Branched Hamiltonians and time translation symmetry breaking in equations of the Lienard type
BBranched Hamiltonians and time translation symmetrybreaking in equations of the Li´enard type
A Ghose-Choudhury ∗ Department of Physics, Diamond Harbour Women’s University,D. H Road, Sarisha, West-Bengal 743368, IndiaPartha Guha † SN Bose National Centre for Basic SciencesJD Block, Sector III, Salt LakeKolkata 700098, India
Abstract
Shapere and Wilczek ( Phys. Rev. Lett. 109, 160402 and 200402 (2012)) have re-cently described certain singular Lagrangian systems which display spontaneous breakingof time translation symmetry. We begin by considering the standard Lienard equationfor which a Lagrangian is constructed by using the method of Jacobi Last Multiplier. Thevelocity dependance of the Lagrangian is such that the momentum may exhibit multival-uedness thereby leading to the so called branched Hamiltonian. Next with a quadraticvelocity dependance in the Li´enard equation one can construct a Hamiltonian descrip-tion involving a position dependent mass. We compute the Lagrangian and Hamiltonianof this system and show that the canonical Hamiltonian is single valued . However, wefind that up to a constant shift, the square of this Hamiltonian describes systems givingrise to spontaneous time translation symmetry breaking provided the potential functionis negative.
PACS:
Keywords:
Jacobi Last Multiplier, position dependent mass, multi-valued Hamiltonians,time translation symmetry breaking. ∗ E-mail [email protected] † E-mail: [email protected] a r X i v : . [ n li n . S I] A p r Introduction
Recently Shapere and Wilczek [1, 2] have shown that for certain special Lagrangian systemsthe time translation symmetry can be spontaneously broken in the lowest energy or groundstate. This has revived interest in the study of systems with non-standard and/or non-convexLagrangians especially with regard to spontaneous breaking of time translation symmetry.A direct consequence of the spontaneously broken time translation symmetry in the groundstates is the multivaluedness of the Hamiltonian.A common feature shared by all the models considered by Shapere and Wilczek [1,2] is that the energy function (Hamiltonian) or Lagrangian systems become multivalued interms of the canonical phase space variables. Recently it has become clear that, for specialkinds of mechanical systems, there are choices of Hamiltonian structures in which certainfundamental aspects of classical canonical Hamiltonian mechanics are changed. It has beenexplored in [3, 4, 5], one can change the phase space variables which makes the Hamiltonianand symplectic structures on the phase space simultaneously well defined at the price ofintroducing a non-canonical symplectic structure. Curtright and Zachos [6] displayed somesimple unified Lagrangian prototype systems which, by virtue of non-convexity in their velocitydependence, branch into double-valued (but still self-adjoint) Hamiltonians.It is noteworthy that for systems possessing multiple Hamiltonian descriptions, therehave been discussions in the literature as to find the proper choice of Hamiltonian functions.Furthermore an analysis of such models has even led to speculations about the possibility ofperpetual motion. Shapere and Wilczek papers triggered a new interest on the systems withbranched Hamiltonians.The issue of time independent classical dynamical systems exhibiting motion in theirlowest energy states has been instrumental in the introduction of a time analogue of spa-tial order as in a crystalline substance [1] (the so called time crystals ) and its spontaneousbreaking. It is therefore natural to investigate the issue of time translation breaking fromthe perspective of second-order differential equations within the general framework of La-grangian/Hamiltonian mechanics [3, 4].
Motivation and result :
The motivation for the present work arose originally fromShapere and Wilczek’s observation that the Lagrangians of some mechanical systems displayspontaneous time translation symmetry breaking properties in their lowest energy state, andthe Hamiltonian descriptions of certain singular models involving multi-valuedness and branch-ing point singularities. In a previous article we obtained the Chiellini integrability criterionfor the Li´enard equation by using Jacobi’s last multiplier [15] and derived the bi-Hamiltonianstructure of those equations of the Li´enard type satisfying this particular criterion. More-over we also constructed certain non-natural Lagrangians and Hamiltonians for the Li´enardequation using Jacobi’s last multiplier; consequently it is only natural that we investigate thepossible existence of time translation symmetry breaking of the ground state for such systems.The first case we deal with is that of a second-order ordinary differential equation (ODE) ofthe usual Li´enard type viz ¨ x + f ( x ) ˙ x + g ( x ) = 0 , (1.1)2or which we present specific cases of a double valued Hamiltonian and its branches. This isfollowed up with a quadratic version (the Li´enard-II equation)[7], namely¨ x + f ( x ) ˙ x + g ( x ) = 0 . (1.2)The latter naturally emerges from Newton’s second law when dealing with a system charac-terized by a variable mass (depending on the position coordinate) and also frequently arises inthe context of isochronous systems [8, 9, 10]. By a suitable modification of the Hamiltonian ofthis equation we obtain the locus of the curve of the singular points for which the energy is lessthan the minimum value indicating the spontaneous breaking of time translation symmetry.This paper is organized as follows. We present the branched Hamiltonian description ofthe Li´enard equation in Section 2. We also illustrate the double valuedness of the Hamiltoniandescription. Section 3 is devoted to the hamiltonization of an equation of Lienard type witha quadratic dependence on the velocity, dubbed as Li´enard II equation. We demonstrate howthe time translation symmetry spontaneously broken for Li´enard II system in Section 4. There exists an extensive literature on the Li´enard-I equation ( for example, [13, 14]) and inthis section our attempt is to incorporate the Li´enard-I equation¨ x + f ( x ) ˙ x + g ( x ) = 0 , (2.1)into the branched Hamiltonian framework. It has been shown in [8, 9] how a system of theLi´enard type as given by (1.2) can be embedded into the Hamiltonian formalism. We brieflyrecapitulate the procedure below. Given a second-order ordinary differential equation (ODE)¨ x = F ( x, ˙ x ) (2.2)we define the Jacobi last multiplier M as a solution of the following ODE d log Mdt + ∂F ( x, ˙ x ) ∂ ˙ x = 0 . (2.3)Assuming (2.2) to be derivable from the Euler-Lagrange equation one can show that the JLMis related to the Lagrangian by the following equation M = ∂ L∂ ˙ x . (2.4)From (2.3) a formal solution of the Jacobi last multiplier for (2.1) may be written as M ( t, x ) = exp (cid:18)(cid:90) f ( x ) dt (cid:19) := u /(cid:96) , (2.5)where u is a new nonlocal variable and (cid:96) is a parameter whose value is fixed by the followinglemma once f and g are given. 3 emma 2.1 The Li´enard equation (2.1) can be written as the following system ˙ u = (cid:96)uf ( x ) , ˙ x = u + W ( x ) where W = g/f (cid:96) with the parameter (cid:96) being determined by the following condition ddx (cid:18) gf (cid:19) = − (cid:96) ( (cid:96) + 1) f ( x ) . (2.6) Proof : From (2.5) we have log u = (cid:96) (cid:82) f ( x ) dt , which implies ˙ u = (cid:96)uf ( x ). Setting˙ x = u + W ( x ) we find by differentiating with respect to t ¨ x = ˙ u + W (cid:48) ( x ) ˙ x. Inserting the expression for ˙ u from the previous equation and after eliminating u we find that¨ x = (cid:96)f ( x )( ˙ x − W ) + W (cid:48) ( x ) ˙ x. Comparison with (2.1) then shows W (cid:48) ( x ) = − ( (cid:96) + 1) f ( x ) and W ( x ) = g/(cid:96)f . Consistency nowrequires that ddx (cid:18) gf (cid:19) = − (cid:96) ( (cid:96) + 1) f ( x ) , which represents actually the Cheillini integrability condition for (2.1) ( see [15], for Cheilliniintegrability condition in the context of Li´enard equation).Since the transformation is nonlocal so a mapping to the ( x, u )-plane is not possible andtherefore one cannot really analyse the problem in the local manner of point transformations.However, from (2.4) and (2.5) we have ∂ L∂ ˙ x = (cid:18) ˙ x − (cid:96) gf (cid:19) /(cid:96) , and it may be shown that (2.1) can be derived from the following Lagrangian L = (cid:96) ( (cid:96) + 1)(2 (cid:96) + 1) (cid:18) ˙ x − (cid:96) gf (cid:19) (2 (cid:96) +1) /(cid:96) , (2.7)provided the functions f and g satisfy the Cheillini integrability condition (2.6). Before proceeding to a determination of the Hamiltonian for (2.1) from the above Lagrangianwe note that the curvature ∂ L/∂ ˙ x changes sign at the points where ˙ x = g/f (cid:96) provided (cid:96) isan odd integer or 1 /(cid:96) is an odd integer. The conjugate momentum is given as usual by p = (cid:96)(cid:96) + 1 (cid:18) ˙ x − (cid:96) gf (cid:19) ( (cid:96) +1) /(cid:96) . x as a function of p and x presents us with diffi-culty and is the source of the double valuedness of the resulting Hamiltonian. Formally theHamiltonian is H = p (cid:96) +1 /(cid:96) +1 K ( (cid:96) ) − ( g/(cid:96)f ) p where K ( (cid:96) ) is just a scaling factor.By enlarging the phase space and making use of Dirac’s theory on constrained Hamilto-nian systems Zhao et al [4] presented the Hamiltonian description and formulated a methodto avoid the multivaluedness and the brunching point singularities.We consider the following example to illustrate our point. Example ¨ x + x ˙ x + x − x = 0Here f ( x ) = x and g ( x ) = x − x . One can easily verify that the Cheillini condition is satisfiedwith (cid:96) = 1 and −
2. For (cid:96) = 1 we obtain p = ( ˙ x − x ) /
2. A plot of the variation of theconjugate momentum with x and ˙ x = y is shown below in Fig. 1 . On the other hand uponinversion we have ˙ x = 1 − x ± √ p and a plot of the variation of ˙ x with x and p is depictedin Fig. 1. It is observed that ˙ x = 1 − x ± √ p whence the Hamiltonian is double valued withthe branches: H ± = p (1 − x ± (cid:112) p )The variation of the Hamiltonians are depicted below in Fig 2.Figure 1: 3D plot showing the variation ˙ x = 1 − x ± √ p when (cid:96) = 1, the lower (upper) oneis the negative ne, both meet at p = 0However, when (cid:96) = − p = 2 (cid:113) ˙ x + (1 − x ) leading to ˙ x = p / − (1 − x ) / H = p / − p (1 − x ) /
2, i.e., we have a single valued Hamiltonian.We illustrate the variation of velocity and Hamiltonian when l = − H ± when (cid:96) = 1Figure 3: 3D plot showing the variation ˙ x = p / − (1 − x ) / (cid:96) = − (cid:96) = − For the equation ¨ x + f ( x ) ˙ x + g ( x ) = 0 , (3.1)one can show that a solution of the JLM is given by M ( x ) = e F ( x ) , F ( x ) := (cid:90) x f ( s ) ds. (3.2)6urthermore it follows from (2.4) that its Lagrangian is L ( x, ˙ x ) = 12 e F ( x ) ˙ x − V ( x ) , (3.3)where the potential term V ( x ) = (cid:90) x e F ( s ) g ( s ) ds. (3.4)Clearly the conjugate momentum p := ∂L∂ ˙ x = ˙ xe F ( x ) implies ˙ x = pe − F ( x ) , (3.5)so that the final expression for the Hamiltonian is H = p M ( x ) + (cid:90) x M ( s ) g ( s ) ds, (3.6)where p = M ( x ) ˙ x and M ( x ) = exp(2 F ( x )) with F ( x ) = (cid:82) x f ( s ) ds . The canonical variablesare x and p and they satisfy the standard Poisson brackets { x, p } = 1. In terms of thecanonical Poisson brackets the equations of motion appear as˙ x = { x, H } = pM ( x ) , ˙ p = { p, H } = M (cid:48) ( x )2 M ( x ) p − M ( x ) g ( x )from which we can recover (3.1) upon elimination of the conjugate momentum p . Here wehave purposely written the Hamiltonian H in terms of the last multiplier M ( x ) to highlightthe latter’s role as a position dependent mass term. From (3.6) it is natural that the potential V ( x ) be identified with V ( x ) = (cid:90) x M ( s ) g ( s ) ds. (3.7)As for the existence of a minima of H , considered as a function of x and p , it is necessary that ∂H∂x = 0 and ∂H∂p = 0 (3.8)whose solutions then define the stationary points. The former yields − p M (cid:48) ( x )2 M ( x ) + M ( x ) g ( x ) = 0while the latter implies p/M ( x ) = 0. Therefore the stationary points are characterized by p = 0 and the value(s) of x for which g ( x ) = 0. If x = x (cid:63) denotes a root of g ( x ) = 0 then( x (cid:63) , p = 0) is a stationary point (s.p). For the s.p to be a minimum one requires that theprincipal minors of ∆ = (cid:12)(cid:12)(cid:12)(cid:12) H xx H xp H px H pp (cid:12)(cid:12)(cid:12)(cid:12) s.p
7e positive definite, i.e., g (cid:48) ( x (cid:63) ) > M ( x (cid:63) ) g (cid:48) ( x (cid:63) ) > , and consistency therefore requires M ( x (cid:63) ) >
0. Note that M ( x ), which may be thought of assome kind of ’effective mass’ such as within a spatial crystal, may be negative for x (cid:54) = x (cid:63) .Clearly the fact that p = 0 in the minimum energy state (ground state) of the system precludesthe possibility of any motion. Consider a one-dimensional generalized Hamiltonian system (cid:101) H = F ( H ) with Hamiltonianvector field given in terms of the canonical form X (cid:101) H = ∂ (cid:101) H∂p ∂∂x − ∂ (cid:101) H∂x ∂∂p , { G, (cid:101) H } = ˙ G. In the symplectic coordinates ( x, p ) this is equivalent to canonical Hamiltonian equations˙ x = F ( H ) (cid:48) { x, H } , ˙ p = F ( H ) (cid:48) { p, H } , where F ( H ) (cid:48) > . It may be easily verified that the above set of Hamiltonian equations may be obtained fromthe modified symplectic form ω = F ( H ) (cid:48) dx ∧ dp . Moreover this change of Hamiltonianstructure will not change the partition function, hence all thermodynamic quantities willremain unchanged.Let us consider a new Hamiltonian [4] defined by (cid:101) H = (cid:18) p M ( x ) + (cid:90) x M ( s ) g ( s ) ds (cid:19) + E = H + E , (4.1)where E is an arbitrary constant. As the New Hamiltonian is anticipated to generate adynamics which is distinct from that of H , let us also introduce the following Poisson structure { x, p } = ξ ( x, p ) so that the equations of motion which follow from˙ x = { x, (cid:101) H } , ˙ p = { p, (cid:101) H } (4.2)give ˙ x = 2 ξH pM ( x ) (4.3)˙ p = − ξH (cid:18) − M (cid:48) ( x )2 M ( x ) p + M ( x ) g ( x ) (cid:19) . (4.4)At this point we need to make a clear distinction regarding the two Poisson structures wehave introduced. It will be noticed that if one assumes { x, p } = ξ ( x, p ) = H ( x,p ) then we get8ack the original Li´enard-II equation (3.1), if however we persist with ξ = 1, i.e., assume x and p are canonical then the equation of motion resulting from the Hamiltonian (cid:101) H is of theform ¨ x + 2 H ( f ( x ) ˙ x + g ( x )) = 0 . (4.5)Although (4.5) appears to be different from (3.1) it is interesting to note that (4.5) can bemapped to the original set of Hamiltonian equations by using a (nonlocal) Sundman trans-formation [12] through a transformation of the independent temporal variable t to a newindependent variable s given by ds = 2 Hdt , whence we obtain x (cid:48) = pM ( x ) , p (cid:48) = − (cid:18) − M (cid:48) ( x )2 M ( x ) p + M ( x ) g ( x ) (cid:19) , (4.6)where (cid:48) = dds . In fact such transformations were used by Sundman while attempting to solvethe restricted three body problem.As for the stationary points of the Hamiltonian (cid:101) H , these follow from the solutions of ∂ (cid:101) H/∂x = 0 and ∂ (cid:101) H/∂p = 0. The latter yields either p = 0 or H = 0. If p = 0 then theformer condition gives either H = 0 or g ( x ) = 0, i.e x = x (cid:63) . The pair ( x (cid:63) , p = 0) leads by theprevious analysis to the case (cid:101) H min = (cid:18)(cid:90) x (cid:63) M ( s ) g ( s ) ds (cid:19) + E . (4.7)From the above equation it is clear that the local minimum of (cid:101) H is in general greater thanthe constant E because the potential V ( x (cid:63) ) is not required to vanish at x = x (cid:63) . As thestationary point corresponds to p = 0 the time translation symmetry is not broken and wehave the same situation as previously discussed in section 2.However one also has now the possibility wherein H = 0 which implies that the locusof the stationary points lie on the curve p M ( x ) + (cid:90) x M ( s ) g ( s ) ds = 0 . (4.8)This condition obviously implies that (cid:101) H has a minima with (cid:101) H min = E which is less than thatgiven by (4.7). Now for real values of p it is then necessary that V ( x ) = (cid:90) x M ( s ) g ( s ) ds < . The force dV /dx is clearly not necessarily zero and motion can therefore occur in the groundstate. The existence of motion under such circumstances is indicative of the spontaneousbreaking of the time-translation symmetry [1].To investigate the possible nature of the motion in this scenario let us demand that V ( x ) = (cid:90) x M ( s ) g ( s ) ds = − X ( x ) , (4.9)9here X ( x ) = (cid:82) (cid:112) M ( x ) dx . Such a choice is consistent with the view expressed in [3] thattime translation symmetry may be present in almost all Newtonian mechanical systems witha conservative potential provided the potential can be shifted to acquire a negative value.Furthermore such symmetry breaking occurs in a non-standard Hamiltonian description wherethe new Hamiltonian is the square of the canonical Hamiltonian together with Poisson bracketswhich are nonlinear. Differentiating (4.9)we get M ( x ) g ( x ) = − X ( x ) X (cid:48) ( x ) with X (cid:48) ( x ) = (cid:112) M ( x ) = e F ( x ) so that e F ( x ) g ( x ) = − X ( x ) which after another differentiation with respect to x leads to thecondition g (cid:48) ( x ) + f ( x ) g ( x ) = − , (4.10)in view of the fact that f ( x ) = M (cid:48) ( x ) / M ( x ). Notice that this basically represents motion ina inverted oscillator potential and it is therefore not surprising that the last condition on thefunctions f and g is just the ‘inverted isochronicity’ condition [7]. The notion of an invertedoscillator also appears in the context of de-Sitter gravity. To arrive at concrete models for thefunction f in this case, we note that one may solve (4.10) for f to get f = − g (cid:48) g , which then implies M ( x ) = 1 g ( x ) exp (cid:18) − (cid:90) dxg (cid:19) . (4.11)from (4.9) it follows that X ( x ) = (cid:90) g ( x ) exp (cid:18) − (cid:90) dxg (cid:19) dx (4.12)The points of minima therefore lie on the curve p = ± (cid:112) M ( x ) X ( x ) = ± g ( x ) exp (cid:18) − (cid:90) dxg (cid:19) (cid:90) g ( x ) exp (cid:18) − (cid:90) dxg (cid:19) dx. We end this section with a couple of examples:
Example 1
Let g ( x ) = x then we have M ( x ) = 1 x , X ( x ) = − x, and p = ∓ x , the singular nature of M ( x ) at x = 0 forces us to confine ourselves to the half line. It isevident that the particle can at any instant of time have only one of the two possible valuesfor the momentum. The particular choice of any one of these two possible values thereforebreaks the time translation symmetry. Example 2 If g ( x ) = 1 /x then we obtain M ( x ) = x e − x , X ( x ) = − e − x and p = ∓ xe − x . Conclusion
We have shown that nonlinear ODEs of the Li´enard type it is easy to recast them into theLagrange/Hamilton formalism and the basic results of Shapere-Wilczek are apparently appli-cable to such a differential system. In particular, we have studied the branched Hamiiltonianand multivaluednes of momentum of this equation. Our analysis is based on (cid:101) H = H + E .Actually when we consider such kind of generalized Hamiltonian the number of critical pointsis changed drastically, and most of the critical points of the generalized Hamiltonian are notthe images of the critical point of the original Hamiltonian. Careful readers might have noticedthat this (quadratic) Hamiltonian connected to (exotic) Lagrangian via Legendre transforma-tion injected the multivaluedness of the momentum.Shapere and Wilczek found that the direct consequence of this multivaluedness is thatthe time translation symmetry is spontaneously broken in the ground states. The phenomenonof spontaneous symmetry breaking was hitherto mostly restricted to the quantum domain.The most outstanding example being that of the Higgs boson besides superconductors, ferro-magnets and liquid crystals. The fact that such a phenomenon may also occur in the classicalregime is tantalizing at least from the theoretical point of view if nothing else. The intro-duction of the associated concept of time crystals by Shapere and Wilczek is not withoutcontroversy especially regarding their experimental realization. While the examples consid-ered by them as also by L. Zhao et al are drawn from classical mechanics and field theoryour motivation in this note is to extend this notion to nonlinear ordinary differential equa-tions. We have shown that nonlinear ODE of the Li´enard type it is easy to recast them intothe Lagrange/Hamilton formalism and the basic results of Shapere-Wilczek are apparentlyapplicable to such a differential system. Acknowledgement
We are extremely grateful to Professor Liu Zhao for his his interest and valuable suggestions.We would also like to thank Ankan Pandey for the diagrams. Finally, we would like tothank the anonymous referee for carefully reading our manuscript and for giving constructivecomments.
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