Closed stable orbits in a strongly coupled resonant Wilberforce pendulum
Misael Avendaño-Camacho, Alejandra Torres-Manotas, José A Vallejo
CClosed stable orbits in a strongly coupledresonant Wilberforce pendulum
Misael Avenda˜no-Camacho, Alejandra Torres-Manotasand Jos´e A VallejoSeptember 30, 2020
Abstract
We prove the existence of closed stable orbits in a strongly coupledWilberforce pendulum, for the case of a 1 : 2 resonance, by usingtechniques of geometric singular symplectic reduction combined withthe more classical averaging method of Moser.
The Wilberforce pendulum is a physical device composed of a long suspendedspring with a mass attached at the lower end, which is free to rotate aroundthe vertical axis, twisting the spring through a non-linear coupling.1 a r X i v : . [ m a t h - ph ] S e p t can be found in any Physics laboratory, where it is used to demonstratethe periodic motion arising from transference of energy between the two mainmodes of oscillation. If the spring is initially stretched, with a certain initialtorsion, and then released from rest, the motion will start being dominatedby an ‘up and down’ swinging, which gradually converts itself into a purelyrotational oscillation of the hanging mass.This striking motion is crucially related to the non-linear coupling be-tween both oscillating modes. In the usual setting, that coupling is as weakestas possible, being given by a quadratic term in the generalized coordinates.To be more specific, consider the spring to me massless, with elastic and tor-sional constants being κ and ρ , respectively. Let the moment of inertia of thehanging mass m be I . Finally, denote by x the elongation of the spring andby y the torsion angle. The complete Lagrangian in the case of a quadraticcoupling is then L = 12 m ˙ x + 12 I ˙ y − (cid:18) κx + 12 ρy + εxy (cid:19) . (1)This case is well known, as general references we can cite [K¨opf(1990),Berg and Marshall(1991), Plavˇci´c et al(2009)]. In this work, we will be in-terested in the case of a stronger coupling, namely, one given by a quarticnon-linear term. The Lagrangian (1) has then to be modified to read L = 12 m ˙ x + 12 I ˙ y − (cid:18) κx + 12 ρy + εx y (cid:19) , with its corresponding Hamiltonian H = 12 m p x + 12 I p y + 12 κx + 12 ρy + εx y , (2)and our goal is to study the existence of closed stable motions.As stated above, the problem falls within the reach of perturbation theory.On physical grounds, it was to be expected that a mild non-linear couplingwould lead to periodic motions, ‘inherited’ from the two independent oscilla-tion modes that would exist in the absence of coupling, but the situation isnot so clear in the presence of a strong non-linear coupling, as these interac-tions typically lead to a chaotic evolution, as shown in many textbooks suchas [Ott(2002), Strogatz(2015)].The standard tool for determining the onset of chaos in any dynami-cal system (in particular, perturbed ones) is the construction of a suitable2oincar´e section in the phase space; we do this in the next section, showingthe progressive destruction of the integrable tori associated with increasingvalues of the perturbation parameter. In the case in which the non-perturbedsystem is integrable, the Kolgomorov-Arnold-Moser (KAM) theorem guar-antees the persistence of some of these tori, thus proving the existence ofstable periodic orbits winding around some of the corresponding orbits ofthe unperturbed system in the absence of resonances . In this work, however,we will be interested in the 1 : 2 resonance (although the ideas presented willwork for an arbitrary m : n resonance), so the KAM theorem will not beapplicable.Aside from the KAM theorem, there are other techniques in perturba-tion theory that are well suited to the kind of problem at hand. Our pathhere will be to put first the system into normal form by using the Lie-Depritmethod (see [Deprit(1969)]), and then to apply a singular geometric reduc-tion (as in [Cushman(1994), Cushman and Bates(1997)]) to pass to a reducedphase space where closed stable orbits can be detected either by applyingMoser’s theorem on reduction (see [Moser(1970), Churchill et al(1983)]) orby determining the fixed points on a suitable Poincar´e surface. This ap-proach has been applied successfully to a qualitatively different system, thePais-Uhlenbeck oscillator, in [Avenda˜no-Camacho et al(2017)] so it can beconsidered to be of a general nature.Transforming the original perturbed Hamiltonian (2) H = H + εH ,where H = 12 m p x + 12 I p y + 12 κx + 12 ρy is the sum of two independent oscillators, into another one in normal form N = H + (cid:80) ∞ j =1 ε j N j , where { H , N j } = 0 for each j ∈ N (with {· , ·} thecanonical Poisson bracket on the algebra of smooth functions on the phasespace, C ∞ ( R )), requires solving a set of equations known (following V. I.Arnold) as the homological equations. To this end, it is convenient to makeuse of the averaging method, based on two averaging operators, (cid:104)·(cid:105) and S ,acting on tensor fields defined on the phase space manifold (in particular,on smooth functions on phase space). This allows us to obtain recursiveformulas for computing the sub-Hamiltonians N j , j ∈ N . Moser’s theoremextracts information about the existence of closed stable orbits from the crit-ical points of the first-order normal perturbation N on the reduced phasespace, but when there are degeneracies (as it turns out to be our case) onehas to go over N (by considering N = H + ε ( N + εN ) as the new perturbed3amiltonian). We will use the explicit expressions for N and N deducedin [Avenda˜no-Camacho et al(2013)], which have the advantage of not relyingon the introduction on action-angle variables; on the contrary, they only de-pend on the averaging operators mentioned above, and the canonical Poissonbracket. Finally, from the study of the critical points of N and N on thereduced phase space, we will be able to conclude the existence of closed stableorbits in the Wilberforce pendulum for any fixed value of the perturbationparameter ε > weakly cou-pled Wilberforce pendulum, see [de Bustos et al(2016)]. There, the authorsparametrize the periodic solutions of that perturbed system by the simplezeros of an associated system of nonlinear equations. To make explicit the characteristic frequencies of the system w , w , let usintroduce them through κ = mw and ρ = Iw . The Hamiltonian is H = H + εH = 12 (cid:0) p x + mw x + p y + Iw y (cid:1) + εx y . (3)The corresponding Hamilton equations of motion are given by the first-order system ˙ x = p x , ˙ p x = − (cid:0) mω x + 2 εxy (cid:1) , ˙ y = p y , ˙ p y = − (cid:0) Iω y + 2 εyx (cid:1) . (4)We solve this system numerically with the symplectic velocity Verlet4ethod, see [Holmes(2007)]. The numerical scheme in this case is x i +1 = x i + k ( p x ) i + k F ( x i , y i ; ε ) , ( p x ) i +1 = ( p x ) i + k F ( x i +1 , y i +1 ; ε ) + F ( x i , y j ; ε )) ,y i +1 = y i + k ( p y ) i + k F ( x i , y i ; ε ) , ( p y ) i +1 = ( p y ) i + k F ( x i +1 , y i +1 ; ε ) + F ( x i , y j ; ε )) , where k > F ( x, y ; ε ) = − mω x − εxy , F ( x, y ; ε ) = − Iω y − εyx .The resulting dynamics in the xy plane is displayed in Figure 1, where wehave taken as initial condition (1 , , , m = I = w = 1, w = 2, which will be assumed in what follows. Notice that the expectedLissajous figure appears when the coupling is switched off, and the patternbecomes fuzzier around ε = 0 . σ transversal to the flow of (4) constructedin the following way : First, we fix an energy value H = h in (3), and write p in terms of ( p , q , q , h ): p = ± (cid:113) (cid:0) h − ε ( q q ) (cid:1) − ( p + q + 4 q ) . (5)Next, we restrict the solution to the level set Σ h := { ( p , q , q ) : H = h } .The Poincar section is then σ = { ( p , q ) ∈ Σ h : q = 0 } .Initial conditions are taken of the form ( j/ , . , p , . j is arandom number in [ − , p is given in (5). In all cases, the value h = 3 has been chosen. Again, the evolution is computed with the velocityVerlet method, recording the points that cross σ by looking at sign changes.Notice that two set of solutions are obtained, one for each sign of (5), whichare superimposed to get the final Poincar map for each value of the parameter ε . The results are shown in Figure 2; as stated above, the destruction of theintegrable tori is quite visible here. However, certain ‘islands of stability’ Of course, this is just a choice. There are many possibilities for constructing a Poincar´esurface, but the idea of the numerical procedure is the same in all of them. Our choice isdetermined by reasons of graphic cleanliness. a) ε = 0 (b) ε = 0 . ε = 0 . ε = 0 . ε = 0 . ε = 0 . Figure 1: Strongly coupled Wilberforce pendulum dynamics for differentvalues of the parameter ε . The resonance 1:2 is shown.6urvive (as in the non-resonant case describes by KAM theorem), and in thenext sections we prove their existence analytically. Given a Poisson manifold ( M, {· , ·} ), consider a perturbed Hamiltonian ofthe form H = H + εH , where H is supposed to be integrable. Hamilton’sequations for H are a coupled non-linear system of differential equationswhose solutions, in general, do not have a closed form. The Lie-Deprit ap-proach to this problem substitutes the system of Hamiltonian equations bya simpler one, suitable to be studied by analytic tools, while providing somecriterion to determine the degree of accuracy of the approximation. Theperturbed Hamiltonian H is said to admit a normal form of order n if thereexist a near-identity canonical transformation on phase space such that H istransformed into H = n (cid:88) i =0 ε i N i + R H , (6)where N = H and { N i , H } = 0 , for all 1 ≤ i ≤ n . (7)The truncated function N = (cid:80) ni =0 ε i N i is the normal form (of order n ) of H .This approach is based on the fact that whenever (cid:107) H − N (cid:107) = (cid:107) R H (cid:107) is small ina suitable norm, the trajectories of N provide us with good approximations tothe true trajectories of H . In particular, closed orbits for H can be detectedthrough the existence of closed orbits for N .The normal form is obtained from a family of canonical transformationsdepending on the parameter ε , x (cid:55)→ y ( x ; ε ) (where x denotes collectively thecoordinates on M ), such that y ( x ; 0) = x . To assure that these transforma-tions are canonical, they are derived from a generating function S = S ( ε ): ∂y j ∂ε = { S, y j } = L X S y j , for j ∈ { , . . . , dim M } . (8)with X S = { S, ·} the Hamiltonian vector field determined by S . Geomet-rically, X S is the ‘ ε − flow generator’, much in the same way as H is thetime-flow generator. 7 a) ε = 0 . ε = 0 . ε = 0 . ε = 0 . ε = 0 . ε = 0 . ε = 1 Figure 2: Poincar maps of the strongly coupled Wilberforce pendulum fordifferent values of ε . The resonance 1 : 2 is shown.8he Lie-Deprit method proceeds by developing S in a formal series S = (cid:80) nj =0 ε j S j and translating the condition of being a generating function forcanonical transformations into a set of equations, one for each term S j , havingthe structure L X H S j = F j − ( j + 1) N j +1 j ≥ , (9)where the F j functions are determined by quantities already calculated in pre-vious steps. What is remarkable (see [Deprit(1969)]) is that these equations(called the homological equations ) have a recursive structure (the Deprit’striangle) and they can be solved in terms of H and the sub-Hamiltonians N j . The usual method of solution is based on the introduction of action-anglecoordinates, thus having a local character and requiring a symplectic phasespace. To avoid these issues here we follow [Avenda˜no-Camacho et al(2013)],where a global method of solution is presented in the case of a system admit-ting a U (1) − action such that the Hamiltonian vector field X H has periodicflow, as is the case with the Wilberforce pendulum.In a general setting, if we have a phase space which is a Poisson manifold( M, {· , ·} ), given the Hamiltonian H = H + εH we can set up the homo-logical equations (9). Now, suppose that the vector field X H = { H , ·} iscomplete and has periodic flow Fl tX H . The periodicity condition means thatthere exists a period function T : M → R such that Fl tX H ( p ) = Fl t + T ( p ) X H ( p ).This flow induces a U (1) − action by putting ( t, p ) (cid:55)→ Fl t/w ( p ) X H ( p ), where w = 2 π/T > U (1) − action is given by the vector fieldΥ = 1 w X H ∈ X ( M ) . Now, for any function f ∈ C ∞ ( M ), its U (1) − averaging is defined in terms ofthe pullback by the flow: (cid:104) f (cid:105) = 1 T (cid:90) T (Fl t Υ ) ∗ f d t . Also, an S operator, mapping C ∞ ( M ) into itself, is defined as S ( f ) = 1 T (cid:90) T ( t − π )(Fl t Υ ) ∗ f d t . N = (cid:104) H (cid:105) = 1 T (cid:90) T (Fl t Υ ) ∗ H d t , (10)and N = 12 (cid:28)(cid:26) S (cid:18) H w (cid:19) , H (cid:27)(cid:29) . (11) Consider the harmonic oscillator with two degree of freedom on T ∗ R with co-ordinates ( q , p , q , p ) and the Poisson bracket induced by the usual canon-ical symplectic structure, whose Hamiltonian is H ( q , p , q , p ) = 12 ( p + ω q + p + ω q ) . (12)The associated Hamiltonian vector field is readily found to be X H = p ∂∂q − ω q ∂∂p + p ∂∂q − ω q ∂∂p . The integral curves of X H , c : I ⊂ R → T ∗ R , can be parametrized as c ( t ) = ( q ( t ) , p ( t ) , q ( t ) , p ( t )), and satisfy the decoupled system (where thedots denote time derivatives) (cid:40) ¨ q + ω q = 0¨ q + ω q = 0 . Hence, we have an action on T ∗ R (cid:39) R given by the (linear) flow of X H :Fl tX H q p q p = q cos ω t + p ω sin ω t − ω q sin ω t + p cos ω tq cos ω t + p ω sin ω t − ω q sin ω t + p cos ω t . ω and ω are commensurable so, by a suitablerescaling in time, we actually have a U (1) action. In particular, if ω , ω ∈ Z are coprime, as in the case of the 1 : 2 resonance that we will consider (thatis, ω = 1, ω = 2), then Fl tX H is already 2 π − periodic. Notice that periodicorbits will be invariant sets under the action of this flow, so we expect tobe able of finding them by studying the invariant functions. In fact, we willrestrict our attention to the set of invariant polynomials under the action ofthis flow; the reason is that any other invariant will be a smooth function ofthese, as we will see below.It is well know that the algebra of invariant polynomials (under the ac-tion of the Hamiltonian flow of X H ) is finitely generated, see for example[Churchill et al(1983), Cushman and Bates(1997)]. Moreover, the generatorscan be chosen as the so-called the Hopf variables : ρ = z z = ω q + p ρ = z z = ω q + p ρ =Re ( z ω z ω ) = Re (( p + iω q ) ω ( p − iω q ) ω ) ρ =Im ( z ω z ω ) = Im (( p + iω q ) ω ( p − iω q ) ω ) . For instance, in the case of the 1 : 2 resonance we get ρ = q + p ρ =4 q + p ρ = p ( p − q ) + 4 p q q ρ =2 q ( p − q ) − q p p . (13)There exists a certain algebraic relation satisfied by the ρ variables,namely: ρ + ρ = ρ ω ρ ω , ρ , ρ ≥ , which is the equation of a singular algebraic surface in R . For the particularcase of the 1 : 2 resonance, this is ρ + ρ = ρ ρ , ρ , ρ ≥ . (14)Since (by a suitable rescaling) the action on T ∗ R (cid:39) R of the flow of X H can be seen as a smooth U (1) − action, the group U (1) is compact, andthe orbit space R /U (1) only contains finitely many orbit types (we willconsider the geometric structure of this orbit space later on), we can apply11he result in [Schwarz(1975)], which tells us that the smooth observablesinvariant under the action of U (1) are smooth functions of the polynomialgenerators ( ρ , ρ , ρ , ρ ). In order to prove analytically the existence of periodic orbits for the Wilber-force pendulum and determine their stability, we compute the second ordernormal form in the case of a quartic interaction and 1 : 2 resonance: H ( q , p , q , p ) = H + εH (15)= 12 ( p + q + p + 4 q ) + εq q , Since { H , N i } = L X H N i = 0, the first and second order normal formsare invariant under the U (1) − action induced by the flow of H ; we will takethe quotient of the phase space by this action and get the correspondingHamiltonian on the reduced phase space in the next section. An importantfeature of this reduction process is that this reduced Hamiltonian will be afunction of only three among the invariant generators ( ρ , ρ , ρ , ρ ). Previousto reduction, we compute in this section the expressions of N and N .Notice that the Hamiltonian flow Fl tX H in this case is given byFl tX H q p q p = q cos t + p sin t − q sin t + p cos tq cos 2 t + p sin 2 t − q sin 2 t + p cos 2 t (16)and it is 2 π − periodic. The second-order normal form of the Wilberforceoscillator is H + εN + ε N , where N and N are given by (10), (11).The computations are straightforward but tedious, and are best done using acomputer algebra system (CAS). We have found the CAS Maxima very usefulin this regard, and we have written a small Maxima package for this kindof computations, called pdynamics , which is available at https://github.com/josanvallejo/pdynamics . 12he resulting normal form sub-Hamiltonians, already written in the Hopfvariables, are as follows: H = 12 ( ρ + ρ ) , for the unperturbed part, and N = (cid:104) H (cid:105) = 116 ρ ρ (17)and N = (cid:104){ S ( H ) , H }(cid:105) = − ρ ρ + 4 ρ + 16 ρ ) , (18)for the first and second-order perturbations, respectively.We will make use of these explicit expressions in the following sections,to determine the existence of stable periodic orbits in the dynamics of theWilberforce oscillator. We begin by identifying the geometry of the the reduced phase space. Then,we find an explicit expression for the reduced Hamiltonian, that is, the normalform Hamiltonian N = H + εN + ε N + O ( ε ) restricted to the reducedphase space. We follow the technique described in [Churchill et al(1983),Cushman(1994)] to prove that (14) and the condition of constant energy H = h >
0, give the algebraic description of the reduced phase space. We usea result in [Po`enaru(1976)], which states that the basic invariant polynomialsseparate the orbits of the Hamiltonian flow Fl tX H . In our case this implies that the equality ( ρ ( q, p ) , . . . , ρ ( q, p )) = ( ρ ( q (cid:48) , p (cid:48) ) , . . . , ρ ( q (cid:48) , p (cid:48) )) holds ifand only if ( q, p ) and ( q (cid:48) , p (cid:48) ) belong to the same orbit. Thus, it is enoughto prove that for every ( u , u , u , u ) such that u + u = u u , its inverseimage under the map ( q, p ) (cid:55)→ ( ρ ( q, p ) , . . . , ρ ( q, p )) is precisely a single orbitof the flow Fl tX H . For instance, if u = 0 then ρ ( q, p ) = 0 and necessarily q = 0 = p (from (13)). This, in turn, implies that ρ = 0 = ρ so we havethe inverse image of ( u , , , u ≥
0, which is the set { ( q , p , , ∈ R : q + p = u } , and this is clearly an orbit of Fl tX H . The remaining cases Here we collectively denote ( q , p , q , p ) by ( q, p ). (cid:40) ρ + ρ = ρ ρ , ρ , ρ ≥ ,ρ + ρ = 2 h , that is, ρ + ρ = ρ (2 h − ρ ) , ≤ ρ ≤ h . (19)As mentioned above, (19) is the equation of a singular algebraic surface S ∈ R . Topologically, this surface is a pinched sphere with a singularity atthe point ( ρ , ρ , ρ ) = (0 , ,
0) (see Figure 3).Figure 3: Reduced phase space of the 1:2 resonance.One of the most important results in the theory is a theorem by Moser(see [Moser(1970), Churchill et al(1983)]), which can be stated as follows:Let H = H + εH be a perturbed Hamiltonian, with S the hypersurface H = h . Suppose that the orbits of the Hamiltonian flow Fl tX H are allperiodic with period 2 π and let M h be the quotient with respect to theinduced U (1) − action on S . Then, to every non-degenerate critical point p ∈ M h of the restricted averaged perturbation N | S = (cid:104) H (cid:105)| M h correspondsa periodic trajectory of the full Hamiltonian vector field X H , that branchesoff from the orbit represented by p and has period close to 2 π .In order to apply this result, we must first characterize the critical pointsof Hamiltonian vector fields in the the reduced space. First, observe that the14ommutator relations among generators ( ρ , ρ , ρ , ρ ) are given by { ρ , ρ } = 0 , { ρ , ρ } = − ρ , { ρ , ρ } = 4 ρ , { ρ , ρ } = 4 ρ , { ρ , ρ } = − ρ , { ρ , ρ } = − ρ ( ρ − ρ ) . (20)Renaming the variables ρ = x , ρ = y , and ρ = z , these relations inducea Poisson bracket on the three dimensional Euclidean space R = { ( x, y, z ) } given by { f, g } = 2 (cid:104)∇ g, ∇ f × ∇ F (cid:105) , (21)where F is the function F ( x, y, z ) = x + y − z (2 h − z ) , (22)and the symbols (cid:104)· , ·(cid:105) , × , ∇ stand for the usual inner product, cross productand nabla operator in R , respectively. Hence, for any f ∈ C ∞ ( R ), itsHamiltonian vector field is given by X f = 2 ∇ f × ∇ F . (23)It follows directly from definition (21) that the function F ( x, y, z ) (22) is aCasimir of the Poisson structure (21). Thus, the symplectic leaves of thecorresponding foliation are precisely the connected components of level setsof F . If we define the mapping P : R → R by P ( ρ , ρ , ρ , ρ ) = ( ρ , ρ , ρ ) , we get that P is a Poisson map and P ( H − ( h )) = F − (0) . Moreover, (cid:16) P ◦ Fl tX H (cid:17) q p q p = P ρ ( p , q , p , q ) ρ ( p , q , p , q ) ρ ( p , q , p , q ) ρ ( p , q , p , q ) . Therefore, the reduced space is contained in a symplectic leaf of F − (0) ⊂ R . Let us denote by M h the reduced space. Then, a realization of it as a smoothmanifold is given by M h = F − (0) and 0 < z ≤ h . (24) Notice that the condition z > f ∈ C ∞ ( R ) defines a Hamiltonian vector field (cid:101) X f on M h bythe restriction of (23): (cid:101) X f := (2 ∇ f × ∇ F ) | M h . It also follows from (23) that the Hamiltonian vector field (cid:101) X f has a criticalpoint at the point p ∈ M h if and only if either ∇ f ( p ) is orthogonal at p tothe reduced space M h , or ∇ f ( p ) = 0.Next, we describe how to obtain the reduced Hamiltonian vector fieldcorresponding to a function G ∈ C ∞ ( R ) such that { H , G } = 0. Asdiscussed above, G can be expressed in terms of the Hopf variables: G = G ( ρ , ρ , ρ , ρ ). Writing ρ = z, ρ = 2 h − z, ρ = x and ρ = y , we obtainthe function Q ( x, y, z ) = G ( z, h − z, x, y ). Thus, the reduced Hamiltonianvector field associated to G is the vector field (cid:101) X G = (2 ∇ Q × ∇ F ) | M h . This expression allows us to compute the critical points of the reduced vectorfield associated to the first-order normal form N ( ρ , ρ , ρ , ρ ) given in (17).Letting as above K ( x, y, z ) = N ( z, h − z, x, y ), we get K ( x, y, z ) = 116 (2 hz − z ) . (25)Hence, the reduced vector field is (cid:101) X N = (2 ∇ F × ∇ K ) | M h . (26)As we pointed out above, the critical points of (26) are those points p ∈ M h such that either ∇ K ( p ) = 0 or ∇ K ( p ) is orthogonal to M h (parallel to ∇ F ( p )). It is immediate to calculate ∇ K = (cid:18) , ,
18 ( h − z ) (cid:19) . It follows form here that ∇ K ( p ) is orthogonal to M h at the point (0 , , h )and that ∇ K ( p ) = 0 if z = h . Thus, the reduced vector field (cid:101) X N hasa critical point at (0 , , h ) and a curve of critical points given by Γ h = { ( x, y, z ) | x + y = h and z = h } , see Figure 4 (where the singular point(0 , ,
0) is also shown). 16igure 4: Critical points of the first-order normal Hamiltonian N .Consider the critical point (0 , , h ). By a straightforward computation,we get ∂F∂z (0 , , h ) = 4 h (cid:54) = 0 . By the implicit function theorem, z = ψ ( x, y ) with ψ a smooth function at(0 ,
0) satisfying ψ (0 ,
0) = 2 h and F ( x, y, ψ ( x, y )) = 0. Therefore, the function K in (25) has the form (cid:101) K = K ( ψ ( x, y )) in a neighborhood of (0 , , h ).Another immediate computation shows thatHess( (cid:101) K (0 , h > . Thus, the critical point (0 , , h ) is non-degenerate and Moser’s theorem (seethe version presented as Theorem 6.4 in [Churchill et al(1983)]) implies that,for small enough ε , the Wilberforce oscillator has a unique stable periodicorbit γ ε with energy h through each point p ( ε ), sufficiently close to (0 , , h ),with period T ( ε ), such that H ( p ( ε )) → h and T ( ε ) → π . We can not apply Moser’s theorem to the curve of critical points of thepreceding section, Γ h , because they are degenerate (non-isolated). In order17o determine if some periodic orbits arise from some points of Γ h we mustresort to the second-order normal form of the Hamiltonian (15), which willbe regarded as a perturbed Hamiltonian H + ε ( N + εN ) on its own.Thus, we consider the ε − dependent function K ε ( x, y, z ) = ( N + εN ) | M h .By arguments similar to those used in the case of Moser’s theorem, we read-ily see that a given point of Γ h generates a periodic orbit of the Wilberforcependulum if it is a non-degenerate critical point of the Hamiltonian vectorfield X K ε = 2 ∇ K ε × ∇ F for all ε . Therefore, we need to look for points p ∈ Γ h such that, either ∇ K ε ( p ) = 0, or ∇ K ε ( p ) is parallel to ∇ F ( p ) for all ε . A straightforward computation gives ∇ K ε = (cid:18) − εx , − εy , h − z − ε
768 (4 h − hz + 3 z ) (cid:19) . For this vector to vanish, its third component must be zero independentlyof ε , that is, both the independent term and the coefficient of ε must vanishseparately. These conditions would lead to z = h and h = 0, so the onlypossibility is the point (0 , ,
0) on the particular surface M , which is the caseof the singular point that we consider in the next section. It follows fromhere that ∇ K ε never vanishes for ε (cid:54) = 0 and h (cid:54) = 0, and it is easy to checkthat it is parallel to ∇ F only at the points (0 , , h ) and (0 , , h ), whichdo not belong to the curve Γ h . Consequently, we conclude that no points ofΓ h (aside from the singular point (0 , ,
0) in the case h = 0) can generate aperiodic orbit. Recall that, in order to impose a smooth structure on the reduced space, weleft aside the singular point (0 , , normal mode , γ ). The existence of closed orbits will be proved byfinding fixed points on a suitable Poincar´e section.Let f ( p , q , p , q ) = ( p + 4 q ). The Hamiltonian vector field withrespect to the canonical symplectic structure on R , X f , has periodic flowwith periodic T = π . This flow generates a free and proper U (1) − action on( R − (0 , × R . For every fixed h >
0, the level set f − ( h ) is foliated by peri-odic orbits of X f and a the reduced space is given by M h = f − ( h ) /U (1). Let18s make the following change of variables from ( p , q , p , q ) to ( p , q , L, θ ):Ψ( p , q , L, θ ) = ( p , q , −√ L sin θ, √ L cos θ ) , with L > , < θ < π/ω . In these coordinates, the canonical symplecticform on the domain R × ( R − (0 , p ∧ d q + d p ∧ d q , becomesd p ∧ d q + d L ∧ d θ , and the Hamiltonian of the Wilberforce oscillator is H ( p , q , L, θ ) = 12 ( p + q ) + 2 L + εLq cos θ. (27)Consider the restriction to the level set Σ h = { ( p , q , L, θ ) | L = h } . Sincethis level set is foliated by orbits of X f , the Hamiltonian equations of (27)are ˙ θ = 2 + εq cos θ , ˙ p = − q − εhq cos θ, ˙ q = p . (28)We now construct the cross section σ = { ( p , q , h, θ ) ∈ Σ h : θ = 0 } , and fixthe point a = (( p , q , h, a is: θ ( t ) = 2 t + ε (cid:82) t q cos θ d τ ,p ( t ) = p cos t − q sin t − εh (cid:82) t q cos θ d τ ,q ( t ) = p sin t + q cos t . (29)Let T ( a, ε ) be the time elapsed between two consecutive intersections of σ .From equations (29), we get4 π = 2 T ( a, ε ) + ε (cid:90) T ( a,ε )0 q cos θ d t , so T ( a, ε ) has the form T ( a, ε ) = 2 π − ε π q ) + O ( ε ) . (30)Substituting (30) in (29), we obtain the following expression for the Poincar´emap determined by σ : p ( T ( a )) = p + επq (cid:18)
12 ( q ) − h (cid:19) + O ( ε ) ,q ( T ( a )) = q + εp π q ) + O ( ε ) .
19n order to prove that there exists periodic orbits for the Wilberforce oscilla-tor in Σ h , we must show that, for each ε small enough, there exist p ( ε ) and q ( ε ) such that we get a fixed point: p ( T ( p ( ε ) , q ( ε ) , − h, , ε )) = p ( ε ) ,q ( T ( p ( ε ) , q ( ε ) , − h, , ε )) = q ( ε ) . To this end, we define the following function F : R → R , F p q ε = (cid:18) πq (cid:0) ( q ) − h (cid:1) + O ( ε ) p π ( q ) + O ( ε ) (cid:19) . First, we note that F (0 , √ h,
0) = 0 . A straightforward computation showsthat det (cid:32) ∂F∂p ∂q (cid:12)(cid:12)(cid:12)(cid:12) (0 , √ h, (cid:33) = det (cid:18) πh πh (cid:19) > . By the implicit function theorem, there exists δ >
0, an open neighborhood U of (0 , √ h ), and a function g : ( − δ, δ ) → U , g ( ε ) = ( p ( ε ) , q ( ε )), suchthat g (0) = (0 ,
0) and F ( g ( ε ) , ε ) = 0. Therefore, p ( T ( g ( ε ) , − h, , ε )) = p ( ε ) ,q ( T ( g ( ε ) , − h, , ε )) = q ( ε ) . This fact proves that for each sufficiently small ε , the Wilberforce oscillatorhas a unique stable periodic orbit γ ε , with energy h , which branches off fromthe normal mode γ . Acknowledgements : MAC was partially supported by a Mexican CONA-CyT Research Project code CB-258302, ATM was supported by a MexicanCONACyT graduate student grant, and JAV was partially supported by aMexican CONACyT Research Project code A1-S-19428.
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M. , Avenda˜no,
Departamento de Matem´aticas, Universidad de Sonora(M´exico), Hermosillo, Son. 83000
E-mail address , M. , Avenda˜no: [email protected]
A. Torres,
Facultad de Ciencias, Universidad Aut´onoma de San Luis Potos´ı(M´exico), San Luis Potos´ı, SLP 78295
E-mail address , A. Torres: [email protected]
J. A. Vallejo (Corresponding author),
Facultad de Ciencias, Universidad Aut´onomade San Luis Potos´ı (M´exico), San Luis Potos´ı, SLP 78295
E-mail address , J.A. Vallejo: [email protected]@fc.uaslp.mx