(1+1) Newton-Hooke Group for the Simple and Damped Harmonic Oscillator
aa r X i v : . [ m a t h - ph ] M a y (1 + 1) Newton–Hooke Group for the Simpleand Damped Harmonic Oscillator
Przemys law Brzykcy Institute of Physics, Lodz University of Technology,W´olcza´nska 219, 90-924 L´od´z, Poland.
July 24, 2018
Abstract
It is demonstrated that, in the framework of the orbit method, a simple anddamped harmonic oscillators are indistinguishable at the level of an abstract Liealgebra. This opens a possibility for treating the dissipative systems within the orbitmethod. In depth analysis of the coadjoint orbits of the (1+1) dimensional Newton-Hooke group are presented. Further, it is argued that the physical interpretation iscarried by a specific realisation of the Lie algebra of smooth functions on a phasespace rather than by an abstract Lie algebra.
Sidney Coleman famously said “The career of a young theoretical physicist consists oftreating the harmonic oscillator in ever-increasing levels of abstraction”. The accuracyof this dictum is striking when one considers the abundance of scientific papers devotedto this subject across many branches of physics. It has long been known that in theframework of the orbit method [1–3] the oscillator is described by the Newton–Hooke(
N H ) group. The
N H type groups first appeared in the classification of the possiblekinematical groups [4]. The thorough study of (3 + 1) dimensional
N H group was pre-sented in [5]. The orbit method was employed in [6] to study the centrally extended
N H group in (2 + 1) dimensions. The coadjoint orbits and the irreducible represen-tations were calculated therein. Besides the orbit method a planar system with exoticNewton–Hooke symmetry was constructed by the technique of nonlinear realisation [7],the analysis therein included the chiral decomposition. The idea of chiral decompositionwas later applied to the non-commutative Landau problem [8, 9] and to the rotation-less E-mail address: [email protected] H symmetry of the 3D anisotropic oscillator [10]. Some work on the anisotropic (2+1)dimensional N H group was presented in [11]. The extended conformal
N H type sym-metries were also studied in connection to the Pais–Uhlenbeck oscillator [12–14]. Thisshort overview is far from being complete but it shows, how telling a study of this simplesystem can be. It is worth to mention that orbit method was also successfully used toanalyse systems with Galilei and Poincar´e type symmetries both in the free case and withexternal electromagnetic fields [15–25].In this paper an accessible yet illuminating example of harmonic oscillator is examinedin the framework of the orbit method [1–3]. In the case of simple harmonic oscillator theLie algebra of (1 + 1)
N H group is derived from the standard Hamiltonian descriptionby a technique encouraged by [21]. Detailed analysis of the coadjoint action provides afull understanding of the physical interpretation. Clearly, analysis becomes more involvedfor the dissipative systems. However, a proper canonical transformation may allow for asignificant simplification. For example, the damped harmonic oscillator can be describedby the same Lie algebra as the undamped case. This simplification comes at a priceof using rather elaborate coordinates. Consequently, at the level of the Lie algebra thedamped harmonic oscillator is indistinguishable from the undamped one. This exampleillustrates a possible way of treating the dissipative systems within the framework of theorbit method. Apparently an abstract Lie algebra does not carry the physical interpreta-tion of the system. The question arrises how to use the orbit method so that the physicalinterpretation is not lost. The current paper is devoted to just this investigation.This paper is structured as follows. In Section 2, starting with the Hamiltonian of theharmonic oscillator, the Lie algebra of the (1 + 1) dimensional Newton–Hooke group isderived to set the scene for the further analysis. Section 3 provides the coadjoint actionof the group under investigation. Also, the symplectic structure on the coadjoint orbit of(1 + 1) Newton–Hooke group are given. The in depth analysis of the coadjoint orbits ispresented in Section 4. Section 5 is devoted to the damped harmonic oscillator and showsthat it may be described by the same abstract Lie algebra as the undamped case. Thepaper closes with conclusions in Section 6 where also some outlooks are provided.
To focus the attention take the Hamiltonian of the simple harmonic oscillator h ( p, x ) = p m + mω x , (2.1)where p is the kinematic momentum and x is the displacement of the oscillator. Exploitingthe canonical Poisson bracket { F ( p, x ) , G ( p, x ) } = ∂F∂x ∂G∂p − ∂F∂p ∂G∂x (2.2)one arrives at the well known equations of motion˙ p = − mω x, ˙ x = pm (2.3)2hich when put together read ¨ x = − ω x . In order to use the orbit method to describe thesimple harmonic oscillator an appropriate Lie algebra is needed. This algebra should besuch that the equations of motion on its coadjoint orbits are equivalent to (2.3). Hereinsuch a Lie algebra is constructed starting with the algebra of smooth functions on thephase space equipped with the Poisson bracket (2.2).The method of constructing such a Lie algebra is based on Poisson’s theorem statingthat the Poisson bracket of two quantities that are constants of motion is also a constantof motion. The Hamiltonian (2.1) i.e. the total energy of the system is the only integralof motion. This system also admits constants of motion e.g f ( p, x, t ) = t − ω arctan ωkp which at p = 0 has to be understood in the sense of the limit. It is mentioned here forthe sake of completeness, however will not be utilised in the present paper because thereis no need to consider the time dependent generators. Therefore, h should be included inthe set of generators. Inasmuch as some coordinates are needed, one just checks whether p and x could do the job. To this end calculate the Poisson bracket (2.2) for all the pairsselected from the set { h, p, x } and find that the non-vanishing brackets are { h, p } = mω x, { h, x } = − m p, { x, p } = 1 . (2.4)Quick conclusion is that, in order to have a closed algebra, the { h, p, x } ought to beaugmented by a constant function equal to 1. Even more elegantly one may replace x with k = mx and use a constant function equal m . In which case the Hamiltonian (2.1)becomes h ( p, k ) = p m + ω k m , (2.5)and the Poisson bracket (2.2), by the chain rule, reads now { F ( p, k ) , G ( p, k ) } = m (cid:18) ∂F∂k ∂G∂p − ∂F∂p ∂G∂k (cid:19) . (2.6)The non-vanishing Poisson brackets are { h, p } = ω k, { h, k } = − p, { k, p } = m. (2.7)Therefore the functions J = m, J = h ( p, k ) , J = p, J = k span the Poisson algebraunder the Poisson bracket (2.6). Note that J is a central generator. What was describedabove is known as the Lie algebra of the (1 + 1) dimensional Newton-Hooke group. Atthe abstract level it is a four dimensional Lie algebra spanned by J = M, J = H, J = P, J = K characterised by the following nonzero structure constants c = ω , c = − c = −
1, in the above numbering of the basis, which will be kept throughout this paper.3
Coadjoint action and dynamics
The matrices of the adjoint action m adJ i corresponding to the generators J , . . . , J aregiven by ( m adJ i ) jk = c ikj where c ikj are the structure constants. Explicitlym AdJ = −
10 0 ω , m AdJ = −
10 0 0 00 0 0 00 − ω , m AdJ = and m AdJ is the zero matrix since m is a central generator. The generic element g of ourgroup can be written as g = ( η, b, a, v ) = e ηM e bH e aP e vK ∈ G . The coordinates for thedual to our Lie algebra g ∗ are realised by m, h, p, k . The matrix of the coadjoint action ofan element g ∈ G is then given by M coAdg = e − v m AdK e − a m AdP e − τ m AdH e − η m AdM which explicitlyreads M coAdg = v + a ω − v cos bω − aω sin bω a cos bω − vω sin bω − v cos bω sin bωω aω − ω sin bω cos bω . (3.1)An element of g ∗ is represented as a row vector ξ = [ m, h, p, k ]. Then the coadjoint actionof g ∈ G is calculated by matrix multiplication of ξ by g ∗ on the right, which yields thefollowing explicit form of the coadjoint action m ′ = m,h ′ = h + mv + ma ω − vp + aω k,p ′ = ( p − mv ) cos bω − ω ( ma + k ) sin bω,k ′ = ( ma + k ) cos bω + p − mvω sin bω. (3.2)More detailed analysis of the action (3.2) will be presented in the following section. Thenext step is to calculate the invariants of the coadjoint action i.e. smooth functions C on g ∗ , such that ∀ g ∈ G ∀ ξ ∈ g ∗ C (coAd g ( ξ )) = C ( ξ ) . They are solutions to the following set ofdifferential equations [26–30] ω k − p − ω k − m p m ∂C∂m∂C∂h∂C∂p∂C∂k = . (3.3)There are two solutions to (3.3), namely C = m, C = k − mhω + p ω . (3.4)The first one is a trivial consequence of m being a central generator. Consider a map C : g ∗ → R ,ξ ( C ( ξ ) , C ( ξ )) . (3.5)4t each point ξ ′ = coAd g ( ξ ), g ∈ G of the orbit through ξ the value of (3.5) is constant.Moreover, mapping (3.5) is of a constant and maximal rank, therefore the preimage ofa point is a submanifold in g ∗ . Each of its compact components is precisely a coadjointorbit through ξ . In the present case, the orbit, denoted O C ,C , admits a single globalparametrisation ϕ : ( p, k ) (cid:18) m = C , h = ˜ h = p m + ω k m − ω C m , p, k (cid:19) (3.6)so, in principle, it might be covered by a single map e.g. ϕ − . Note that, in the thisexample, the hamiltonian ˜ h ( p, k ) = p m + ω k m − ω C m (3.7)which was derived from the invariants of the coadjoint action is, up to an additive constant,equivalent to the initial Hamiltonian (2.1). For the sake of completeness note that, theJacobian of the map ( m, h, p, k ) ( C , C , p, k ) is − mω so there is a singularity for m = 0.This case shall not be discussed in depth for it is of no physical interest. It suffices tosay that, for the fixed m = 0, there is one invariant of the coadjoint action C = k + p ω , and since h is unrestricted, the orbits resemble the flatten cylinders. In what follows, m = 0 shall be assumed. The Poisson tensor Λ on the orbit O C ,C , written in the chart( O C ,C , ϕ − ) reads Λ ij = (cid:20) − mm (cid:21) , (3.8)which is equivalent to (2.6). Since m = 0 on the orbit O C ,C the Poisson structureis non-degenerate and one quickly finds, by the techniques presented in [31], that thecorresponding symplectic two-form is ω = − m d p ∧ d k. (3.9)Therefore, employing the Hamiltonian (3.7), one finds the equations of motion to be˙ p = − ω k, ˙ k = p (3.10)which, when combined with k = mx , are equivalent to (2.3). Further insight into the structure of the coadjoint orbit can be gained by examining, oneby one, the coadjoint actions of the group elements that correspond to the generators.Consider a test solution of (3.10) for example p = − mωA sin ( ωt ) , k = mA cos ( ωt ) thatis, at t = 0 the displacement is maximal and momentum is zero. The energy is constantand at any time is given by (3.7) ( E = ˜ h ( p, k )), furthermore m = C is also fixed. Thetrajectory of the system, as time flies, is then given by ξ ( t ) = [ m, E = ˜ h ( p, k ) , p = − mωA sin ωt, k = mA cos ωt ] . (4.1)5he trajectory (4.1) lies on the coadjoint orbit characterised by C = m and a fixed C .Coadjoint action of a group element g = exp ( τ H ) generated by H on (4.1) is: m ′ = m , E ′ = E and p ′ = − mωA sin( ω ( t + τ )) ,k ′ = mA cos ( ω ( t + τ ))that is to say m and E remain constant and p , k follow the elliptic trajectory. Clearly, H generates temporal shifts. Moreover as the system evolves, it stays on the same orbit.Next, let us consider a group element g = exp ( lP ) generated by P . Its coadjointaction on (4.1) is: m ′ = m , p ′ = p and E ′ = E + 12 mω x ′ + mx ′ ω A cos ωt,k ′ = k + ml. The displacement ( x = km ) is increased by l and energy is changed exactly in such a way,that the system stays on the same orbit which means that P generates spatial shifts ofthe initial conditions. p k h k p A BCDA BCD
Figure 1: Ilustration of the coadjoint action of the elements of the group
N H (1 + 1)corresponding to the generators. The time evolution brings the system along the ellipticaltrajectory e.g. from point A to C or B to D . Starting from the point A the system canbe moved to point B by the spatial shift generated by P . Performing a boost generatedby K brings the system from the point C to D .6inally, a group element g = exp ( uK )generated by K acts on (4.1) as: m ′ = m , k ′ = kE ′ = E + 12 mu + muωA sin ωt,p ′ = p − mu i.e. m and k are constant, momentum p is decreased by mu and energy is adjusted so thatthe system remains on the orbit. Clearly, K generates the momentum shifts. For the sakeof completeness it is worth mentioning that the action of group elements generated by M is identity. The examples of above-described actions are presented in Figure 1 which also,by a small leap of imagination, allows us to visualise the coadjoint orbit. A slightly more complex system which can be investigated by similar techniques is adamped harmonic oscillator. Take the following time dependent Hamiltonian h ( P , q, t ) = P m e − γt + 12 mω e γt x (5.1)where 2 γ = βm with β being the friction coefficient and ω is the undamped frequencyof the oscillator. Note that P = m ˙ xe γt i.e. the canonical momentum does not coincidewith the kinetic momentum p = m ˙ x . The Hamiltonian (5.1) with the canonical Poissonbracket yields the following equations of motion˙ x = P m e − γt , ˙ P = − mω e γt x (5.2)or equivalently ¨ x + 2 γ ˙ x + ω x = 0 . It is an easy exercise to check that the procedure thatwas carried out in Section 2 for the undamped oscillator fails in the present case. Indeed,introducing the new coordinate k = mx one finds that the hamiltonian (5.1) becomes h ( P , k, t ) = P m e − γt + ω k m e γt (5.3)and, by the chain rule, new the Poisson bracket is just (2.6). Then, one quickly finds that { h, P} = ω e γt k, { h, k } = − e − γt P , { k, P} = m (5.4)which fails to constitute a Lie algebra because there is an undesired time dependency ofthe structure constants. One way to deal with this problem is to use a generating functionmethod to bring the Hamiltonian (5.1) to a more convenient form (see e.g. [32]). Considerthe following generating function of the second kind F ( k, P, t ) = e γt kP − mγe γt x . (5.5)The transformation rules for the coordinates are P = ∂F ∂x = e γt P − mγe γt x, Q = ∂F ∂P = e γt x (5.6)7urthermore, the old h and the new H Hamiltonians obey H − h = ∂F ∂t = γe γt xP − mγ e γt x = γQP − mγ Q . (5.7)The relations between old ( P , x ) and new ( P, Q ) coordinates can be written as (cid:20) P x (cid:21) = (cid:20) e γt − mγe γt e − γt (cid:21) (cid:20) PQ (cid:21) , (cid:20) PQ (cid:21) = (cid:20) e − γt mγe γt e γt (cid:21) (cid:20) P x (cid:21) . (5.8)By the chain rule ∂∂ P = ∂P∂ P ∂∂P + ∂Q∂ P ∂∂Q = e − γt ∂∂P and ∂∂x = ∂P∂x ∂∂P + ∂Q∂x ∂∂Q = mγe γt ∂∂P + e γt ∂∂Q so one quickly finds that the Poisson bracket (2.2) becomes { F ( P, Q ) , G ( P, Q ) } = ∂F∂Q ∂G∂P − ∂F∂P ∂G∂Q (5.9)i.e. the transformation (5.7) is canonical but, since H = h it is not a symmetry. Finally,the transformed Hamiltonian takes the following form H ( P, Q ) = P m + 12 m ( ω − γ ) Q (5.10)which, functionally is just (2.1) with ω = ω − γ . Therefore, in the new coordinates P and Q the procedure of constructing the Lie algebra as in Section 2 can be carriedout. The resulting algebra is exactly (1+1) Newton-Hooke algebra as it was for theundamped oscillator therefore, at the level of the abstract Lie algebra the two systemsare indistinguishable. The difference lies in the realisation of the generators as smoothfunctions of the phase space coordinates. It is important to stress that the Hamiltonianderived from the invariants of the coadjoint action would be functionally equivalent (up toan additive constant) to (5.10) not to the initial Hamiltonian (5.1) as the interpretationof the coordinates has changed. It was shown that, in the framework of the orbit method, a simple and damped harmonicoscillators can be described by the same abstract Lie algebra. The simple, yet strikingexample presented here shows that, when the dynamics on the coadjoint orbits are con-sidered, a simple knowledge of an abstract Lie algebra does not suffice to provide thephysical interpretation. What describes the system is rather a specific realisation of theLie algebra in terms of the smooth functions on the classical phase space.Particularly, the result presented in the current paper stresses the importance of keep-ing track of the physical interpretation when constructing the dynamics on the coadjointorbits which might be crucial when the deformation quantisation on the coadjoint orbitis considered [33–39]. One way to achieve that is to derive a relevant Lie algebra startingfrom the Hamiltonian formulation as was done in the current paper for the harmonicoscillator or in [21] for extended Galilei group also known in the literature as the Galilei-Maxwell group [24]. It is the intention of the author to follow with the application of thecurrent results in case of the Poincar´e-Maxwell group soon.8 eferences [1] A. A Kirillov,
Elements of the Theory of Representations , Springer-Verlag, BerlinHeidelberg 1976.[2] A. A. Kirillov,
Bull. Amer. Math. Soc. (N.S. ) , 433 (1999).[3] A. A. Kirillov, Lectures on the Orbit Method , American Mathematical Soc. 2004.[4] H. Bacry and J. M. L´evy-Leblond,
J. Math. Phys. , 1605 (1968).[5] J. R. Derome and J. G. Dubois, Il Nuovo Cim. B , 351 (1972).[6] O. Arratia, M. A. Mart´ın, and M. A. del Olmo, Int. J. Theor. Phys. , 2035 (2011).[7] P. D. Alvarez, J. Gomis, K. Kamimura, and M. S. Plyushchay, Ann. Phys. , 1556(2007).[8] P. D. Alvarez, J. Gomis, K. Kamimura and M. S. Plyushchay,
Phys. Lett. B ,906 (2008).[9] P. M. Zhang and P. A. Horv´athy,
Ann. Phys. , 1730 (2012).[10] P. M. Zhang, P. A. Horv´athy, K. Andrzejewski, J. Gonera and P. Kosi´nski,
Ann.Phys. , 335 (2013).[11] L. Todjihounde, A. Ngendakumana and J. Nzotungicimpaye, arXiv:math-ph/1102.0718.[12] A. Galajinsky and I. Masterov,
Phys. Lett. B , 190 (2013).[13] K. Andrzejewski, A. Galajinsky, J. Gonera and I. Masterov,
Nucl. Phys. B , 150(2014).[14] K. Andrzejewski,
Phys. Lett. B , 405 (2014).[15] J. Lukierski, P. C Stichel and W. J. Zakrzewski,
Ann. Phys. (N. Y.) , 224 (1997).[16] C. Duval and P. A. Horv´athy,
Phys. Lett. B , 284 (2000).[17] R. Jackiw and V. P. Nair,
Phys. Lett. B , 237 (2000).[18] C. Duval and P. A. Horv´athy,
J. Phys. A , 10097 (2001).[19] P. A. Horv´athy, Ann. Phys , 128 (2002).[20] P. A. Horv´athy and M. S. Plyushchay,
J. High Energy Phys. , 033 (2002).[21] P. A. Horv´athy, L. Martina and P. C. Stichel,
Phys. Lett. B , 87 (2005).[22] S. Ghosh,
Phys. Lett. B , 350 (2006).923] M. A. del Olmo and M. S. Plyushchay,
Ann. Phys. (N. Y.) , 2830 (2006).[24] M. A. del Olmo, J. Negro and J. Tosiek,
J. Math. Phys. , 033508 (2006).[25] P. A. Horv´athy, L. Martina and P. C. Stichel, arXiv:hep-th/1002.4772.[26] E. G Beltrametti and A, Blasi, Phys. Lett. , 62 (1966).[27] L. Abellanas and L. Martinez Alonso, J. Math. Phys. (N.Y.) , 1580 (1975).[28] J. Patera, R. T. Sharp, P. Winternitz and H. Zassenhaus, J. Math. Phys. (N.Y.) ,986 (1976)[29] V. Boyko, J. Patera and R. Popovych, J. Phys. A , 5749 (2006).[30] L. ˇSnobl and P. Winternitz, Classification and Identification of Lie Algebras , CRMMonograph Series. American Mathematical Society, (2014)[31] J. F. Cari˜nena, J. A. Gonz´alez, M. A. del Olmo and M. Santander,
Fortschr. Phys. , 681 (1990).[32] W. Greiner, Classical Mechanics: systems of particles and Hamiltonian dynamics,
Springer, Berlin Heidelberg 2009.[33] M. Gadella, M. A. Mart´ın, L. M. Nieto and M. A. del Olmo,
J. Math. Phys. , ,1182 (1991).[34] A. Ballesteros, M. Gadella and M. A. del Olmo, J. Math. Phys. , 3370 (1992).[35] O. Arratia, M. A. Martin and M. A. del Olmo, arXiv:quant-ph/9611055.[36] M. A Mart´ın and M. A del Olmo, J. Phys. A , 689 (1996).[37] M. A. Lled´o, Int. J. Mod. Phys. B , 2397 (2000).[38] R. Fioresi and M. A. Lled´o, Pacific J.Math. , 411 (2001).[39] G. Dito and F. J. Turrubiates,
Physics Letters A ,352