields(H − (pKy + qKz)eHΛ − p2 + q2H)((K2y +K2z)e2HΛ − pKy + qKzHeHΛ +H2 + p2 + q2H2)5/2= (Λ′)2(H2(K2y +K2z ) + (qKy − pKz)2)e2HΛ,which is a first-order ordinary differential equation on Λ.3.2.2 The general caseLet us now assume that H = 0 and C = 0. From there, and (5), we know that p + Hy′ andq+Hz′ do not vanish simultaneously. Let us suppose that p+Hy′ = 0. Then, from equation (5)we can write the derivative of z′ in terms of the derivative of y′ asdz′dx=q +Hz′p+Hy′dy′dx− CHv3p+Hy′, (7)and then we can write this equation as a differential equation for z′(y′):dz′dy′=q +Hz′p+Hy′− CHv3(dy′/dx)(p+Hy′). (8)Substitution of (7) in (6) then yields(dy′dx)2 [(pz′ − qy′)2 + (Hz′ + q)2 + (hy′ + p)2] + 2dy′dxCHv3[y′(pz′ − qy′) − (Hz′ + q)]+C2H2v6(y′2 + 1) − (H − py′ − qz′)(Hy′ + p)v5 = 0,whose solutions aredy′dx=v3 (y′(pz′ − qy′) − (Hz′ + q)) ± ∆1/2A, (9)where∆ = −C2H2v8(Hy′ + p)2 + v5(Hy′ + p)(H − py′ − qz′)A,A = (qy′ − pz′)2 + (Hy′ + p)2 + (Hz′ + q)2.We have thus reduced the Hamilton equations to a pair of first-order ordinary differential equa-tions, (8) and (9). Also note that for C = 0 and H = 0, the above system is non-singularwhere it is defined so that the singularities of the system can only arise in the particular casespreviously studied.We remark that in order to find out where the solutions of (9) are defined, we have to studythe behaviour of the discriminant ∆. It is clear that where H − py′ − qz′ < 0 the discriminantis negative and hence no solution is defined in that region. On the half-plane H − py′ − qz′ > 0,∆ is positive if and only if(H − py′ − qz′)2 ((qy′ − pz′)2 + (Hy′ + p)2 + (Hz′ + q)2)2 > C4H4 (1 + y′2 + z′2)3 .The analysis of the sign of this expression has turned too long to be stated here, and we leaveit for a future work.176 J. Muñoz Masqué and L.M. Pozo CoronadoReferences[1] Constantelos G.C., On the Hamilton–Jacobi theory with derivatives of higher order, Nuovo Cimento B,1984, V.84, 91–101.[2] Giaquinta M. and Hildebrandt S., Calculus of Variations II: The Hamiltonian Formalism, Springer-Verlag,Berlin, 1996.[3] Goldschmidt H. and Sternberg S., The Hamilton–Cartan formalism in the calculus of variations, Ann. Inst.Fourier (Grenoble), 1973, V.23, N 1, 203–267.[4] Griffiths P.A., Exterior Differential Systems and the Calculus of Variations, Birkäuser, Boston, 1983.[5] Langer J. and Singer D.A., The total squared curvature of closed curves, J. Diff. 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Hamiltonian Formulation and Order Reduction for Nonlinear Splines in the Euclidean 3-Space

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Hamiltonian Formulation and Order Reduction for Nonlinear Splines in the Euclidean 3-Space

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