Hamilton-Jacobi approach for linearly acceleration-dependent Lagrangians
aa r X i v : . [ h e p - t h ] J a n Hamilton-Jacobi approach for linearly acceleration-dependent Lagrangians
Alejandro Aguilar-Salas ∗ Facultad de Matem´aticas, Universidad Veracruzana,Cto. Gonz´alo Aguirre Beltr´an s/n, Xalapa, Veracruz 91000, M´exico
Efra´ın Rojas † Facultad de F´ısica, Universidad Veracruzana, Cto. Gonz´alo Aguirre Beltr´an s/n, Xalapa, Veracruz 91000, M´exico
We develop a constructive procedure for arriving at the Hamilton-Jacobi framework for the so-called affine in acceleration theories by analysing the canonical constraint structure. We find twoscenarios in dependence of the order of the emerging equations of motion. By properly defininggeneralized brackets, the non-involutive constraints that originally arose, in both scenarios, may beremoved so that the resulting involutive Hamiltonian constraints ensure integrability of the theoriesand, at the same time, lead to the right dynamics in the reduced phase space. In particular, when wehave second-order in derivatives equations of motion we are able to detect the gauge invariant sectorof the theory by using a suitable approach based on the projection of the Hamiltonians onto thetangential and normal directions of the congruence of curves in the configuration space. Regardingthis, we also explore the generators of canonical and gauge transformations of these theories. Further,we briefly outline how to determine the Hamilton principal function S for some particular setups.We apply our findings to some representative theories: a Chern-Simons-like theory in (2 + 1)-dim,an harmonic oscillator in 2 D and, the geodetic brane cosmology emerging in the context of extradimensions. PACS numbers: 02.30.Jr, 04.20.Fy, 11.10.Ef
I. INTRODUCTION
There is a broad class of relevant Lagrangian func-tions that are not built from the standard prescription L = T − V . Closely related to this set are the second-order derivative theories that in many cases are consid-ered extensions, or corrections, to the usual physical the-ories, possessing an intricate gauge freedom which is re-sponsible for not allowing the highest derivatives to besolved algebraically in the equations of motion. In thissense, particular attention has been focused on the so-called affine in acceleration theories [1–3], mainly moti-vated by model building associated with some relativisticacceleration phenomena including general relativity, elec-tromagnetism, or modified gravity theories. These arecharacterized by a linear dependence on the accelerationsof the systems and are necessarily singular giving rise ingeneral to both gauge ambiguities and gauge symmetries.In fact, these theories have been widely studied by meansof the Lagrangian and Hamiltonian formalism. However,such analyses can be supplemented and, in many casesimproved, by another scheme defined for singular systemsas it is the case of the Hamilton-Jacobi formalism.In this paper we construct in a consistent man-ner a Hamilton-Jacobi (HJ) framework for linearlyacceleration-dependent Lagrangians. In providing thisscheme we also furnish with a great alternative to ana-lyze the phase space constraint structure of this type oftheories. The HJ framework we are interested in, intro- ∗ [email protected] † [email protected] duced by Carath´eodory for regular systems [4] and fur-ther strengthened from the physical point of view for sin-gular systems by G¨uler and many authors [5–11], is esen-tially based on the variational principle named equivalentLagrangians method and the first-order partial differen-tial equations theory. This scheme is obtained directlyfrom the Lagrangian formalism without going throughthe usual approximation of the canonical transformationsof the Hamiltonian formalism, even in the case of singulartheories, so it becomes an adequate shorcut for arrivingat the HJ framework. For regular systems we can arrivestraightforwardly at the commonly known HJ partial dif-ferential equation. On the contrary, for singular systemsconstraints arise as necessary conditions to ensure the ex-istence of extremes for a given action but, without ceasingto be partial differential equations from which the charac-teristic system of equations turn out to depend on severalindependent variables, named parameters, which may berelated with the gauge information of the systems [12].The complete set of HJ partial differential equations mustobey geometric conditions in order to guarantee their in-tegrability. These conditions are equivalent to the con-sistency conditions developed in the Dirac-Bergmann ap-proach for constrained systems [13–15]. In consequence,the HJ integrability analysis separates the constraints ininvolutive and non-involutive under an extended Poissonbracket [12, 16]. The existence of non-involutive con-straints signals a dependence between the parameters ofthe theory which leads to a redefinition of the symplecticstructure of the phase space in order to have control ofthe right dynamics of the theory.In contrast to the Dirac-Bergmann approach, this HJframework possesses robust geometric foundations anddoes not need support from the split of first- and second-class constraints to build the right evolution of physicalconstrained systems as well as to obtain the gauge sym-metry information. For theories being linear on the ac-celerations non-involutive constraints are always present.To unravel this situation within this geometric frame-work it is mandatory to introduce a generalized bracket(GB) structure that leaves the complete set of constraintsin involution since the so-called Frobenius integrabilityconditions are fulfilled [12], thus solving the problem ofintegrability. This framework also links the complete setof involutive HJ equations with the canonical and gaugesymmetries. Regarding this, we find that theories thatonly have non-involutive constraints are characterized byhaving matrices N µν and M AB being non-singular, (4)and (53), respectively, so that the evolution in phasespace is only generated by the canonical Hamiltonian H under a properly defined GB. On the other hand, whenmatrix M AB turns out to be singular, the evolution inphase space of any observable involves several parame-ters, treated on an equal footing as the time parameterwhere some of them are related to the gauge invarianceproperties of the Lagrangian function behind that the-ory. At this particular point we highlight the uselfulnessin taking advantage of the use of the zero-modes of thematrix M AB in order to obtain the Hamiltonians thathelp to identify the reduced physical phase space. Ourfindings shed light on previously overlooked geometricalaspects of HJ approaches for affine in accelerations orvelocities theories and can serve as the basis for quan-tization schemes by using a WKB approximation. Wewant to emphasize here that, despite of being warnedthat every higher-order theory can be written down as afirst-order theory by appropriately enlarging the config-uration space by introducing auxiliary fields, [1, 10, 11],the point of view we wish to implement here is strictlythat emerging from the higher-order theory. Indeed, inmany cases it is more appropriate to analyze the originalinformation of the system while maintaining all the orig-inal symmetries thus avoiding the extension of both thenumber of variables and constraints in the new configu-ration which, due to the cumbersome notation involvedmay hide the geometrical structure of a model.The paper is organized as follows. In Sect. II we glancean overview of the kinematical description of the affine inacceleration theories and introduce some important geo-metrical structures. After established the notation andconventions, in Sect. III we construct the HJ frameworkfor the affine in acceleration theories that yield third-order equations of motion. The HJ scheme when second-order equations are present is developed in Sec. IV wherea couple of possible scenarios is discussed. In Sec. V ourdevelopment is illustrated with some examples: a Chern-Simons-like theory in (2 + 1)-dimensions, an harmonicoscillator in 2 D and, the RT cosmology in extra dimen-sions. We conclude in Sec. VI with some remarks on ourdevelopment. Further, here we outline the obtaining ofthe Hamilton principal function S and comment on pos- sible extensions of the work. In an Appendix we showthe geometric role played by the matrix M µν , (54), inour development. II. HAMILTON-JACOBI FORMALISM FORAFFINE IN ACCELERATION THEORIES
We are interested in physical systems with a finitenumber of degrees of freedom described by Lagrangianfunctions, including only terms linear in accelerations,and described by the action S [ q µ ] = Z dt L ( q µ , ˙ q µ , ¨ q µ , t ) µ = 1 , , . . . , N ; (1)where L ( q µ , ˙ q µ , ¨ q µ , t ) = K µ ( q ν , ˙ q ν , t ) ¨ q µ − V ( q ν , ˙ q ν , t ) . (2)Summation over repeated indices is henceforth under-stood. According to the higher-order derivative theoriesviewpoint the configuration space C N is spanned by the N coordinates q µ and their N velocities ˙ q µ . An overdotstands for the derivative with respect to the time param-eter t so that ˙ q µ = dq µ /dt and so on. Here, K µ and V are assumed to be smooth functions defined on C N .Guided by the Hamilton’s principle adapted to ac-tions of the form (1), the optimal trajectory q µ = q µ ( t )parametrized by t , is obtained by solving the Euler-Lagrange (EL) equations of motion (eom) ∂L∂q µ − ddt (cid:20) ∂L∂ ˙ q µ − ddt (cid:18) ∂L∂ ¨ q µ (cid:19)(cid:21) = 0 . Explicitly, these are given by N µν ... q ν − M µν ¨ q ν + F µ = 0 , (3)where N µν := ∂ K ν ∂ ˙ q µ − ∂ K µ ∂ ˙ q ν = − N νµ , (4) M µν := ∂ K ν ∂q µ − ∂∂ ˙ q ν (cid:18) − ∂ V ∂ ˙ q µ − ∂ K µ ∂q ρ ˙ q ρ + N µρ ¨ q ρ (cid:19) , (5) F µ := ∂∂q ν (cid:18) − ∂ V ∂ ˙ q µ − ∂ K µ ∂q ρ ˙ q ρ + N µρ ¨ q ρ (cid:19) ˙ q ν + ∂ V ∂q µ . (6)It is worth noting that N µν = N µν ( q ρ , ˙ q ρ , t ), M µν = M µν ( q ρ , ˙ q ρ , ¨ q ρ , t ) and F µ = F µ ( q ρ , ˙ q ρ , ¨ q ρ , t ). The form (3)for the eom proves to be fairly useful and allows the the-ory to be better understood in the HJ scheme to be de-veloped.Following the original Carath´eodory’s equivalent La-grangians approach [4–6], later extended to second-orderin derivatives theories [7, 8], in order to have an extremeconfiguration of the action (1) the necessary conditionsare associated to the existence of a family of surfaces de-fined by a generating function , S ( q µ , ˙ q µ , t ), such that itsatisfies ∂S∂ ˙ q µ = ∂L∂ ¨ q µ =: P µ , (7) ∂S∂q µ = ∂L∂ ˙ q µ − ddt (cid:18) ∂L∂ ¨ q µ (cid:19) =: p µ , (8) ∂S∂t + ∂S∂q µ ˙ q µ + ∂S∂ ˙ q µ ¨ q µ − L = 0 , (9)where, on physical grounds, P µ denotes the conjugatemomenta to the velocities ˙ q µ while p µ are the conjugatemomenta to the coordinates q µ .The HJ framework in which we are interested in,emerges from (7), (8) and (9) considered as partial dif-ferential equations (PDE) for S . Indeed, for non-singularsystems it is straightforward to convert (9) into a PDEfor S by solving for ¨ q µ in terms of q µ , ˙ q µ and partialderivatives of S what is obtained by appropriately in-verting (7). However, for singular physical systems thisis not possible as the Hessian matrix with elements H µν associated to (2) vanishes identically, H µν := ∂ L∂ ¨ q µ ∂ ¨ q ν = 0 . (10)This fact defines what is known as an affine in accelera-tion theory [1–3]. The rank of the Hessian matrix is zerowhich causes that the manifold C N is fully spanned by R = N − N variables, ˙ q µ , all of them related to thekernel of H µν . In this sense, within the HJ scheme weare going to discuss below we have to treat all the gener-alised velocities as free parameters [5–7]. Clearly, we cannot invert any of the accelerations ¨ q µ in favour of the co-ordinates, the velocities ˙ q µ and the momenta P µ throughthe partial derivative of the function S , fact that entailsthe presence of N constraints given by the definition of P µ itself. Indeed, as one can infer from (7-9), the con-straints form a set of PDE of first-order for S which, inorder to be integrable, must obey the so-called Frobeniusintegrability conditions , [4, 12], (see (29) below). On theother hand, relying on the structure of the momenta p µ ,(see (12) below) we could or not solve for the accelera-tions ¨ q µ in favour of the remaining variables of the phasespace which could or not give rise to more contraints.Indeed, such a dependence rests heavily on the natureof the quantities K µ which in turn promote two possiblescenarios characterized by the order of the eom. As amatter of fact, the momenta P µ and p µ written out infull are P µ = K µ ( t, q ν , ˙ q ν ) , (11) p µ = − ∂ V ∂ ˙ q µ − ∂ K µ ∂q ν ˙ q ν + N µν ¨ q ν . (12)Note the linear dependence on the accelerations in (12).The eom in this framework are written as total differen-tial equations known as characteristic equations , [5, 6],whose solutions are trajectories in dependence of inde-pendent variables, t α , in a reduced phase space, as we will see below. For the further analysis, due to the ex-istence of several parameters in the development, in duecourse it will necessary to relabel the indexes associatedwith the main geometric quantities. For that reason, itis mandatory to discuss sistematically the two feasiblescenarios. III. THEORIES WITH THIRD-ORDEREQUATIONS OF MOTION
According to the Jacobi’s theorem in matrix algebra,the determinant of any antisymmetric matrix of odd or-der has determinant equal to zero. Certainly, by observ-ing (3), not necessarily all coordinates q µ will obey third-order eom in some sector of the configuration space. Inorder to ensure the existence of independent third-ordereom we confine ourselves to consider an even number ofvariables q µ . Hence, we set N = 2 n implying thus that N µν is non-vanishing and that det( N µν ) = 0 in this sce-nario. Under these conditions, to determine the timeevolution it is needed to consider 6 n initial conditions forthe quantities q µ , ˙ q µ , ¨ q µ with µ = 1 , , . . . , n at initialtime t = t . Additionally, from (12) it should be notedthat accelerations can be solved in favour of the rest ofthe phase space variables.From the result (10), all the generalized velocities havethe status of parameters [10] so, it is reasonable to intro-duce the notation: t := tt µ := ˙ q µ and H Pµ := − ∂L∂ ¨ q µ = −K µ ( t, q ν , t ν ) , (13)as well as the set of coordinates t I in the following order t I := q I := ( t , t µ ) , I = 0 , , , . . . , n. (14)As a result, relationship (7) reads ∂S∂t µ + H Pµ (cid:18) t, t ν , q ν , ∂S∂t ν (cid:19) = 0 . (15)In the same spirit, by introducing the Hamilton function H := ∂S∂q µ t µ + ∂S∂t µ ˙ t µ − L ( t , q µ , t µ , ˙ t µ ) , (16)which does not depend explicitly on ˙ t µ , as it may bechecked straightforwardly, one finds that the expres-sion (9) becomes ∂S∂t + H (cid:18) t , t µ , q µ , ∂S∂t µ , ∂S∂q µ (cid:19) = 0 , (17)that is the common expression of the Hamilton-Jacobiequation. In terms of the original notation, the Hamiltonfunction reads H = p µ ˙ q µ + V ( q ν , ˙ q ν , t ) . (18)We can express (15) and (17) as a unified set of PDEfor the generating function S . To do this, it is useful toassume that P := ∂S∂t is conjugate canonical momentumto t . We then find ∂S∂t I + H I (cid:18) t J , q µ , ∂S∂q µ , ∂S∂t J (cid:19) = 0 , I,J =0 , , ,..., n., (19)where H I := ( H , H Pµ ). In the following, the 2 n + 1 rela-tions (19) will be referred to as the Hamilton-Jacobi par-tial differential equations (HJPDE). Bearing in mind (7),we can also write (15) and (17) in the form H := P + H ( t , t µ , q µ , P µ , p µ ) = 0 , (20) H Pµ := P µ + H Pµ ( t , t ν , q ν ) = 0 , (21)which acquire the compact constrained Hamiltonian fash-ion H I ( t J , q ν , P J , p ν ) := P I + H I ( t J , q ν , P J , p ν ) = 0 , (22)where H I := ( H , H Pµ ) and P I := ( P , P µ ). These expres-sions have thus acquired the well-known form of canon-ical Dirac constraints. The constraints written in theform (22) are also referred to as Hamiltonians in this HJscheme. In a sense, this HJ approach replaces the anal-ysis of the 2 n canonical constraints, H Pµ = 0, with theanalysis of the (2 n + 1) HJPDE given by relations (19).Some remarks are in order. We do not have a furtherHJPDE relating the momenta p µ because from (12) weobserve the linear dependence on ¨ q µ which allows us towrite the accelerations in favour of the momenta thatcan be inserted in the Legendre transformation defin-ing the canonical Hamiltonian. We shall prove this inshort; in fact, within the Dirac-Bergmann approach forconstrained systems, this feature signals the presence ofsecond-class constraints. On the other hand, the Hamil-tonian H , (20), is said to be associated with the timeparameter t while the Hamiltonians H Pµ , (21), are asso-ciated with the remaining parameters t µ related to thevelocities of the system.The equations of motion, known as characteristic equa-tions (CE), associated to the Hamiltonian set (22), aregiven as total differential equations [5, 6]. At this initialstage these are given by dq µ = ∂ H I ∂p µ dt I dq I = ∂ H J ∂P I dt J , (23) dp µ = − ∂ H I ∂q µ dt I dP I = − ∂ H J ∂q I dt J . (24)We would like to emphasize that t µ = ˙ q µ have a statusof independent evolution parameters, on an equal foot-ing to t . To prove this it is enough to evaluate (23) for˙ q µ ; indeed, d ˙ q µ = ( ∂ H /∂P µ ) dt +( ∂ H Pν /∂P µ ) dt ν = dt µ .On mathematical grounds, within this HJ formalism it issaid that t I are the independent variables or parameters of the theory. In fact, the number of parameters is de-termined not only by the rank of the Hessian matrix butalso by the integrability conditions. On physical grounds, the parameters encode the local symmetries and gaugetransformations (see below for details). The solution ofthe first equations in (23) and (24) leads to a congruenceof parametrized curves in the configuration space C N +1 ,given by q µ = q µ ( t I ). In a like manner, the generatingfunction S ( t, q µ , ˙ q µ ) is satisfying dS = ∂S∂t I dt I + ∂S∂q µ dq µ = − H I dt I + p µ dq µ , (25)where (8) and (22) have been considered.For two arbitrary functions F, G ∈ Γ N +1 := T ∗ C N +1 ,that is, functions in the extended phase space spannedby the variables ( t I , q µ ) and their conjugate momenta( P I , p µ ), we introduce the extended Poisson bracket (PB) { F, G } = ∂F∂t I ∂G∂P I + ∂F∂q µ ∂G∂p µ − ∂F∂P I ∂G∂t I − ∂F∂p µ ∂G∂q µ . (26)We may therefore express evolution in Γ N +1 as follows dF = { F, H I } dt I , (27)where the t I play the role as parameters of the Hamilto-nian flows generated by the Hamiltonians H I . In passing,the CE (23) and (24) may be obtained from (27) by eval-uating F for any of the phase space variables. In this HJframework, the dynamical evolution is provided by (27)which is referred to as the fundamental differential . A. Integrability conditions
With the intention of integrating the HJPDE (19), itis convenient to rely in the method of characteristics [4].On physical grounds, it is completely unclear whether ornot all coordinates are relevant parameters of the theory,so it is crucial to find a subspace among the parameters t I where the system becomes integrable. Regarding this,the matrix occurring in (3) N µν = ∂ K ν ∂ ˙ q µ − ∂ K µ ∂ ˙ q ν = {H Pµ , H Pν } , (28)plays an important role to unravel under what conditionsthe eom associated with the action (1) will be integrable.The complete solution of (19) (or (22)) is given by afamily of surfaces orthogonal to the characteristic curves.In this sense, the fulfillment of the Frobenius integrabilityconditions [4, 12] {H I , H J } = C KIJ H K , (29)ensures the existence of such a family where C KIJ are thestructure coefficients of the theory. This means that theHamiltonians must close as an algebra. Accordingly, itmust be imposed that d H and d H Pµ are vanishing iden-tically d H I = 0 . (30)Guided by the aforementioned Jacobi theorem, we havenon-involutive constraints since we have N µν = 0 anddet( N µν ) = 0, that is, a regular case. Thence, nonew Hamiltonians arise from the realization of d H I =0 but a relation of dependence between the param-eters of the theory. Indeed, we have that dt µ = − ( N − ) µν {H Pν , H } dt where ( N − ) µν denotes the in-verse matrix of N µν such that N νρ ( N − ) ρµ = δ µν or( N − ) µρ N ρν = δ µν . In such a case it is often enoughto consider that t is the independent parameter of thetheory. From (27) we infer now that the evolution of F ∈ Γ N +1 is provided by dF = { F, H } ∗ dt , (31)where { F, G } ∗ := { F, G } − { F, H Pµ } ( N − ) µν {H Pν , G } . (32)In this HJ spirit, the remaining variables, q µ , are referredto as dependent variables . Note that the N independentHamiltonians (20) and (21) will fix the dynamics on thephase space in a unique way.The bracket structure introduced in (32) is referred toas the generalized bracket (GB) which has all the knownproperties of the standard Poisson bracket. In the presentcase, this redefines the dynamics by eliminating the pa-rameters t µ with exception of t . Accordingly, the non-involutive Hamiltonians have been absorbed in the GB.As a matter of fact, the GB is closely related to the Diracbracket arising in the Dirac-Bergmann Hamiltonian ap-proach for constrained systems [13–15]. Therefore, un-der the fundamental differential (31), the Hamiltonianspreserve the condition (30) in the reduced phase spacedefined by H Pµ = 0, and we have as a result an integrableset of Hamiltonians. B. Characteristic equations
The characteristic equations may now be computedfrom (27). First, dq µ = { q µ , H } ∗ dt = ˙ q µ dt , (33)which is a trivial identity. Second, dt µ = d ˙ q µ = { ˙ q µ , H } ∗ dt , = ( N − ) µν (cid:18) p ν + ∂ V ∂ ˙ q ν + ∂ K ν ∂q ρ ˙ q ρ (cid:19) dt , (34)provides the accelerations of the mechanical system aswe observe from (12). Third, dp µ = { p µ , H } ∗ dt , = − ∂ V ∂q µ dt + ∂ K ν ∂q µ dt ν , (35)represents the equations of motion provided by (3) bydirect substitution of the previous characteristic equation and (12). Finally, dP µ = { P µ , H } ∗ dt , = (cid:18) − p µ − ∂ V ∂ ˙ q µ (cid:19) dt + ∂ K ν ∂ ˙ q µ dt ν . (36)This is nothing but the definition of the momenta p µ ,namely p µ = ∂L/∂ ˙ q µ − dP µ /dt , once we insert (34), inagreement with (8).Regarding the Hamilton principal function we get dS = { S, H } ∗ dt , = P dt + p µ dq µ + P µ dt µ . In other words, dS = p µ dq µ − H I dt I , (37)where the Hamiltonians (22) have been introduced. Thisexpression is in agreement with (25). On the other hand,bearing in mind the nature of the matrix (4), we mustrecall that the only independent parameter is t so that dS = p µ dq µ + P µ d ˙ q µ − H dt . (38)Some comments are in order. First, it has been inferredthat the solutions to the characteristic equations, in thecomplete phase space are given by expressions of the form q µ = q µ ( t I ), which represent congruences of curves inthe ( N + 1) parametric space, where the t I play the roleof coordinates. From a physical point of view, the ac-tual dynamics of the system is achieved in a reducedphase space and dictated by the fundamental differen-tial (31). Hence, the solutions to the characteristic equa-tions in the physical sector of phase space are given by q µ = q µ ( t ), representing congruences of one-parametercurves. In comparison with the Ostrogradski-Hamiltonanalysis for this type of theories [20, 21], we note that wehave a second-class system as observed from (28) since N µν = 0. IV. THEORIES WITH SECOND-ORDEREQUATIONS OF MOTION
It is readily inferred from (3) that to obtain second-order equations of motion the matrix N µν must vanishidentically. Unlike the previous case, the matrix M µν becomes symmetric. This fact can be proved from (5)with support with the property ∂ K µ /∂ ˙ q ν = ∂ K ν /∂ ˙ q µ .From the facts that now F µ = F µ ( q ν , ˙ q ν , t ) and M µν = M µν ( q ρ , ˙ q ρ , t ), the eom (3) specialize to Newton-likeequations of the form M µν ¨ q ν = F µ µ, ν = 1 , , . . . , N ; (39)where M µν can be interpreted as the components of amass-like matrix while the term F µ may be intepreted asa force vector. In fact, the matrix M µν corresponds to theHessian matrix of a first-order equivalent Lagrangian L d (see Appendix A for details). For this particular case, theHamiltonian constraints (20) and (21) hold. Further, atthe initial stage, the evolution in phase space is dictatedby the fundamental differential (27). The next step, un-der the new setup, is to test the integrability conditionsfor the Hamiltonians (20) and (21)as we will discuss inshort. A. Integrability conditions
This particular setting determines a fully constrainedsystem. Indeed, to prove this statement we must firstmention that the expression for the PB (26) holds andthen we must to proceed to test the integrability condi-tion for H Pµ by using (27) d H Pµ = {H Pµ , H } dt + {H Pµ , H Pν } dt ν , = − (cid:18) p µ + ∂ V ∂t µ + ∂ K µ ∂q ν t ν (cid:19) dt , where we have considered N µν = 0 as we observefrom (28). Then, we identify new Hamiltonian con-straints given by H pµ := p µ + ∂ V ∂t µ + ∂ K µ ∂q ν t ν = 0 . (40)The integrability conditions have to be tested with theseHamiltonians as well. As before, when separating t fromthe remaining parameters t µ we find d H pµ = {H pµ , H } dt + {H pµ , H Pν } dt ν , = − F µ dt + C µν dt ν = 0 , (41)where we have introduced the antisymmetric matrix C µν := {H pµ , H Pν } = − C νµ , (42)and recognize that {H , H pµ } = F µ as defined in (6).Clearly, the variations (41) do not vanish identically, andin consequence, the idea to promote them as new con-straints of the theory is deceptive. These are mere depen-dence relationships between the parameters of the theory.We then have a complete set of Hamiltonians H = P + H ( t , t µ , q µ , p µ ) = P + p µ t µ + V , (43) H Pµ = P µ + H Pµ ( t , t ν , q ν ) = P µ − K µ , (44) H pµ = p µ + H pµ ( t , t ν , q ν , p ν ) = p µ + ∂ V ∂t µ + ∂ K µ ∂q ν t ν (45)Certainly, the Hamiltonians (43), (44) and (45) are non-involutive constraints. To satisfy the integrability con-dition it is required to remove the non-involutive con-straints by redefining the fundamental differential trougha generalised bracket structure. To perform this, the con-straints (40) must enter the game accompanied of newparameters. In order to further analyse the integrabilityconditions it is necessary to introduce a convenient no-tation and a relabeling of the indices. In agreement with the scheme previously outlined in (13), introducing thequantities¯ t ¯ µ := q ¯ µ and H p ¯ µ := − ∂L∂ ˙ q µ + ddt (cid:18) ∂L∂ ¨ q µ (cid:19) , (46)with the understanding that ¯ µ = N + 1 , N + 2 , . . . , N .The ¯ t ¯ µ are expected to be in relation to the generalizedcoordinates q µ . Additionally, we introduce the completeset of parameters t I in the order t I := ( t , t µ , ¯ t ¯ µ ) I = 0 , . . . , N, N + 1 , . . . , N., (47)as well as the notation H I := ( H , H Pµ , H p ¯ µ ), H I =( H , H Pµ , H p ¯ µ ) and P I = ( P , P µ , p µ ), respecting that or-der, with I, J = 0 , , , . . . , N, N + 1 , . . . , N . Surely,we can also express the Hamiltonians in a constrainedHamiltonian fashion according to (22) H I ( t I , P I ) = P I + H I ( t I , P I ) = 0 , . (48)As before, the 2 N + 1 relations (48) represent HJPDE.Similarly, as in previous section, it will be said that theHamiltonians H p ¯ µ , (40), are associated with the param-eters ¯ t ¯ µ that are in relation with the coordinates of thephysical system.Now, the evolution of the theory is derived from thefundamental differential dF = { F, H } dt + { F, H Pµ } dt µ + { F, H p ¯ µ } d ¯ t ¯ µ , (49)where the space of parameters has been expanded. Atthis point it is worthwhile to remark that the integrabilityof the system now should be tested by using (49). Atthis stage, we should be able to elucidate if the systemis integrable either in a complete or in a partial manner.By an appropriate relabeling of the indices, when usingthe fundamental differential (49) as well as separating thetime parameter from the remaining ones, the condition d H I = {H I , H J } dt J = 0 on the Hamiltonians (48) iswritten as d H I = {H I , H } dt + {H I , H A } dt A . (50)Here, I, J = 0 , A with A = µ, ¯ µ = 1 , , . . . , N, N +1 , . . . , N and H A = ( H Pµ , H p ¯ µ ) being the Hamiltoniansorganized in that suitable order. Explicitly d H = {H , H A } dt A = 0 , (51) d H A = {H A , H } dt + M AB dt B = 0 , (52)where we have introduced the antisymmetric matrix com-ponents M AB := {H A , H B } . This is a 2 N × N parti-tioned matrix M = (cid:18) − M µ ¯ ν M ¯ µν Q ¯ µ ¯ ν (cid:19) (53)decomposed in terms of the symmetric matrix M ¯ µν andthe antisymmetric one Q ¯ µ ¯ ν defined as {H Pµ , H p ¯ ν } =: − M µ ¯ ν = − M ¯ νµ , (54) {H p ¯ µ , H p ¯ ν } =: Q ¯ µ ¯ ν = − Q ¯ ν ¯ µ . (55)Notice that (54) agrees with the expression defining thematrix (5). If all the parameters are independent, then {H A , H } = 0 and M AB = 0 are required. These condi-tions lead to the only possible solution of the equationsof motion and the system becomes integrable. On thecontrary, it could happen that the Hamiltonians do notobey (51) and (52) so that the fulfillment of the integra-bility conditions leads to assume a linear dependence onthe parameters t I , leading us to define generalized brack-ets. For that reason, the discussion must be addressedon the case where M AB is different from zero. Certainly,from (52) we get M AB dt B = −{H A , H } dt . (56)Therefore, at this new stage, the integrability analysisbifurcates. • If M AB is non-singular then det( M AB ) = 0. Insuch a case the inverse matrix ( M − ) AB exists sothat ( M − ) AC M CB = δ AB or M AC ( M − ) CB = δ AB . Indeed, we will have that M − = (cid:18) ( M − QM − ) µν ( M − ) µ ¯ ν − ( M − ) ¯ µν (cid:19) . (57)The realization of d H I = 0 leads to consider that dt and dt A are dependent. We infer from (56) that t = t is the independent parameter of the theory dt A = − (cid:0) M − (cid:1) AB {H B , H } dt . (58)In this manner, when substituting (58) into (50), weobserve that the evolution of F ∈ Γ N +1 is providedby dF = { F, H } ∗ dt , (59)where { F, G } ∗ := { F, G } − { F, H A } ( M − ) AB {H B , G } . (60)This structure helps to redefine the dynamics byeliminating the parameters t A with exception of t . In passing, we would like to mention that the com-plete set of Hamiltonians H I are in involution withthe bracket structure (60). In a sense, this set upis similar to one outlined in the previous section. • If M AB is singular then det( M AB ) = 0. This prop-erty is related to the existence of a gauge symmetryin the Lagrangian (2). The rank of M AB being,say R = 2 N − r , implies the existence of r left(or right) null eigenvectors λ A ( α ) , or zero-modes , of M AB such that M AB λ B ( α ) = 0 where ( α ) labelsthe independent zero-modes with α = 1 , , . . . , r .In such a case the original configuration mani-fold C N , not including the time parameter, be-comes splitted in two submanifolds: C r spanned by r coordinates, t α , related to the kernel of M AB ,and C R spanned by R coordinates t A ′ , with A ′ = r +1 , r +2 , . . . , N ., associated with the regular partof M AB . Under these conditions, we ensure the ex-istence of a R × R submatrix, say M A ′ B ′ , such thatdet( M A ′ B ′ ) = 0 so that we have an inverse matrix( M − ) A ′ B ′ satisfying ( M − ) A ′ C ′ M C ′ B ′ = δ A ′ B ′ or M B ′ C ′ ( M − ) C ′ A ′ = δ A ′ B ′ . According to this split-ting we set A = α, A ′ = 1 , , . . . , r, r + 1 , . . . , N .The expansion of the condition (56) reads −{H A , H } dt = δ A ′ A M A ′ B ′ dt B ′ + δ αA M αB ′ dt B ′ + δ A ′ A M A ′ β dt β + δ αA M αβ dt β . (61)We must inspect two casesa) If A = A ′ it follows that dt A ′ can be expressedin terms of dt α and dt . Indeed, from (61) weget M A ′ B ′ dt B ′ =: −{H A ′ , H α ′ } dt α ′ , where we have introduced the label α ′ =0 , , , . . . , r . Thence, dt A ′ = − ( M − ) A ′ B ′ {H B ′ , H α ′ } dt α ′ . (62)We have therefore a relation of dependencebetween the parameters dt A ′ and dt α ′ .b) If A = α , from (61) again, it is inferred that − {H α , H } dt − {H α , H β } dt β − {H α , H A ′ } ( M − ) A ′ B ′ {H B ′ , H } dt − {H α , H A ′ } ( M − ) A ′ B ′ {H B ′ , H β } dt β (63)where we have inserted (62). Based on thelinear independence between dt and dt α , it follows that {H α , H } = {H α , H A ′ } ( M − ) A ′ B ′ {H B ′ , H } , (64) {H α , H β } = {H α , H A ′ } ( M − ) A ′ B ′ {H B ′ , H β } , (65)should be considered as conditions that fix thesubspace of parameters where the system be-comes integrable.The aforementioned dependence on the vari-ables, (62), when inserted into (50), allows to de-termine the evolution of F ∈ Γ N +1 as follows dF = { F, H α ′ } ∗ dt α ′ , (66)where { F, G } ∗ := { F, G }−{ F, H A ′ } ( M − ) A ′ B ′ {H B ′ , G } . (67)As already mentioned, the variables t α ′ are the in-dependent parameters of the theory whereas the re-maining variables t A ′ are the dependent variables .Clearly, under this split of the variables t A into t α and t A ′ , each of them will be in relation to bothcoordinates and velocities of the system.The bracket structure introduced either in (60) or in (67),is also referred to as the generalized bracket . As a mat-ter of fact, this structure is closely related to the Diracbracket arising in the Dirac-Bergmann approach for con-strained systems [13–15]. Therefore, the dynamical evo-lution of the theory depends on α ′ parameters, t α ′ .On the other hand, it may happens that the constraints H I = 0 do not satisfy d H I = 0 identically when (59)or (66) are considered as fundamental differentials. Insuch a case, the integrability condition leads us to obtainequations of the form f ( q µ , ˙ q µ , p µ , P µ ) = 0 which shouldalso be considered as constraints of the system. In a likemanner, the integrability conditions must be tested for f , which could also generate more Hamiltonians. Oncewe have found the complete set of involutive Hamiltoni-ans, it is mandatory to incorporate them within the HJframework where some of them must be considered asgenerators of the dynamics. This incorporation must beaccompanied by the introduction of more parameters tothe theory, these related to the new constraints that gen-erate dynamics, derived from the integrability analysis.Thereupon, the space of parameters has been increasedwhere, every arbitrary parameter is in relation to the gen-erators of the dynamics [11, 12]. As a result, in this newscenario, the final form of the fundamental differentialreads dF = { F, H α } ∗ dt α , (68)with the understanding that t α denotes the complete setof independent parameters where the index α spans overthe entire set of these parameters. Thence, the funda-mental differential (68) must be used to obtain the rightevolution in the reduced phase space through the GB.
1. Integrability analysis based on zero-modes
A useful procedure to identify the inversible subma-trix M A ′ B ′ as well as for the search of gauge informa-tion of the physical systems, under the present condi-tions, is based on the zero-modes of M AB . To develop this, consider a non-singular transformation G actingon the differentials of the parameters, except t , givenby dt A = G AB dt B . By expanding the indices coveringthe ker( M AB ) and its complement subspace, we have dt A = G Aα dt α + G AA ′ dt A ′ . We can choose now as asuitable basis of the linear transformation the zero-modes λ A ( α ) with α = 1 , , . . . , r., and a set of R vectors λ A ( A ′ ) ,with A ′ = r + 1 , r + 2 , . . . , N , chosen in a way such thatthey do not depend on the zero-modes or on one anotherand on the condition that det( G AB ) = 0. Relative tothis basis we have that G Aα = λ A ( α ) and G AA ′ = λ A ( A ′ ) sothat we can build the evolution of the system by usingthe fundamental differential dF = { F, H } dt + { F, H α } dt α + { F, H A ′ } dt A ′ , (69)where H α := H A λ A ( α ) = 0 , (70) H A ′ := H A λ A ( A ′ ) = 0 . (71)From this viewpoint we must now work with the equiv-alent set of 2 N + 1 Hamiltonians given by (20), (70)and (71). According to this, the integrability of the sys-tem should be tested by considering (69). The integra-bility condition applied to the projected Hamiltoniansbecome d H = {H , H α } dt α + {H , H A ′ } dt A ′ = 0 , (72) d H α = {H α , H } dt = 0 , (73) d H A ′ = {H A ′ , H } dt + M A ′ B ′ dt B ′ = 0 , (74)where we have defined M A ′ B ′ := M AB λ A ( A ′ ) λ B ( B ′ ) . (75)This is an antisymmetric non-singular matrix that iseverywhere invertible on the reduced phase space sothe existence of an inverse matrix, say ( M − ) A ′ B ′ , isensured. From (74) we identify a dependence amongsome of the original parameters. Indeed, we have that dt A ′ = − ( M − ) A ′ B ′ {H B ′ , H } dt so that when insertedinto (69) it follows straightforwardly that the evolutionmust be given by dF = { F, H α ′ } ∗ dt α ′ , (76)where { F, G } ∗ := { F, G } − { F, H A ′ } ( M − ) A ′ B ′ {H B ′ , G } , (77)with the understanding that H α ′ := ( H , H α ) and α ′ =0 , , , . . . , r . In arriving to the new fundamental differ-ential (76) we have considered that {H A ′ , H α } = 0. Onthe other hand, from (73) and by inserting dt A ′ , deducedfrom (74), into (72) we have that d H = {H , H α } dt α = 0 , (78) d H α = {H α , H } dt = 0 , (79)are identically vanishing. To prove this, observe that {H , H α } dt = {H , H A } λ A ( α ) dt ; now, from (56) we findthat {H , H α } dt = −M BA λ A ( α ) dt = 0 so that it is in-ferred that {H , H α } = 0. Therefore, the fundamen-tal differential (76) provides the ( r + 1)-parameter evo-lution in the phase space in which the Hamiltonians H α ′ are the generators. It should be noted that Hamiltoni-ans (20), (70) and (71) become involutive under the GBgiven by (77) so that the theory becomes integrable. Wewould like to mention that this alternative approach issuitable for systems exhibiting a neatly geometric struc-ture as it is the case, for instance, for extended objectsevolving in an ambient spacetime [17]. B. Characteristic equations
The characteristic equations that arise in the regularcase governed by the fundamental differential (59), aresimilar to those obtained in the previous section. On theother hand, for the singular case, having at our disposalthe GB (77), the dynamics of the system is provided bythe fundamental differential (76) by evaluating F for thephase space variables. In this spirit, it will be convenientto write the characteristic equations in terms of the com-ponents of the zero-modes λ A ( α ) . First, for F = q ¯ µ wehave dq ¯ µ = { q ¯ µ , H } ∗ dt + { q ¯ µ , H α } ∗ dt α , = h ˙ q µ − λ ¯ µ ( A ′ ) ( M − ) A ′ B ′ {H B ′ , H } i dt , + λ ¯ µ ( α ) dt α , (80)where we have considered that {H B ′ , H α } = 0. Second,for F = ˙ q µ we obtain d ˙ q µ = − ( M − ) A ′ B ′ ∂ H A ′ ∂P µ {H B ′ , H } dt + λ µ ( α ) dt α . (81)For F = P µ , we get dP µ = (cid:20) − p ¯ µ − ∂ V ∂ ˙ q µ + ( M − ) A ′ B ′ ∂ H A ′ ∂ ˙ q µ {H B ′ , H } (cid:21) dt − ∂ H α ∂ ˙ q µ dt α . (82)Finally, for F = p ¯ µ dp ¯ µ = (cid:20) − ∂ V ∂q ¯ µ + ( M − ) A ′ B ′ ∂ H A ′ ∂q ¯ µ {H B ′ , H } (cid:21) dt − ∂ H α ∂q ¯ µ dt α . (83)From this viewpoint, the characteristic equations pro-vide on the one hand the time evolution whereas, on theother hand, provide the canonical and gauge transforma-tions by analysing the evolution of the system at fixedtime δt along the remaining parameters. To correctlyreproduce the equations of motion, it will be necessary to choose appropriate parameters t α . This is so sincewhen calculating the characteristic equations from theoriginal form (49), we can observe that the differentials dt µ are arbitrary so, under the conditions in this scenario,the right dynamics in the physical phase space is fixed bychoosing some values for the parameters.Regarding the Hamilton principal function, S = S ( q ¯ µ , ˙ q µ , t ), we have that dS = { S, H } ∗ dt + { S, H α } ∗ dt α , = ∂S∂q ¯ µ ˙ q ¯ µ dt + (cid:18) ∂S∂q ¯ µ λ ¯ µ ( A ′ ) + ∂S∂ ˙ q µ λ µ ( A ′ ) (cid:19) dt A ′ + (cid:18) ∂S∂q ¯ µ λ ¯ µ ( α ) + ∂S∂ ˙ q µ λ µ ( α ) (cid:19) dt α . Taking into account (7), (8) and (9) projectedalong the λ A ( α,A ′ ) vectors, and collected into P I =( P , p α , p A ′ , P α , P A ′ ) we have dS = − H α ′ dt α ′ + p A ′ dt A ′ + P A ′ dt A ′ , (84)where we have considered (48). In summary, (80-84) define a reduced phase space with coordinates( q A ′ , ˙ q A ′ , p A ′ , P A ′ ). As we already mentioned, given someinitial conditions, the solution to the characteristic equa-tions will be dynamical trajectories restricted to the vari-ables q A ′ whose parametric equations will be of the form q A ′ = q A ′ ( t, t α ). C. Generator of gauge symmetries
Once the complete set of involutive Hamiltonians, H α ,satisfying {H α , H β } ∗ = C γαβ H γ , has been obtained,these must be considered as generators of infinitesimalcanonical transformations of the form, [12] δz ¯ A = { z ¯ A , H α } ∗ δt α , (85)where z ¯ A = ( q A ′ , P A ′ ). These are referred to as the char-acteristic flows of the system. Here, δt α := ¯ t α − t α = δt α ( z ¯ A ). In particular, when these transformations aretaken at constant time, δt = 0, the expression (85) de-fines a special class of transformations δz ¯ A = { z ¯ A , H ˙ α } ∗ δt ˙ α , (86)which, by observing that they remain in the reducedphase space, T ∗ C P , form the so-called infinitesimal con-tact transformations in the spirit of the constrainedHamiltonian framework by Dirac [13]. In this sense, thetransformations (86) do not alter the physical states ofthe system. Thus, t ˙ α denotes the set of all independentparameters where t is excluded. Clearly, transforma-tions (86) are generated by G := H ˙ α δt ˙ α , (87)0so that (86) is equivalent to δ G z ¯ A = { z ¯ A , G } ∗ . (88)Thus, δ G z ¯ A is the specialization of (85) to T ∗ C r where G is the generating function of the infinitesimal canonicaltransformation. In the spirit of the theory of gauge fields,transformations (88) stand for the gauge transformationsof the theory, [12]. V. APPLICATIONS
Having disposal of the general aspects of our develop-ment we can now turn to consider some examples thatillustrate how the above schemes work.
A. Galilean invariant (2 + 1) -dim model with aChern-Simons-like term
Consider the effective Lagrangian of a non-relativisticsystem [18] L ( q µ , ˙ q µ , ¨ q µ , t ) = − kǫ µν ˙ q µ ¨ q ν + m q µ, ν = 1 , ., (89)where k is a parameter of the theory, m is a constantand ǫ µν is the Levi-Civita antisymmetric metric with ǫ = 1. This one-particle model with second-orderderivatives, describes a free motion in the D = 2 spacewith non-commutating coordinates and internal struc-ture described by oscillator modes with negative energies.Once we identify the basic structures K µ = kǫ µν ˙ q ν and V = − ( m/
2) ˙ q , from (4), (5) and (6), it is straightfor-ward to compute N µν = − kǫ µν , M µν = − mδ µν and F µ = 0 , (90)respectively. Hence, from (3) the eom are given by2 kǫ µν ... q ν − mδ µν ¨ q ν = 0 , (91)which are of third-order in the derivatives. On the otherhand, from (11) and (12), the momenta for this theoryreads P µ = kǫ µν ˙ q ν , (92) p µ = m ˙ q µ − kǫ µν ¨ q ν . (93)The canonical Hamiltonian H = p µ ˙ q ν + P µ ¨ q µ − L canbe readily obtained H = p µ ˙ q µ − m q . (94)Now, from (19) (or (22)), we have the HJPDE H = ∂S∂t + ∂S∂q µ t µ − m δ µν t µ t ν , (95) H Pµ = ∂S∂t µ − kǫ µν t ν = 0 , (96) or, H = P + p µ ˙ q µ − m δ µν t µ t ν , (97) H Pµ = P µ − kǫ µν t ν = 0 , (98)in a constrained Hamiltonian fashion. By noticing thatthe inverse matrix of (4), for the present case, is given by( N − ) µν = (1 / k ) ǫ µν , the GB (32) reads { F, G } ∗ = { F, G } − k { F, H Pµ } ǫ µν {H Pν , G } , (99)where F and G are phase space functions. Defining Q µ := 2 ˙ q µ , Q := 2 /k and K := k/
2, we are able tofind the non-vanishing fundamental generalized bracketsof the theory { q µ , p ν } ∗ = δ µν { Q µ , Q ν } ∗ = Q ǫ µν , { Q µ , P ν } ∗ = δ µν { P µ , P ν } ∗ = Kǫ µν . (100)We thus find that ( q µ , p ν ) and ( Q µ , P ν ) are the canonicalpairs of the theory under the GB structure. In pass-ing, we observe that under the GB (99) the coordinates Q µ are non-commutative. Regarding the characteris-tic equations, by considering the results of IV we havethat (33) is a mere identity. In the same spirit, (34)leads to dt µ = d ˙ q µ = (1 / k ) ǫ µν ( p ν − m ˙ q ν ) dt which is inagreement with (93). On the other hand (35) yields to dP µ = − ( p µ − m ˙ q µ ) dt − kǫ µν dt ν which is also in agree-ment with (92). Finally, (36) leads to dp µ = 0. Thislast fact plays a double duty. On one hand, this leads tothe equation of motion (91) once we insert the expres-sion (93). On the other hand, this yields the fact thatthe momenta p µ are constant.It is worthwhile to mention that in [18] this systemhas been analysed using the equivalent canonical Hamil-tonian given by H = − m k P + 1 k ǫ µν P µ p ν , (101)where the authors focused on the quantum properties ofthe system. B. Harmonic oscillator in 2D
A case where the matrix (53) is regular is provided bythe Lagrangian [19] L ( q µ , ˙ q µ , ¨ q µ , t ) = − m q µ ¨ q µ − k q µ, ν = 1 , . (102)where m and k are constants. This is a non-standardway to study an isotropic harmonic oscillator as we willsee shortly. We readily identify that K µ = − ( m/ δ µν q ν and V = ( k/ δ µν q µ q ν . From (4), (5) and (6) we findthat N µν = 0 M µν = − m δ µν and F µ = k q µ . (103)1Consequently, from (3) we have the second-order equa-tions of motion m ¨ q µ + kq µ = 0 , (104)which is nothing but the well known Hooke’s law. Re-garding the momenta, from (11) and (12), we get P µ = − m q µ and p µ = m q µ . (105)For this pedagogical case, the corresponding Legendretransformation yields H = p µ ˙ q µ + k q . (106)On account of the expressions (43), (44) and (45), theHamiltonians of the theory are H = P + p µ ˙ q µ + k q = 0 , (107) H Pµ = P µ + m q µ = 0 , (108) H pµ = p µ − m q µ = 0 . (109)These expressions, together with (26), allow us to quicklydetermine that the matrix Q µν , (55), vanishes. Thus,the partitioned matrix (53) turns out to be non-singular.This, and its inverse, are given by M = m − − M − = 1 m − −
11 0 0 00 1 0 0 , (110)respectively. Under these conditions, t = τ is the onlyindependent parameter of the model and the evolutionin the phase space is provided by the fundamental differ-ential given by (59) where, on account of (60), the GB isgiven by { F, G } ∗ = { F, G } + 1 m { F, H Pµ } δ µν {H pν , G }− m { F, H pµ } δ µν {H Pν , G } . (111)Having at our disposal this GB and by defining π µ := 2 p µ and Π µ := 2 P µ we find that the non-zero fundamentalgeneralized brackets are { q µ , π ν } ∗ = δ µν { q µ , ˙ q ν } ∗ = m δ µν { ˙ q µ , Π ν } ∗ = δ µν { π µ , Π ν } ∗ = mδ µν . (112)Apparently ( q µ , π ν ) and ( ˙ q µ , Π ν ) are the canonical pairsof the theory but this fact is deceptive. This is definitelyan example of the fact that the theory described by (102)needs a boundary term in order to restore the originalproperties of a harmonic oscillator without the need todescribe it in terms of second-order terms. Regarding thecharacteristic equations, when using (59) and (112), allthe dynamical and geometrical information is correctlyreproduced, as stated at the beginning of IV B. C. Geodetic brane cosmology with a cosmologicalconstant
Consider the cosmological effective Lagrangian for ageodetic brane-like universe governed by the Regge-Teitelboim (RT) model [28–36] L ( a, ˙ t, ˙ a, ¨ t, ¨ a, τ ) = a ˙ tN (cid:0) ˙ t ¨ a − ˙ a ¨ t (cid:1) + aN Υ , (113)where Υ := t − N a ¯Λ . Here, and in what fol-lows, ¯Λ := Λ / α where Λ and α are constants and N := p ˙ t − ˙ a represents the lapse function that com-monly appears when we perform an ADM decomposi-tion of the RT model. Further, a dot stands for deriva-tive with respect to the time parameter τ where q µ =( t ( τ ) , a ( τ )). This model is invariant under reparameteri-zations of the coordinates (for more details see [22, 28]).For this theory we readily identify K µ = a ˙ t ( − ˙ a, ˙ t ) /N and V = − ( a/N )Υ with µ = 1 , µ = 3 ,
4. Fromthese, we compute the basic structures (4), (5) and (6): N µν = 0,( M µ ¯ ν ) = a Φ N (cid:18) ˙ a − ˙ t ˙ a − ˙ t ˙ a ˙ t (cid:19) , ( Q ¯ µ ¯ ν ) = ˙ t Θ N (cid:18) −
11 0 (cid:19) , (114)and ( F ¯ µ ) = − ˙ a Θ N (cid:0) − ˙ a ˙ t (cid:1) , (115)where Θ := ˙ t − N a ¯Λ , (116)Φ := 3 ˙ t − N a ¯Λ . (117)Note that det( M µ ¯ ν ) = 0 where its rank is r = 1. Con-trary to this fact, we have that det( Q ¯ µ ¯ ν ) = 0. By insert-ing (114) and (115) into (3) we find a solely equation ofmotion ddτ (cid:18) ˙ a ˙ t (cid:19) = − N a ˙ t ΘΦ , (118)which is of second-order in derivatives in the variables t ( τ ) and a ( τ ).The momenta of the theory, from (11) and (12), are( P µ ) = a ˙ tN (cid:0) − ˙ a ˙ t (cid:1) and ( p ¯ µ ) = − a Υ N (cid:0) − ˙ t ˙ a (cid:1) . (119)In this spirit, the corresponding Legendre transformationyields H = p t ˙ t + p a ˙ a − aN Υ . (120)On physical grounds, the HJ analysis gets more conve-nient when using the projector approach based on zero-modes. The partitioned matrix (53) is given by M = ¯Φ N − ˙ a ˙ t ˙ a t ˙ a − ˙ t ˙ a − ˙ t ˙ a − N − ˙ t ˙ a ˙ t N , (121)2where ¯Θ := ( ˙ t/N )Θ and ¯Φ := ( a/N )Φ with Θ and Φdefined in (116) and (117), respectively. The rank of thismatrix, being R = 2, signals the presence of two zero-modes. Indeed, guided by the notation introduced inSect. IV and in the obtaining of the vectors λ A ( α,A ′ ) asdepicted in IV A 1 we have that the vectors λ ( α ) = λ (1) = ˙ t ˙ a λ (2) = ¯Θ ˙ a ¯Θ ˙ t − ¯Φ ˙ t − ¯Φ ˙ a λ ( A ′ ) = λ (3) = ˙ a ˙ t λ (4) = a ˙ t , (122)span the kernel of (121) and its complement subspace,respectively, where α = 1 , A ′ = 3 ,
4. Bearing inmind that t = τ , the original HJPDE for the presentcase are given by H = ∂S∂τ + t ¯ µ ∂S∂t ¯ µ − aN Υ = 0 , H Pµ = ∂S∂t µ − a ˙ tN n µ = 0 , H p ¯ µ = ∂S∂t ¯ µ + a Υ N ˙ q ¯ µ = 0 , where we have introduced the vectors˙ q µ = (cid:18) ˙ t ˙ a (cid:19) and n µ = 1 N (cid:18) ˙ a ˙ t (cid:19) , (123)which are orthogonal in the sense that η µν ˙ q µ n ν = 0 where( η µν ) = diag( − , η µν n µ n ν = 1.In fact, these represent both the time vector field and thenormal vector to the brane-like universe. By projectingthe original Hamiltonians along (122), as dictated by (70)and (71), we get H α = ( H = C , H = ¯Θ C − ¯Φ C , H A ′ = ( H = C , H = C , (124)where C := P t ˙ t + P a ˙ a = 0 , (125) C := P t ˙ a + P a ˙ t − a ˙ tN = 0 , (126) C := p t ˙ t + p a ˙ a − aN Υ = 0 , (127) C := p t ˙ a + p a ˙ t = 0 . (128) Note that C = H which identically vanishes as a conse-quence of the reparameterization invariance of the model.In the representation (124) the Hamiltonians split intoinvolutive, H α , and non-involutive, H A ′ , ones. The ex-tended Poisson algebra among the C i , with i = 1 , , , { C , C } = 0 { C , C } = − C { C , C } = − C { C , C } = − C − ¯Φ , { C , C } = − C { C , C } = − ¯Θ . (129)Clearly, the C i determine a non-involutive set of phasespace functions. In this sense, the Hamiltonians (124)obey the extended Poisson algebra {H , H } = −H {H , H } = a ˙ t ˙ aN ¯Φ H − ¯Φ H + aN ¯Φ ( ¯ΦΥ − a ˙ t ¯Θ) H {H , H } = 0 {H , H } = − t N ¯Φ H + t N ¯Φ ( ˙ t ¯Θ − N a ¯Λ ¯Φ) H {H , H } = − H {H , H } = H − ¯Θ¯Φ H − ¯Φ , (130)what verifies that H A ′ = ( H , H ) are non-involutiveHamiltonians. From (75) and (122) we can find the regu-lar submatrix embedded in the partitioned matrix (121)as well as its inverse M = ¯Φ (cid:18) −
11 0 (cid:19) and M − = 1¯Φ (cid:18) − (cid:19) (131)respectively. Under these conditions the complete dy-namics of the theory is dictated by the fundamental dif-ferential (76) where the corresponding GB is given by { F, G } ∗ = { F, G } − { F, H }{H , G } + 1¯Φ { F, H }{H , G } . (132)Once we have reduced the phase space by constructingthe appropriate GB, (132), we can extract physical in-formation of the theory. In this spirit, the non-vanishingfundamental GB between the phase space variables are3 { t, ˙ t } ∗ = − ˙ a ¯Φ { a, ˙ t } ∗ = − ˙ t ˙ a ¯Φ { ˙ t, P a } ∗ = − ˙ ap t ¯Φ { t, ˙ a } ∗ = − ˙ t ˙ a ¯Φ { a, ˙ a } ∗ = − ˙ t ¯Φ { ˙ a, P t } ∗ = − ˙ tp a ¯Φ { t, p t } ∗ = 1 { a, p a } ∗ = 1 − t ¯Φ { ˙ a, P a } ∗ = 1 − ˙ tp t ¯Φ { t, p a } ∗ = − t ˙ a ¯Φ { a, P t } ∗ = ˙ tP a − ˙ aP t ¯Φ { p a , P t } ∗ = − tp a Φ { t, P t } ∗ = ˙ aP a ¯Φ + a ˙ a N ¯Φ { a, P a } ∗ = tP t ¯Φ { p a , P a } ∗ = − tp t Φ { t, P a } ∗ = aP t ¯Φ { ˙ t, P t } ∗ = 1 − ˙ ap a ¯Φ { P t , P a } ∗ = ap t Φ (133)Clearly, the pair ( t, p t ) is the unique canonical one sowe have the presence of only a physical degree of free-dom in this theory. It is worthwhile to mention that theHamiltonians H α obey a truncated Virasoro algebra asdiscussed in [22, 26, 27].
1. Characteristic equations
Guided by (76), (132) and (80-83) the evolution alongthe complete set of parameters is given as follows. First, dt = ˙ t dτ − ¯Φ ˙ t dt d ˙ t = − ¯Θ¯Φ ˙ a dτ + ˙ t dt + ¯Θ ˙ a dt ,da = ˙ a dτ − ¯Φ ˙ a dt d ˙ a = − ¯Θ¯Φ ˙ t dτ + ˙ a dt + ¯Θ ˙ t dt , (134)where we have used the values of the zero-modes givenby (122). Second, dp t = 0 dp a = (cid:18) − a ˙ tN ¯Θ¯Φ + ¯Θ˙ t (cid:19) dτ − a ˙ t ¯ΘΥ dt (135)and dP t = (cid:20) − p t − a ˙ tN (cid:0) ˙ t − N (2 − a ¯Λ ) (cid:1) + ¯Θ¯Φ a N ( ˙ t + ˙ a ) (cid:21) dτ + a ˙ t ˙ a N dt − a ˙ tN (cid:2) ( ˙ t + ˙ a )Θ − a Φ (cid:3) dt ,dP a = (cid:20) − p a + a ˙ aN (cid:0) ˙ t + N a ¯Λ ) (cid:1) − ¯Θ¯Φ 2 a ˙ t ˙ aN (cid:21) dτ − a ˙ t N dt + 2 a ˙ t ˙ aN (Θ − Φ) dt . (136)By choosing dt = (2 / ˙ t ¯Φ)( ¯Φ d ˙ t + ¯Θ da ) and dt = dt ( τ )we are able to obtain the eom (118) as well as to recoverthe definition for the momenta P µ . It is worthwhile tomention that the momenta p t is a constant of motionwhich is a result of the invariance under reparameteriza-tions of this brane theory. Indeed, this corresponds to theconserved bulk energy conjugate to the time coordinate t ( τ ).
2. Gauge transformations
As stated above, the Hamiltonians H α ′ are responsiblefor generating the complete dynamics along the direc- tions of the parameters t α ′ . Regarding the gauge trans-formations for this theory, the generator function G canbe constructed from (87), G = H α δt α α = 1 ,
2; (137)where these transformations are taken at constant time, δt . It is therefore inferred, from (88), that the gaugetransformations are given by δ G z ¯ A = { z ¯ A , G } ∗ = { z ¯ A , H α } ∗ δt α . (138)From (124) we have that δ G t = − ¯Φ ˙ t δt δ G ˙ t = ˙ t δt + ¯Θ ˙ a δt ,δ G a = − ¯Φ ˙ a δt δ G ˙ a = ˙ a δt + ¯Θ ˙ t δt . (139)Similarly, regarding the momenta, we obtain δ G p t = 0 δ G p a = − a ΥΘ N δt , (140)and δ G P t = − P t δt − (cid:20) P a − p t − a N ¯Θ+ 2 a ˙ tN ¯Φ(1 − a ¯Λ ) (cid:21) δt δ G P a = − P a δt − (cid:18) ¯Θ P t − ¯Φ p a + a ˙ aN ¯Φ a ¯Λ (cid:19) δt (141)Therefore, by considering the definitions ǫ := ¯Φ δt and2 ǫ := δt , one is able to show that δL = 0 whenever ǫ ( τ ) is vanishing at the extrema located at τ = τ and τ = τ . This brane model has recently been studiedalso using an HJ approach but, using auxiliary variableswhere the number of Hamiltonians grows enormously.[22]. Regarding the gauge symmetries, it was shown in[22] that, by considering ǫ and ǫ , δ G t and δ G a reflectthe presence of the invariance under reparameterizationsof the model (113) while δ G ˙ t and δ G ˙ a reflect the presenceof an inverse Lorentz-like transformation in the velocities. VI. CONCLUDING REMARKS
In this paper we have analyzed the integrability ofthe linearly acceleration-dependent theories whithin theHamilton-Jacobi framework for second-order singular4systems. Our construction entails mainly two differentscenarios according to the nature of the equations of mo-tion. In both cases it was shown the presence of a GBstructure as a consequence that the original theory, fromthe beginning, contains non-involutive constraints. Un-like the Dirac-Bergmann approach for constrained sys-tems, whithin the HJ framework it is not mandatory tosplit the constraints into first- and second-class althoughthey are closely related.On the other hand, when the partitioned matrix (53)turns out to be singular and when the involutive con-straints can be identified, these determine the so-calledcharacteristic flows (86) which are in connection with thegauge transformations that leave the action (1) invariant.Along this line, in order to illustrate our HJ developmentsome theories were considered where the obtained resultsare in agreement with previous analyzes. Although is outof the scope of the paper, the obtaining of the Hamiltonprincipal function is an element that need to be exploredin detail. Referring to this, despite that the abstractprocedure is clear regarding the calculation process, oneshould always be cautious. On the one hand, one may touse the PDE techniques in order to solve the HJPDE or,on the other hand, when one knows the form of the co-ordinates and momenta as functions of the parameters itis possible to perform the integration of the correspond-ing characteristic equation which in general is rather in-volved. Attempts in this direction have been done in[37]. The analysis here achieved has been carried out, forsimplicity, for systems with a finite number of degrees offreedom. For field theories we believe that it is possi-ble to extend our analysis as long as we are careful withthe functional analysis. In a sense, our development pavethe way to be applied to relativistic second-order geomet-ric systems characterized by a linear dependence on theaccelerations as in the case of the so-called Lovelock-likebrane models that pursue explanation of cosmological ac-celeration phenomena [23–25, 38] where we will explorethe WKB approximation for this type of cosmologicaltheories. This subject will be reported elsewhere.
ACKNOWLEDGMENTS
The authors thank Alberto Molgado for the criticalreading and enlightening remarks that improved the workand also thank Rub´en Cordero for suggestions and com-ments received. ER thanks the Departamento de F´ısicade la Escuela Superior de F´ısica y Matem´aticas del I.P.N,M´exico, where part of this work was developed dur-ing a sabbatical leave. ER acknowledges encouragmentfrom ProDeP-M´exico, CA-UV-320: ´Algebra, Geometr´ıay Gravitaci´on. Also, ER thanks the partial support fromSistema Nacional de Investigadores, M´exico. AAS ac-knowledges support from a CONACyT-M´exico doctoral fellowship.
Appendix A: On the matrix M µν The matrix M µν contains important geometric infor-mation so it is not just a notational resource. This isnothing but the Hessian matrix associated to a first-orderLagrangian, L d , where a surface term, L s , in the La-grangian (2) is identified. To prove this, suppose that weare able to write the Lagrangian (2) in terms of a dy-namical Lagrangian and a surface Lagrangian as follows L ( q µ , ˙ q µ , ¨ q µ , t ) = L d ( q µ , ˙ q µ , t ) + L s ( q µ , ˙ q µ , ¨ q µ , t ) , (A1)where L s := dh ( q µ , ˙ q µ , t ) dt , (A2)for a smooth function h ( q µ , ˙ q µ , t ). From theOstrogradski-Hamilton viewpoint, the momenta aregiven by P µ = ∂L∂ ¨ q µ = ∂L s ∂ ¨ q µ , = K µ , (A3) p µ = ∂L∂ ˙ q µ − ddt (cid:18) ∂L∂ ¨ q µ (cid:19) = ∂L d ∂ ˙ q µ + ∂L s ∂ ˙ q µ − ddt (cid:18) ∂L s ∂ ¨ q µ (cid:19) , = − ∂ V ∂ ˙ q µ − ∂ K µ ∂q ν ˙ q ν . (A4)Clearly, the momenta conjugate to the coordinates canbe written in terms of the corresponding momenta arisingfrom L d and L s , p µ = p d µ + p s µ . On the one hand we have that ∂p µ ∂ ˙ q ν = ∂p d µ ∂ ˙ q ν + ∂p s µ ∂ ˙ q ν = W µν + ∂p s µ ∂ ˙ q ν , (A5)where W µν := ∂p d µ /∂ ˙ q ν is the Hessian matrix associatedwith the Lagrangian L d . On the other hand, we have that ∂p µ ∂ ˙ q ν = ∂∂ ˙ q ν (cid:18) − ∂ V ∂ ˙ q µ − ∂ K µ ∂q ρ ˙ q ρ (cid:19) = ∂ K ν ∂q µ − M µν , (A6)where we have substituted (5) defining the matrix M µν .Hence, by considering (A3), from (A5) and (A6) it followsthat M µν = − W µν , provided that ∂ K ν ∂q µ = ∂p s µ ∂ ˙ q ν . (A7)We have therefore proved that, when a surface term isexisting in the Lagrangian (2), the matrix M µν is noth-ing but the Hessian matrix associated to the dynamicalLagrangian L d whenever the relationship (A7) holds.5 [1] J. F. Cari˜nena, J. Fern´andez-N´u˜nez and M. F. Ra˜nada,J. Phys. A: Math. Gen. , 174-185 (2011).[3] M. Cruz, R. G´omez-Cort´es, A. Molgado and E. Rojas, J.Math. Phys. , 062903 (2016).[4] C. Carath´eodory, Calculus of variations and partial dif-ferential equations of first-order (Holden-Day, San Fran-cisco, 1967).[5] Y. G¨uler Y, Il Nuovo Cimento B , 1389 (1992).[6] Y. G¨uler Y, Il Nuovo Cimento B , 1143 (1992).[7] B. M. Pimentel and R. G. Teixeira, Il Nuovo Cimento
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