Constrained Dynamics: Generalized Lie Symmetries, Singular Lagrangians, and the Passage to Hamiltonian Mechanics
aa r X i v : . [ m a t h - ph ] J un Constrained Dynamics: Generalized Lie Symmetries, SingularLagrangians, and the Passage to Hamiltonian Mechanics
Achilles D. Speliotopoulos
Department of Physics, University of California, Berkeley, CA 94720 USA ∗ (Dated: June 5, 2020) bstract Guided by the symmetries of the Euler-Lagrange equations of motion, a study of the constraineddynamics of singular Lagrangians is presented. We find that these equations of motion admit ageneralized Lie symmetry, and on the Lagrangian phase space the generators of this symmetry lie inthe kernel of the Lagrangian two-form. Solutions of the energy equation—called second-order, Euler-Lagrange vector fields (SOELVFs)—with integral flows that have this symmetry are determined.Importantly, while second-order, Lagrangian vector fields are not such a solution, it is alwayspossible to construct from them a SOELVF that is. We find that all SOELVFs are projectable to theHamiltonian phase space, as are all the dynamical structures in the Lagrangian phase space neededfor their evolution. In particular, the primary Hamiltonian constraints can be constructed fromvectors that lie in the kernel of the Lagrangian two-form, and with this construction, we show thatthe Lagrangian constraint algorithm for the SOELVF is equivalent to the stability analysis of thetotal Hamiltonian. Importantly, the end result of this stability analysis gives a Hamiltonian vectorfield that is the projection of the SOELVF obtained from the Lagrangian constraint algorithm. TheLagrangian and Hamiltonian formulations of mechanics for singular Lagrangians are in this wayequivalent.
I. INTRODUCTION
The Lagrangian phase space formulation of mechanics [1–4], with its roots in differen-tial geometry, provides an especially fruitful framework with which to analyze dynamicalsystems of singular Lagrangians L . Instead of trajectories q ( t ) = ( q ( t ) , . . . , q D ( t )) on a D -dimensional configuration space Q that are solutions of the Euler-Lagrange equations ofmotion, trajectories in the Lagrangian phase space formulation u ( t ) = ( q ( t ) , . . . , q D ( t ) , v ( t ) , . . . , v D ( t )) , (1)are on a D -dimensional Lagrangian (or velocity) phase space P L = T Q embodiedwith a Lagrangian two-form Ω L . They are determined by vector fields X E ∈ T u P L that are ∗ Also at Division of Physical, Biological and Health Science, Diablo Valley College, Pleasant Hill, CA94523, USA; [email protected] energy equation i X E Ω L = d E, (2)with E being the energy of the system. For regular Lagrangians, Ω L is symplectic. Thesolution to Eq. (2) is unique, and is a second-order, Lagrangian vector field (SOLVF)[1] (alsocalled a second-order dynamical equation in the literature). For singular Lagrangians, onthe other hand, Ω L is presymplectic [5]. The solution to Eq. (2) is not unique, need notbe a SOLVF, nor need it even exist [6]. Nevertheless, with few exceptions in the literature[7], focus has been placed on solutions of Eq. (2) that are SOLVFs. This is done for phys-ical reasons: the condition ˙ q a = v a , for a = 1 , . . . , D , immediately follows for trajectoriesdetermined by such fields. This focus on SOLVFs has consequences, however.The presence of a singular Lagrangian often predicates the existence of a Lagrangianconstraint submanifold of P L , and for solutions of Eq. (2) to exist, trajectories of these dy-namical systems must lie on this submanifold. Algorithms—called a constraint algorithm ora stability condition—for constructing such solutions have been developed [6, 8–14]. How-ever, irrespective of the one used, the end result of these algorithms is a vector field that hasa number of troubling attributes. First, this vector field need not be a SOLVF, even thoughphysical arguments were used to restrict the starting point of these algorithms to such fields.This is called the second order problem, first noted within a different context by Künzle [15],and emphasized by Gotay and Nester [9]. Currently, it is known that requiring the end resultof these algorithms be a SOLVF is very restrictive, and additional conditions may need to beimposed [16]. Second, the fibre derivative (Legendre transform) L for singular Lagrangiansis singular, and thus the rank of the Hessian of L is not maximal. Because of this the passagefrom Lagrangian to Hamiltonian mechanics is problematic. The ability to map dynamicalstructures from the Lagrangian to the Hamiltonian phase space has long been studied forSOLVFs [4, 8, 11, 12, 16–24]. It is found that a general SOLVF—even after the application ofthe constraint algorithm, and even under the weak projectability condition [11]—is not pro-jectable [11, 16]. (Examples of systems for which a SOLVF is projectable, and for which itis not are given in [11].) One immediate consequence of this non-projectability is dynamicalsystems for which the Hamiltonian flow field determined through constrained Hamiltonianmechanics as described in [25, 26] (see [27, 28] for more modern approaches)—even after itsrestriction to the primary constraint submanifold—need have little relation to the SOLVF3btained from the Lagrangian constraint algorithm. Third, it is known that the Lagrangianconstraints obtained while a constraint algorithm is being imposed on a SOLVF also neednot be projectable [4, 11, 16, 18, 19, 22, 29]. Combined, this means that dynamics on theLagrangian phase space and dynamics on the Hamiltonian phase space can take place ontwo inequivalent submanifolds, be determined by two inequivalent vector fields, resulting intwo different families of trajectories on the configuration space Q for the same dynamicalsystem with the same initial data.We take a different starting point in our analysis of singular Lagrangians, one that isrooted in the generalized Lie symmetries of the Euler-Lagrange equations of motion. Theanalysis is guided by the observation that if a symmetry of the dynamical system has beendetermined through Lagrangian mechanics, it must be present in the Lagrangian and Hamil-tonian phase space descriptions of motion as well. We emphasize, however, that while thesesymmetries play an important role, this role is nevertheless supportive. One of the main goalsof this work is the construction of algebraic-geometric structures within differential geometrythat will then be used to implement these symmetries; to characterize the relevant geometryof the Lagrangian phase space; to determine the structures needed to decribe dynamics onthis phase space; and to show the equivalence of the Lagrangian and Hamiltonian phasespace formulation of mechanics. In doing so, we are led to construct the second-order,Euler-Lagrange vector field (SOELVF). These fields avoid the second-order problem, areprojectable to the Hamiltonian phase space, and lie on Lagrangian constraint submanifoldsthat also are projectable. Importantly, the projection of the SOELVF is the Hamiltonianflow field of the total Hamiltonian obtained from constrained Hamiltonian mechanics. (Wefollow the terminology in [26], and call the result of augmenting the canonical Hamiltonianwith the primary constraint functions the total Hamiltonian .)The generalized Lie symmetry [30] of the Euler-Lagrange equations of motion is generatedby second-order prolongation vectors in the tangent space T M (2) of the second-order jetspace M (2) . This symmetry is reflected in the Lagrangian phase space description of motion,and the projection of T M (2) to T P L maps these prolongation vectors into the kernel of Ω L .Surprisingly, it is not the vertical vector fields of the kernel that generates this symmetry,as may have been expected. Also surprisingly, the corresponding symmetry group Gr S ym isnot a symmetry group for SOLVFs; action on a SOLVF by Gr S ym results in a vector fieldthat is no longer a SOLVF, nor need it even be a solution of Eq. (2) . It is, however, always4ossible to construct from a SOLVF vector fields that do have Gr S ym as a symmetry group,and are solutions to Eq. (2). These vector fields are the SOELVFs, and they resolve theissues listed above for the SOLVF.That a SOELVF is projectable is a natural consequence of having Gr S ym as a symmetrygroup. Moreover, all the Lagrangian constraints—both those due to the energy equation andthose introduced through the application of the constraint algorithm to the SOELVF—areprojectable as well. We find also that there is a choice of a basis for the kernel of Ω L thatis projectable, and that the primary Hamiltonian constraints can be constructed from theirimage. Indeed, all the dynamical structures needed to describe evolution on the Lagrangianphase space are projectable, and their image corresponds to the dynamical structures neededto describe evolution on the Hamiltonian phase space obtained through constrained Hamilto-nian mechanics [14, 25–28, 31–33]. In this way the Lagrangian and Hamiltonian fomulationsof mechanics are equivalent even for singular Lagrangians.Analysis of the symmetries of Lagrangian systems (both regular and singular) have beendone before. However, such analyses have been focused on time-dependent Lagrangians(and in particular their Noether symmetries) [10, 34–40]; on systems of first-order evolutionequations on either the Lagrangian or Hamiltonian phase space [41–45]; or on general solu-tions of Eq. (2) [7] (see also [46] for an analysis of particle motion on curved spacetimes).Importantly, the great majority of these analyses have been done using first-order prolon-gations on first-order jet bundles with a focus on the Lie symmetries of first-order evolutionequations. Our interest is in the symmetries of the Euler-Lagrange equations of motion—asystem of second-order differential equations—that come from singular Lagrangians. Thisnaturally leads us to consider generalized Lie symmetries and second-order prolongations.Such symmetry analysis of the Euler-Lagrange equations of motion has not been done be-fore. (Although the framework for k th -order prolongations on k th -order jet bundles havebeen introduced before [10, 40, 53, 54], they have not been applied to the Euler-Lagrangeequations of motion.)In this paper we only consider autonomous Lagrangians for which the rank of Ω L isconstant on P L . We also require such Lagrangians to have a fibre derivative that is asubmersive map of P L to the Hamiltonian phase space P C ; the rank of the Hessian of suchLagrangians is necessarily constant on P L . In addition, the preimage of ( q, p ) = L ( u ) , u ∈ P L ,must be a connected submanifold of P L (see also [33] where this condition is relaxed). These5agrangians are called almost regular Lagrangians in the literature [2, 4, 8, 16, 24], andwe also use this terminology.Some of the results on the equivalence of singular Lagrangian and Hamiltonian me-chanics presented in this paper have been presented elsewhere. However, the approachesused previously often rely on such dynamical structures from constrained Hamiltonian me-chanics as the primary Hamiltonian constraints; pullbacks of their derivatives to the La-grangian phase space are used to construct such mappings as the time-evolution operator K [12, 17, 19, 20, 22, 38], for example. We take a different approach, one that starts with theLagrangian phase space formulation, and, with restrictions imposed by Gr S ym, is one whichshows that the dynamical structures on the Hamiltonian phase space necessary to describethe dynamical system can be obtained directly from those on the Lagrangian phase space.The importance of establishing the equivalence between the Lagrangian and Hamiltonianformulations of mechanics for singular Lagrangians can been seen from the starting pointof any physically relevant system: the action, and through it, its Lagrangian. For systemswith local gauge or diffeomorphism symmetry, the Lagrangian is singular, and dynamics havetraditionally been analysed using Hamiltonian constraints [26]. This is done by first usingthe fibre derivative to construction the Hamiltonian from the Lagrangian, and then usingHamiltonian stability analysis to determine both the total Hamiltonian and the Hamiltonianconstraint surfaces. However, as the Lagrangian was the starting point, and as the fibrederivative is not invertable for such Lagrangians, a natural question to ask is whether thedynamics described by the total Hamiltonian has any relation to the original dynamics givenby the Lagrangian. With the equivalence between the Lagrangian and the Hamiltonianphase space formulations of dynamics demonstrated, we have shown that they are. Thisequivalence is even more important for the path integral formulation of quantum mechanicsand quantum field theory. Both are based on the action, and integration is over paths onthe configuration space. For systems with local gauge or diffeomorphism symmetries theseintegrals must be restricted, which for non-abelian gauge theories leads to the use of BRSTsymmetries. These symmetries have traditionally been constructed using the Hamiltonianand Hamiltonian constraint analysis (see [26]).Although we freely use the tools and language of differential geometry, we are aware thatinterest in constrained dynamics is often due to its application to quantum field theories.In these applications, the ability to calculate and determine symmetries is paramount. To6nsure that the tools and methodologies given in this paper can be so applied to the analysisof quantum field-theoretic systems, we have also written a number of the expressions givenin this paper in terms of local coordinates using a notation that is both familiar and usefulfor calculations. In particular, the general solution to i K Ω L = 0 given in Section III B isgiven in terms of local coordinates as are the construction of second-order Lagrangian andEuler-Lagrangian vector fields.The rest of the paper is arranged as follows. In
Section II we show that the Euler-Lagrange equations of motion have a generalized Lie symmetry, and determine the existenceconditions for the generators of this symmetry. In
Section III the vectors that lie inthe kernel of Ω L are found, and the role they play in generating Gr S ym is determined.Physically relevant solutions of the energy equation are characterized, and the SOELVFis defined and constructed. First-order Lagrangian constraints are also constructed, anda constraint algorithm for SOELVFs is presented. In Section IV focus is on the passagefrom the Lagrangian to the Hamiltonian phase space. The projectability of functions on P L is reviewed, and a new result on the projectability of vector fields in T P L is presented.The dynamical structures needed to describe evolution with SOELVFs are shown to beprojectable, and the primary Hamiltonian constraints are constructed. The equivalence ofthe constraint algorithm presented in Section III with the usual Hamiltonian stabilityanalysis is shown. In
Section V application of the analysis given here to three differentdynamical systems with singular Lagrangians is presented. Concluding remarks can be foundin
Section VI , with the crucial role that the vertical vector fields in the kernel of Ω L playsummarized. II. GENERALIZED LIE SYMMETRIES AND LAGRANGIAN MECHANICS
We begin with Lagrangian mechanics, and an analysis of the generalized Lie symmetry[30] of the Euler-Lagrange equations of motion. The existence conditions for the generatorsof this symmetry will be established.The Euler-Lagrange equations of motion are the system of D second-order differentialequations M ab ¨ q b = − ∂E∂q a − F ab ˙ q b , (3)7here E ( q, ˙ q ) := ˙ q b ∂L ( q, ˙ q ) ∂ ˙ q b − L ( q, ˙ q ) , (4)is the energy, while M ab ( q, ˙ q ) := ∂ L ( q, ˙ q ) ∂ ˙ q a ∂ ˙ q b , and F ab ( q, ˙ q ) := ∂ L ( q, ˙ q ) ∂ ˙ q a ∂q b − ∂ L ( q, ˙ q ) ∂ ˙ q b ∂q a . (5)Here, Einstein’s summation convention is used.For almost regular Lagrangians the rank of the Hessian M ab ( u ) , where u = ( q, ˙ q ) [47], isconstant on P L . However, as the rank of M ab ( u ) = D − N , with N = dim ( ker M ab ( u )) ,this rank is not maximal, and thus Eq. (3) cannot be solved for a unique ¨ q . Instead, a chosenset of initial data u = ( q , ˙ q ) given at t = t determines a family of solutions to Eq. (3) thatevolve from the same u . These solutions are related to one another through a generalizedLie symmetry [30].Following Olver [30], we define ∆ a ( q, ˙ q, ¨ q ) := ∂E ( q, ˙ q ) ∂q a + F ab ( q, ˙ q ) ˙ q b + M ab ¨ q b , (6)which reduces to Eq. (3) on the surfaces ∆ a ( q, ˙ q, ¨ q ) = 0 . The set O ( u ) := (cid:8) q ( t ) \ ∆ a ( q, ˙ q, ¨ q ) = 0 with q ( t ) = q , ˙ q ( t ) = ˙ q (cid:9) , (7)is the family of solutions to Eq. (3) that evolve from u . Consider two such solutions q a ( t ) and Q a ( t ) . From Eq. (3) there exists a z a ( u ) ∈ ker M ( u ) such that ¨ Q a − ¨ q a = z a ( u ) . As z a depends on both q a and ˙ q a , we are led to consider generalized Lie symmetry groups generatedby g := ρ ( u ) · ∂∂ q , (8)with a ρ ( u ) that does not depend explicitly on time. This gives the total time derivative dd t := ˙ q · ∂∂ q + ¨ q · ∂∂ ˙ q , (9)with ˙ ρ := d ρ/ d t .With this g , the second-order prolongation vector, pr g := ρ · ∂∂ q + ˙ ρ · ∂∂ ˙ q + ¨ ρ · ∂∂ ¨ q , (10)8n the second-order jet space M (2) = { ( t, q, ˙ q, ¨ q ) } with pr g ∈ T M (2) can be constructed.Its action on the Euler-Lagrange equations of motion gives pr g [∆ a ( q, ˙ q, ¨ q )] = − ∂ ¨ q b ∂q a M bc ρ c + ddt (cid:2) F ab ρ b + M ab ˙ ρ b (cid:3) , (11)on the ∆ a = 0 surface. However, because the rank of M ab ( u ) is not maximal, the solution for ¨ q on this surface is not unique. For g to generate the same symmetry group for all the trajec-tories in O ( u ) , we must require ρ ( u ) ∈ ker M ab ( u ) . It then follows that pr g [∆ a ( q, ˙ q, ¨ q )] = 0 if and only if (iff) b a = F ab ρ b + M ab ˙ ρ b for some function b a ( u ) where ˙ b a = 0 . However, becauseall the solutions in O ( u ) have the same initial data, ρ a ( u ) = 0 and ˙ ρ a ( u ) = 0 , and thus b a = 0 . This leads to the following new result. Lemma 1 If g is a generalized infinitesimal symmetry of ∆ a , then ρ a ( u ) ∈ ker M ab ( u ) and ˙ ρ a ( u ) is a solution of F ab ( u ) ρ b ( u ) + M ab ( u ) ˙ ρ a ( u ) . (12)We denote by g the set of all vector fields g that satisfy Lemma , and by pr g := { pr g \ g ∈ g } the set of their prolongations. This pr g is involutive [30].The conditions under which pr g generates a generalized Lie symmetry group are wellknown [30]. However, because our Lagrangians are singular, three additional conditionsmust be imposed:1. While ρ a = 0 and ˙ ρ a = z a for any z ∈ ker M ab ( u ) is a solution of Eq. (12) , we requirethat ˙ ρ a = d ρ a / d t , and they must be removed.2. If ˙ ρ a is a solution of Eq. (12) , then ˙ ρ a + z a is a solution of Eq. (12) as well. The ˙ ρ a are not unique, and this, along with the first condition, leads us to generators that areconstructed from equivalence classes of prolongations.3. For any z a ∈ ker M ab ( u ) , Eq. (6) gives, z a (cid:18) ∂E∂q a + F ab ( q, ˙ q ) ˙ q b (cid:19) , (13)on the solution surface ∆ a ( q, ˙ q, ¨ q ) = 0 . If Eq. (13) does not hold identically, it mustbe imposed, leading to the well-known, first-order Lagrangian constraints. As each q ( t ) ∈ O ( u ) must lie on this constraint submanifold, any symmetry transformationof q ( t ) generated by pr g must give a path Q ( t ) that also lies on the constraintsubmanifold. 9ot all vectors in pr g will be generators of the generalized Lie symmetry group for O ( u ) .Determining which vectors are is best done within the Lagrangian phase space framework.This will be done in the next section. For now, we note the following.For any g ∈ g , decompose pr g = k + ¨ ρ · ∂∂ ¨ q . (14)Then for pr g A , pr g B , ∈ pr g , [ pr g A , pr g B ] = [ k A , k B ] + [ pr g A ( ¨ ρ aB ) − pr g B ( ¨ ρ aA )] ∂∂ ¨ q a . (15)Because pr g is involutive, there exists a pr g C ∈ pr g such that pr g C = [ pr g A , pr g B ] .There is then a k C such that k C = [ k A , k B ] , and the collection of vectors K = ( k = ρ ( u ) · ∂∂ q + ˙ ρ ( u ) · ∂∂ ˙ q \ ρ a ( u ) ∈ ker M ab ( u ) , F ab ( u ) ρ b ( u )+ M ab ( u ) ˙ ρ b ( u ) ) , (16)is involutive. Importantly, dim pr g = dim K = 2 N . III. THE LAGRANGIAN PHASE SPACE
In this section we determine the generators of the generalized Lie symmetry found in
Sec-tion II . This is done on the Lagrangian phase space using the tools of differential geometry.These generators are then used to determine the physically relevant solutions of the energyequation, and with them, the constraint algorithm and the Lagrangian constraint subman-ifold. Much of the content in
Sections III A to III D have been established elsewhere.They are gathered here for clarity and coherence of argument, and to establish notationand terminology. On the other hand, much of the construction and the results presented in
Sections III E to III G are new.
A. Passage from Lagrangian mechanics to the Lagrangian phase space
To treat singular Lagrangian dynamics using the methods of differential geometry, tra-jectories t → q ( t ) ∈ Q are replaced by integral flows t → u ( t ) ∈ P L [1], which for the initialdata u = ( q , v ) are given by solutions to d u dt := X ( u ) , (17)10here X is a smooth vector field in the tangent space T P L . As P L = T Q , we have thebundle projections τ Q : T Q → Q and τ T Q : T ( T Q ) → T Q with τ Q ◦ τ T Q : T ( T Q ) → Q (see[6] and [1]). It is also possible to construct T τ Q , the prolongation of τ Q to T ( T Q ) , which is asecond projection map T τ Q : T ( T Q ) → T Q which is determined by requiring that τ Q ◦ τ T Q and τ Q ◦ T τ Q map a point in T ( T Q ) to the same point in Q . The vertical subbundle [ T P L ] v of T ( T Q ) is defined by [ T P L ] v = ker T τ Q [6]; a vector X v ∈ [ T u P L ] v above a point u ∈ P L is called a vertical vector field . The horizontal subbundle [ T P L ] q of T ( T Q ) isdefined by [ T P L ] q = Image T τ Q ; a vector X q ∈ [ T u P L ] q is called a horizontal vector field .Each X ∈ T u P L can be expressed as X = X q + X v with X q ∈ [ T u P L ] q and X v ∈ [ T u P L ] v ,which in terms of local coordinates are, X q := X qa ∂∂ q a , and X v := X va ∂∂ v a . (18)In particular, a second order Lagrangian vector field X L is the solution of Eq. (2) such that T τ Q ◦ X L is the identiy on T Q (see [1]). In terms of local coordinates X L = v a ∂∂ q a + X va ∂∂ v a , (19)where for singular Lagrangians X va is not unique.For a one-form α ∈ T ∗ u P L , where T ∗ P L is the cotangent space of one-forms on P L , and avector field X ∈ T u P L , we define the dual prolongation map T ∗ τ Q by h α | T τ Q X i = h T ∗ τ Q α | X i . (20)Here, we adapt Dirac’s bra and ket notation to denote the action of a k -form ω ( x ) by ω ( x ) : Y ⊗ · · · ⊗ Y k → h ω ( x ) | Y ⊗ · · · ⊗ Y k i ∈ R , (21)where Y j ∈ T u P L . The k -form bundle is Λ k ( P L ) , while the exterior algebra of forms isdenoted by Λ ( P L ) . Then the vertical one-form subbundle [ T ∗ P L ] v of T ∗ P L is definedby [ T ∗ P L ] v := ker T ∗ τ Q ; a one-form α v ∈ [ T ∗ u P L ] v is called a vertical one-form . The horizontal one-form subbundle [ T ∗ P L ] q of T ∗ P L is defined by [ T ∗ P L ] q = Image T ∗ P L ;a one-form α q ∈ [ T ∗ u Q ] q is called a horizontal one-form . Each one-form ϕ ∈ T ∗ u P L canbe expressed as ϕ = ϕ q + ϕ v with ϕ q ∈ [ T ∗ u P L ] q and ϕ v ∈ [ T ∗ u P L ] q . In terms of localcoordinates ϕ q := ϕ qa d q a and ϕ v := ϕ va d v a .11here are a number approaches [1] used in the literature to obtain the Lagrange two-form Ω L . We follow [6, 9], and define Ω L = − dd J L , where d J is the vertical derivative (see[6]). This two-form can be expressed as Ω L := Ω F + Ω M where Ω F ( X , Y ) := Ω L ( T τ Q X , T τ Q Y ) , (22)for all X , Y ∈ T u P L ; this is a horizontal two-form of Ω L . Then Ω M ( X , Y ) = Ω L ( X , Y ) − Ω F ( X , Y ) ; this is a mixed two-form of Ω L . In terms of local coordinates, Ω L = − dθ L , where θ L = ∂L∂v a d q a , (23)while Ω F := 12 F ab d q a ∧ d q b , and Ω M := M ab d q a ∧ d v b . (24)If u ( t ) is to describe the evolution of the dynamical system given by L , then the vectorfield X must be chosen so that its integral flows on P L faithfully represent their trajectorieson Q . For regular Lagrangians this is guaranteed by setting X = X L , and is a uniquesolution of the energy equation Eq. (2) [1].We adopt the general assumption that even for singular Lagrangians there are solutionsof the energy equation that faithfully represent trajectories on Q . For general singularLagrangians neither the existence nor the uniqueness of solutions to Eq. (2) is assured. Foralmost regular Lagrangians, however, a number of general results are available. These resultsdepend on the structure of the family of solutions evolving from u , and in this the kernelof Ω L , ker Ω L ( u ) := { K ∈ T u P L \ i K Ω L = 0 } , (25)plays a defining role. B. Properties of ker Ω L ( u ) In this section we characterize the structure of ker Ω L ( u ) , and determine the vectors thatlie in it.The two-form Ω L gives the lowering map Ω ♭L : T u P L → T ∗ u P L , with Ω ♭L X := i X Ω L . As Ω ♭L = Ω ♭F + Ω ♭M , the action of Ω ♭L on a vector X is given by Ω ♭F : X ∈ T u P L → [ T ∗ u P L ] q ,and, since Ω M is a mixed two-form, by Ω ♭M = Ω v♭M + Ω q♭M , where Ω q♭M : X ∈ T u P L → [ T ∗ u P L ] q Ω v♭M : X ∈ T u P L → [ T ∗ u P L ] v . In terms of local coordinates, Ω ♭F X = F ab X qa d q b , Ω q♭M X = − M ab X va d q b and Ω v♭M X = M ab X qa d v b .For almost regular Lagrangians ker Ω v♭M = C ⊕ [ T u P L ] v while ker Ω q♭M = [ T u P L ] q ⊕ G , where C = { C ∈ [ T q P L ] q \ i C Ω M = 0 } , (26)while G ⊂ { G ∈ [ T q P L ] v \ i G Ω M = 0 } . (27)Moreover, because the rank of M ab ( u ) is constant on P L there exists a basis, n z ( n ) ( u ) = (cid:0) z n ) ( u ) , . . . , z D ( n ) ( u ) (cid:1) \ M ab ( u ) z b ( n ) ( u ) = 0 , n = 1 , . . . , N o , (28)for ker M ab ( u ) at each u ∈ P L . This in turn gives the bases C = span (cid:26) U q ( n ) = z ( n ) · ∂∂ q , n = 1 , . . . , N (cid:27) , G = span (cid:26) U v ( n ) = z ( n ) · ∂∂ v , n = 1 , . . . , N (cid:27) , (29)for C and G . It is well known [2] that for almost regular Lagrangians G is involutive.Furthermore, when the rank of Ω L is constant on P L , ker Ω L ( u ) is involutive as well.Corresponding to U q ( n ) and U v ( n ) we have the one-forms Θ ( n ) q and Θ ( n ) v where h Θ ( n ) q | U q ( m ) i = δ nm and h Θ ( n ) v | U v ( m ) i = δ nm . Then [ T u P L ] q = C ⊕ C ⊥ and [ T u P L ] v = G ⊕ G ⊥ , where C ⊥ := (cid:26) X ∈ [ T u P L ] q \ (cid:10) Θ ( n ) q (cid:12)(cid:12) X (cid:11) = 0 , n = 1 , . . . , N (cid:27) , and G ⊥ := (cid:26) X ∈ [ T u P L ] v \ (cid:10) Θ ( n ) v (cid:12)(cid:12) X (cid:11) = 0 , n = 1 , . . . , N (cid:27) . (30)To determine the vectors in ker Ω L ( u ) , choose a K ∈ ker Ω L ( u ) . Then Ω v♭M K q = 0 , and Ω q♭M K v = − Ω ♭F K q . (31)Solutions of these equations are found with the use of the following theorem from linearalgebra stated without proof (see also [4] where a special case of this theorem was proved). Theorem 2
For linear spaces E and F of dimension D and a linear map A : F → E ∗ ofrank r , the inhomogeneous linear equation, A f = ϕ , (32) has solutions if and only if h ϕ | e i = 0 ∀ e ∈ A , (33)13 here A := { e ∈ E \ h A f | e i = 0 ∀ f ∈ F } . (34)This theorem is applied to Eq. (31) by setting F = [ T u P L ] v , E = [ T u P L ] q , A = Ω q♭M ,and ϕ = − Ω ♭F K q . To make the connection with the results of Section II clear, thisapplication is done locally. The condition that X q ∈ A is D Ω q♭M K (cid:12)(cid:12)(cid:12) X q E = − K qa M ab X qb = 0 ∀ K qa , which requires M ab X qb = 0 . This establishes A = C . Equation (33) reduces to (cid:10) Ω ♭F K q (cid:12)(cid:12) C (cid:11) = F ab K qa C b = 0 ∀ C ∈ C , or equivalently z a ( n ) F ab K qb = 0 . Using the actionof Ω v♭M and Ω q♭M in Eq. (31), we find that M ab K qa = 0 , and thus K q ∈ C . The existencecondition for solutions of Eq. (31) is then, z a ( n ) F ab K qb = 0 , n = 1 , . . . , N . (35)With the basis given in Eq. (29) , we may express K qa = N X m =1 K q ( m ) z a ( m ) , (36)and Eq. (35) becomes, N X m =1 ¯ F nm K q ( m ) = 0 , where ¯ F nm := z a ( n ) F ab z b ( m ) , (37)is the reduced matrix of F ab . Then K q is restricted to the subspace, C := ( C ∈ C \ N X m =1 ¯ F nm C ( m ) = 0 ) ⊂ C . (38) Theorem 3
The vectors K = K q + K v ∈ ker Ω L are given by, K q = C , K v = G + b C , (39) where, C ∈ C , G ∈ G , and b C ∈ G ⊥ is the unique solution of M ab b C b = − F ab C b . Proof.
The horizontal component, K q = C , of K satisfies Eq. (35), and is a solution ofEq. (31). As G ∈ ker Ω q♭M , the general solution of Eq. (31) is K v = G + b C , where b C ∈ G ⊥ .With M ( α )( β ) := z a ( α ) M ab z a ( β ) for α, β = 1 , . . . , D , and with the choice that z ( α ) ∈ ker M ab ( u ) for α = 1 , . . . , N , the components of b C and M satisfy b C ( n ) = 0 , and M ( n )( e ) = M ( f )( m ) = 0 ,when n, m = 1 , . . . , N . Thus M = M , M ( f )( e ) := M ( f )( e ) , (40)14 or e, f = N + 1 , . . . , D , and M is nonsingular. In this basis Eq. (31) becomes M b C = R ,where R ( f ) := − U va ( f ) F ab C b , and R ( n ) = 0 . The nonvanishing components of b C and R arevectors with ( D − N ) -components that satisfy M b C = R. (41) Thus there is a unique solution for b C that belongs to G ⊥ . The constant rank assumption for M ab ( u ) together with the definition of G shows thatthere are N free choices for G at each u ∈ P L . According to Theorem 3 the b C -term isuniquely specified by the choice of C ∈ C . In general, ¯ F is not determined by M ab , and thusthe constant rank of M ab does not guarantee that the rank of ¯ F is constant. We then find dim (ker Ω L ( u )) = N + ¯ D, where ¯ D := dim C ≤ N . (42)The results of Theorem 3 are more general than we need. Because its proof is local andalgebraic, even though our focus is on Ω L with constant rank the theorem would neverthelessstill hold if the rank was not. It would only have to be applied to each region of P L on whichthe rank of Ω L is constant, resulting in a ¯ D that takes on different values on the Lagrangianphase space. C. Projection of K to ker Ω L ( u ) Consider a region U Sol ∈ M (2) on which solutions of the Euler-Lagrange equations ofmotion exist, and a point ( t, q, ˙ q, ¨ q ) ∈ U Sol . Then under the isomorphism ( t, q, ˙ q, ¨ q ) → ( t, q, v, X vaL ) , K → K ′ ⊂ T u P L , with a k ∈ K mapped into a k ′ ∈ K ′ where k ′ = ρ · ∂∂ q + ˙ ρ · ∂∂ v . (43)Now ˙ ρ = L X L ρ , and L is the Lie derivative. From Lemma , K ′ ⊆ ker Ω L ( u ) . But since dim K = 2 N , while dim ker Ω L ( u ) = N + ¯ D ≤ N , it follows that K ′ = ker Ω L ( u ) .Although this conclusion can only be reached on U Sol , the rank of Ω L is constant on P L ,and thus ¯ D = N on all of P L . As such C = C , and for any K ∈ ker Ω L ( u ) , K = C + ˙C + G where now ˙ C = ˙ C a ∂ / ∂ v a . (This expression for K can also be established directly usingEq. (31) .) This result is also new. 15 . First-order Lagrangian constraints For singular Lagrangians solutions of the energy equation X E are not unique. They alsodo not, in general, exist throughout P L , but are rather confined to a submanifold of thespace given by Lagrangian constraints. The first-order constraints, those that come directlyfrom the energy equation, are the focus of this section. Although most of this analysis isdone for a SOLVF, we show later that our results do not depend on this choice.With X L = X qL + X vL , the energy equation can be expressed in terms of a one-form Ψ as Ω q♭M X vL = Ψ . (44)The existence condition for solutions to Eq. (44) is given again by Theorem 2 by identifying A = Ω q♭M , F = [ T u P L ] v , E ∗ = [ T ∗ u P L ] q , and E = [ T u P L ] q . Combining Eq. (34) with the actionof Ω q♭M on vector fields yields A = C ; consequently Eq. (33) requires that h Ψ | C i = 0 ∀ C ∈ C or, after using the basis n U q ( n ) o of C , that γ [1] n := D Ψ | U q ( n ) E = 0 for n = 1 , . . . , N . Interms of local coordinates, γ [1] n = U qa ( n ) (cid:18) ∂E∂q a + F ab v b (cid:19) . (45)The condition γ [1] n = 0 imposes relations on the coordinates q and v , and defines a set ofsubmanifolds of P L . These γ [1] n are called the first-order constraint functions . (Becausethey are obtained through the energy equation, they are also called dynamical constraintsin the literature [11, 16].)While the number of first-order constraint functions in C [1] L := n γ [1]1 , . . . , γ [1] N o is equalto the dimension of C , these functions need not be mutually independent. Let I [1] be thenumber of independent functions in C [1] L . Then I [1] = rank n d γ [1] n o ≤ N , and P [1] L := n u ∈ P L \ γ [1] n ( u ) = 0 , n = 1 , . . . , N o —called the first-order Lagrangian constraintsubmanifold —has dim P [1] L = 2 D − I [1] . We will assume that P [1] L is not empty, i.e. thatthe first-order constraint functions are consistent. Otherwise, there is no SOLVF, and theintegral flows that give the evolution of the dynamical system would not exist.While the energy equation is usually written as d E = i X E Ω L , in doing so we haveimplicitly restricted ourselves to P [1] L . This is too restrictive for our purposes, and in thispaper we introduce the constraint one-form β [ X E ] := d E − i X E Ω L , (46)16see also the approach in [16]). The condition β [ X E ] = 0 then gives both solutions of theenergy equation and the submanifold P [1] L . Furthermore, as i U q ( n ) β = γ [1] n , * β [ X E ] − N X m =1 γ [1] m Θ ( m ) q ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) U q ( n ) + = 0 , n = 1 , . . . , N , (47)and β [ X E ] = N X n =1 γ [1] n Θ ( n ) q + ϑ , (48)where ϑ ∈ T ∗ u P L such that h ϑ | C i = 0 for all C ∈ C . But as β [ X E ] = 0 on P [1] L , we maychoose ϑ = 0 .From Eq. (46) , β [ X E + K ] = β [ X E ] , and the construction of γ [1] n does not depend on ouruse of X L . E. The Generalized Lie Symmetry Group
Our construction of the generalized Lie symmetry group for O ( u ) is guided by the threeconditions listed in Section II , and makes use of the projection of K to ker Ω L ( u ) in SectionIII C . It begins with the vector space,ker Ω L ( u ) := { P ∈ ker Ω L ( u ) \ [ G , P ] ∈ [ T u P L ] v ∀ G ∈ G} , (49)along with the following collection of functions on P L , F := { f ∈ C ∞ on P L \ G f = 0 ∀ G ∈ G} . (50)The following result will be used a number of times in our analysis. Lemma 4
Let X ∈ T u P L and G ∈ G such that [ G , X ] ∈ ker Ω L ( u ) . Then [ G , X ] ∈ [ T u P L ] v iff [ G , X ] ∈ G . Proof.
Since [ G , X ] ∈ ker Ω L ( u ) , from Theorem 3 there exists a C ∈ C and G ′ ∈ G suchthat [ G , X ] = C + b C + G ′ , and we see that [ G , X ] ∈ [ T u P L ] v iff C = 0 . Then b C = 0 , and [ G , X ] ∈ G . It then follows that [ G , P ] ∈ G for all P ∈ ker Ω L ( u ) . As G is involutive and as G ⊂ ker Ω L ( u ) , G ⊂ ker Ω L ( u ) as well, and thus G is an ideal of ker Ω L ( u ) .17 emma 5 There exists a choice of basis for ker Ω L ( u ) that is also a basis of ker Ω L ( u ) . Proof.
Given a basis { G ( n ) } of G , choose a set { K ( n ) } such that { K ( n ) , G ( n ) , n = 1 , . . . , N } form a basis of ker Ω L ( u ) . These { G ( n ) } are also a basis of ker Ω L ( u ) . To show that { K ( n ) } can be chosen to complete this basis, express K ( n ) = C ( n ) + b C ( n ) + G ( n ) . Then as [ G ( m ) , K ( n ) ] = [ G ( m ) , C ( n ) ] + [ G ( m ) , b C ( n ) + G ( n ) ] , we need only show that there exists a choiceof { C ( n ) } such that [ G ( m ) , C ( n ) ] ∈ [ T u P L ] v . This we do by construction.Because ker Ω L ( u ) is involutive, [ G ( m ) , K ( n ) ] ∈ ker Ω L ( u ) , and there exist functions λ lmn on P L such that T τ Q [ G ( m ) , C ( n ) ] = X l =0 λ lmn C ( l ) . (51) Let { C ( n ) , n = 1 , . . . , N } be another choice of basis of C where C ( n ) = N X m =1 ω mn C ( m ) . (52) Requiring T τ Q [ G ( m ) , C ( n ) ] = 0 in turn requires that ω mn be a solution of G ( l ) ω mn + N X k =1 ω kn λ mlk = 0 . (53) This is a linear, first-order Cauchy problem [48]. A solution exists for a given set of boundaryconditions for ω mn given on a surface S as long as G ( l ) is nowhere tangent to S [48]. As N The set of general solutions to the energy equation is S ol := { X E ∈ T u P L \ i X E Ω L = d E on P [1] L } . (58)19mportantly, while a SOLVF X L ∈ S ol, the majority of vectors in S ol are not SOLVFs. Thisis the root cause of the “second order problem” first raised by Künzle [15] (see also [6], [9],and [4]).If u ( t ) is the integral flow of a vector in S ol whose projection onto Q corresponds to atrajectory q ( t ) that is a solution of the Euler-Lagrange equations of motion, then Gr S ymmust map one of such flows into another. However, while L G X L = [ G , X L ] ∈ ker Ω L ( u ) ,in general L G X L / ∈ G . The action of σ P on the flow u X L will in general result in a flow u Y generated by a Y that is not a SOLVF. It need not even be a solution of the energyequation. By necessity, general solutions of the energy equation must be considered. Onlya specific subset of such solutions are physically relevant, however.As i [ X E , P ] Ω L = i P d β [ X E ] , (59)for P ∈ ker Ω L ( u ) and X E ∈ S ol, in general L P X E / ∈ ker Ω L ( u ) . The exception is when P ∈ S ym as well, which leads us to the subset of solutions S ol := { X EL ∈ S ol \ [ G , X EL ] ∈ [ T u P L ] v ∀ G ∈ G} . (60)Moreover, as i [ G , X EL ] Ω L = − L G β = 0 , from Lemma 4 [ G , X EL ] ∈ G . Lemma 6 [ X EL , P ] ∈ ker Ω L ( u ) for all P ∈ S ym. Proof. As P ∈ S ym, from Eq. (59) [ X EL , P ] ∈ ker Ω L ( u ) . Next, for each G ∈ G , there isa G X ∈ G such that G X = [ G , X EL ] . There is also a G P ∈ G such that G P = [ G , P ] . Itthen follows from the Jacobi identity that [[ X EL , P ] , G ] ∈ G , and [ X EL , P ] ∈ ker Ω L ( u ) . The vector fields in S ol generate the family of integral flows O EL ( u ) := (cid:26) u ( t ) \ d u dt = X EL ( u ) , X EL ∈ S ol , and u ( t ) = u (cid:27) . (61)The physical significance of these flows can be seen from the following theorem. Theorem 7 Gr S ym forms a group of symmetry transformations of O EL ( u ) . Proof. Let u X EL ( t, u ) ∈ O EL ( u ) be an integral flow generated by X EL , and let σ P ( ǫ, u ) ∈ Gr S ym be a one-parameter subgroup of Gr S ym generated by P ∈ S ym with σ P (0 , u ) = u .The action of σ P on u X E gives u Y ( t, u ) := σ P ( ǫ, u X EL ( t, u )) , while the choice ǫ = 0 when t = t ensures that u X EL and u Y have the same initial data. The tangent to this path is Y = σ ∗ P ◦ X EL ( σ − P ( ǫ, u X EL )) , (62)20 here σ ∗ P is the pullback map of σ P . As σ P is also a mapping of P L into itself, for a suitablysmall neighborhood about u X EL we may expand Y about ǫ = 0 in the Lie series, Y ( σ ( ǫ, u X EL ( t, u ))) = ∞ X n =0 ǫ n n ! L ( n ) P X E (cid:12)(cid:12)(cid:12) u X EL ( t, u ) . (63) However, from Lemma , L P X EL ∈ ker Ω L ( u ) , and ker Ω L ( u ) is involutive. Then Y ( σ P ( ǫ, u X EL ( t, u ))) = X EL ( u X EL ( t, u )) + ǫ Z ( ǫ, u X EL ( t, u )) , where Z ∈ ker Ω L ( u ) . Itthen follows that Y ∈ S ol, and u Y ( t, u ) ∈ O EL ( u ) . While X L / ∈ S ol, it is possible to construct from X L a vector field X L that is. Choose abasis (cid:8) P ( n ) , G ( n ) , n = 1 , . . . , N (cid:9) of ker Ω L ( u ) , and consider a vector field X L such that X L = X L + N X m =1 f m ( u ) P ( m ) + G , (64)where G ∈ G , and f m ( u ) are functions on P L . For X L ∈ S ol as well, we must have [ X L , G ( n ) ] ∈ G , and thus these functions must be solutions of G ( n ) f m ( u ) = − (cid:10) Θ ( m ) q (cid:12)(cid:12) [ X L , G ( n ) ] (cid:11) . (65)Once again, this is a linear Cauchy problem, and a solution exists with the appropriatechoice of boundary conditions and surfaces.If f m ( u ) is a solution to Eq. (65) , then f m ( u ) + u m ( u ) is as well as long as u m ( u ) ∈ F .This leads us to the second-order, Euler-Lagrange vector field (SOELVF), X EL = X L + N X m =1 u m ( u ) (cid:2) P ( m ) (cid:3) , (66)where { [ P ( n ) ] , n = 1 , . . . , N } is a choice of basis for ker Ω L ( u ) / G . By construction, X EL ∈S ol. Conversely, if Y EL ∈ S ol, then Y EL − X L ∈ ker Ω L ( u ) and Y EL is a SOELVF. Thus, Y EL ∈ S ol iff Y EL is a SOELVF. G. A constraint algorithm for second-order, Euler-Lagrange vector fields For most dynamical systems the flow fields in O EL ( u ) will not be confined to P [1] L , andyet this is the submanifold on which the solutions X EL ∈ S ol of the energy equations exist.21n these cases it is necessary to jointly choose a SOELVF X EL and a submanifold of P [1] L onwhich u X EL will be confined. Doing so requires that L X E β = 0 , (67)which is called the constraint condition . Implementing it involves imposing successiveconditions on X EL . At each step additional constraints may be introduced, giving a suc-cession of submanifolds of P [1] L . It is an iterative process that terminates either when u X EL is confined to the current submanifold under the current generator of time evolution, orwhen the possibility of dynamics on P L is exhausted. This process is called a constraintalgorithm, and has been introduced often in the literature. While such an algorithm willalso be presented here, its purpose is to show that the end result X EL of the algorithm isonce again a SOELVF, and a second-order problem is avoided. Later, it will also be used toshow that both this X EL and the Lagrangian constraints—whether first-order or introducedby the algorithm—are projectable.To present the constraint algorithm we introduce the following notation used in conjectionwith the constraint analysis X [1] EL := X EL , X [1] L := X L , P [1]( n ) := P ( n ) , u m [1] := u m , N [1]0 := N . (68)As both u n [1] , γ [1] n ∈ F , h P [1]( n ) i γ [1] m = P ( n ) γ [1] m . The constraint condition Eq. (67) requires L X E γ [1] n = 0 , which, after using Eq. (66) for a general SOELVF, reduces to N X m =1 Γ [1] nm u m [1] = − D d γ [1] n (cid:12)(cid:12)(cid:12) X [1] L E , with Γ [1] nm := D d γ [1] n (cid:12)(cid:12)(cid:12) P [1]( m ) E . (69)Then r [1] := rank Γ [1] nm of the u m [1] is determined by Eq. (69) , while N [2]0 := N [1]0 − r [1] are not.Moreover, N [2]0 second-order Lagrangian constraint functions γ [2] n [2] := D d γ [1] n [2] (cid:12)(cid:12)(cid:12) X [1] L E , n [2] = 1 , · · · , N [2]0 . (70)are introduced with the conditions γ [2] n [2] = 0 imposed. In general there will be I [2] := rank n d γ [1] n [1] , d γ [2] n [2] o independent functions in C [2] L := C [1] ∪ n γ [2] n [2] \ n [2] = 1 , . . . , N [2]0 o , and P [1] L is reduced to the second-order constraint submanifold , P [2] L := n u ∈ P [1] L \ γ [2][ n ] = 0 , n [2] = 1 , . . . , N [2]0 o , (71)22here dim P [2] L = 2 D − I [2] . At this point, there are two possibilities. If I [2] = I [1] or I [2] = 2 D ,the iterative process stops, and no new Lagrangian constraints are introduced. If not, theprocess continues.For the second step in the iterative process, we choose a basis n [ P [2]( n ) ] o for ker Ω L ( u ) / G and arbitrary functions n u m [2] o such that for m = 1 , . . . , N [2]0 , u m [2] are linear combinations of u m [1] that lie in the kernel Γ [1] nm . Then X [2] EL = X [2] L + N [2]0 X m =1 u m [2] h P [2]( m ) i , (72)with X [2] L = X [1] L + N [1]0 X m = N [2]0 +1 u m [2] h P [2]( m ) i . (73)Here, the functions u m [2] for m = N [2]0 + 1 , . . . , N [1]0 have been determined through the con-straint analysis of γ [1] n .Because for any G ∈ G , G i X [1] E d γ [1] n = L [ G , X [1] E ] γ [1] n = 0 and G Γ [1] nm = L [ G , P [1] m ] γ [1] n = 0 , itfollows that G u m [1] = 0 , as required. Similarly, G γ [2] n = L [ G , X EL ] d γ [2] n = 0 . Clearly γ [2] n ∈ F and we may require u m [2] ∈ F as well. It then follows that h P [2]( n ) i γ [2] m = P [2]( n ) γ [2] m and imposingEq. (67) on γ [2] n , gives N [2]0 X m =1 Γ [2] nm u m [2] = − D d γ [2] n (cid:12)(cid:12)(cid:12) X [2] L E , where Γ [2] nm := D d γ [2] n (cid:12)(cid:12)(cid:12) P [2]( m ) E , n = 1 , . . . , N [2]0 . (74)Then r [2] := rank Γ [2] nm , of the remaining u m [2] functions are determined, up to N [3]0 = N [2]0 − r [2] third-order Lagrangain constraint functions , γ [3] n [3] = D d γ [2] n [3] (cid:12)(cid:12)(cid:12) X [2] L E , n [3] = 1 , . . . , N [3]0 , (75)are introduced with the conditions γ [3] n [3] = 0 imposed. With I [3] := rank n d γ [1]( n [1] ) , d γ [2]( n [2] ) , d γ [3]( n [3] ) o , (76)independent functions in C [3] L := C [2] L ∪ n γ [3] n [3] , n [3] = 1 , . . . , N [3]0 o , we now have the third-order constraint submanifold , P [3] L := n u ∈ P [2] L \ γ [3] n [3] ( u ) = 0 , n [3] = 1 , . . . , N [3]0 o . (77)23nce again, the process stops when I [3] = I [2] or I [3] = 2 D . However, if I [2] < I [3] < D , theprocess continues until at the n F -step either I [ n F ] = I [ n F ] − or I [ n F ] = 2 D .The end result of this algorithm is1. A submanifold P [ n F ] L ⊂ P L on which dynamics takes place.2. A collection C [ n F ] L ⊂ F of constraint functions of order to n F .3. A second-order, Euler-Lagrange vector field X [ n F ] EL = X [ n F ] L + N [ nF ]0 X m =1 u m [ n F ] ( u ) h P [ n F ]( m ) i , (78)with N [ n F ]0 arbitrary functions u m [ n F ] ( u ) ∈ F for m = 1 , . . . , N [ n F ]0 , and X [ n F ] L = X [1] L + N [1]0 X m = N nF +1 u m [ n F ] ( u ) h P [ n F ]( m ) i , (79)where the N [1]0 − N [ n f ]0 functions u m [ n F ] ( u ) ∈ F , m = N [ n F ]0 + 1 , . . . , N [1]0 , have beenuniquely determined through the constraint algorithm.Importantly, the end result of the constraint algorithm X [ n F ] EL is still a SOELVF.As with the first-order constraint manifold P [1] L , we assume that P [ n F ] L is non-empty. Inaddition, we assume that the rank of Γ [ l ] nm is constant on P L for each l = 1 , . . . , n F . IV. THE PASSAGE TO HAMILTONIAN MECHANICS The question of whether and how dynamical structures on the Lagrangian phase spaceare equivalent to such structures on the Hamiltonian phase space has had a long history[9, 29] (see also [12, 17, 19, 20, 22, 38]). These analyses have focused solely on SOLVFs, andoften make use of pullbacks of structures on the Hamiltonian phase space in the constructionof such operators as the evolution operator K and the vector field operator R (see [12, 17,19, 20, 22, 38]) which are used to determine the projectability of Lagrangian constraintsand vector fields on T P L , respectively. However, while primary Hamiltonian constraintsplay a central role in the construction of both operators, the existence of such constraints ispresumed. Moreover, because of the reliance on primary constraints, a number of subtletiesinvolving first- and second-class Hamiltonian constraints must be dealt with.24hese subtleties and their conclusions, present for SOLVFs, are not present for SOELVFs.The approach used here focuses on the symmetry group Gr S ym, and the geometric struc-tures inherent to almost regular Lagrangians. The passage from Lagrangian to Hamiltonianmechanics follows naturally. Much of the content of this section is new. A. Projectability of functions and vector fields on P L The canonical phase space P C := T ∗ Q has the cotangent bundle coordinates s = ( q, p ) with q ∈ Q and p ∈ T ∗ q Q . The fiber derivative is the map L : ( q, v ) ∈ P L → ( q, p = ∂L/∂v ) ∈ P C . For regular Lagrangians, its action on a function f ( u ) ∈ C ∞ on P L gives thefunction f c ( s ) := ( f ◦ L − )( s ) = f ( L − ( s )) = f ( u ) | L ( u )= s , on P C . The action of L on a vectorfield X ∈ T u P L is given by the pushforward map L ∗ : T u P L → T L ( u ) P C , while its action ona one-form σ ∈ T ∗ s P C is given by the pullback map L ∗ : T ∗L ( u ) P C → T ∗ u P L .The situation changes for singular Lagrangians. While the pullback of one-forms simplyinvolves replacing s by L ( u ) , the action of both L and L ∗ involve solving for u in s = L ( u ) .For singular Lagrangians solutions of this equation are not single valued, but instead givesthe preimage of L , L − ( s ) := { u ∈ P L \ L ( u ) = s } , (80)which is a submanifold of P L . As such, the pullback of a function now results in the collectionof functions ( f ◦ L − )( s ) = (cid:8) f ( u ) \ u ∈ L − ( s ) (cid:9) , (81)while the pushforward L ∗ X of X gives the collection of vectors L ∗ X ( s ) = (cid:8) X C ( u ) \ u ∈ L − ( s ) (cid:9) . (82)This ambiguity for Eq. (81) can be avoided by focusing on functions that are constant on L − ( s ) . Then f ( u ) = f C ( s ) ∀ u ∈ L − ( s ) , so that ( f ◦ L − )( s ) = (cid:8) f ( u ) \ u ∈ L − ( s ) (cid:9) = f C ( s ) , (83)and is thus single valued. Following the literature, we say that a function f on P L is projectable when Eq. (83) holds. It is well known [2, 4, 6, 21, 24, 31] that the conditionfor a function f to be projectable is G f ( u ) = 0 , (84)25or all G ∈ G .For vector fields, the ambiguity Eq. (82) is avoided when the components of X C areconstant on the preimage. Then L ∗ X ( s ) = (cid:8) X C ( u ) \ u ∈ L − ( s ) (cid:9) = X C ( s ) , (85)and we say a vector field on T u P L is projectable when Eq. (85) holds. To determine whichvectors are projectable, consider first the collection of vectors in T u P L for which G is anideal, T u P L := { X ∈ T u P L \ [ X , G ] ∈ G ∀ G ∈ G} , (86)Applying the same arguments using Lemma 5 to T u P L as was applied to ker Ω L ( u ) givessimilar results: dim T u P L = 2 D , and T u P L is involutive. The equivalence relation X ∼ X iff X − X ∈ G then follows along with the quotient space T u P L / G . Theorem 8 T u P L / G is projectable. Proof. Choose an open covering U of P L , and a point u in an open neighborhood U u ∈ U such that G u A = 0 , A = 1 , . . . , D , for all G ∈ G . Choose also a X ∈ T u P L / G , and considerthe path u X ( t, u ) given by d u X dt = X ( u X ) , (87) with u X (0 , u ) = u . This open neighborhood can always be chosen small enough such that, u A X ( t, u ) = e t X u A , (88) on U u . Then as G is an ideal of T u P L , e − t X G e t X ∈ G , and G u A X ( t, u ) = 0 in U u . By applyingEq. (88) to a sequence of such open neighborhoods, we can extend this result to any connectedregion R of P L . Importantly, as the path u X ( t, u ) is projectable on R , there is the path s X ( t, L ( u )) = L ( u X ( t, u )) on P C with tangent vector X and initial data s X (0) = L ( u ) . Theintegral flow u X ( t, u ) is unique for a given X and u . Similarly, the integral flow s X ( t, L ( u )) is unique for a given X and initial data s = L ( u ) . As the projection of u X ( t ) to s X ( t, L ( u )) is also unique, we conclude that each X ∈ T u P L / G is projectable with X = L ∗ X . (A coordinate-based proof using Eq. (84) can also be given.) The converse is also true, aswe show in the next section. 26 . Projection of dynamical structures By construction, F is projectable, and as both u m ( u ) and γ ∈ F for any γ ∈ C [ n F ] L ,they also are projectable. In addition, G E = 0 , and E is projectable with its image H C =( E ◦ L − )( s ) being the canonical Hamiltonian. With the exception of the energy, we avoidintroducing new notation, and will represent the projection of any function f ( u ) ∈ F throughits argument: f ( s ) .Both S ol and ker Ω L ( u ) / G are subsets of T u P L / G , and are projectable. Of particularinterest arePrim := L ∗ (cid:16) ker Ω L ( u ) (cid:17) = { P ∈ T s P C \ P = L ∗ [ P ] , ∀ [ P ] ∈ ker Ω L ( u ) / G} , (89)and Flow H T := L ∗ (cid:0) S ol (cid:1) = (cid:8) X H T ∈ T s = L ( u ) P C \ X H T = L ∗ X EL ∀ X EL ∈ S ol (cid:9) . (90)In particular, the general X EL in Eq. (66) gives the general vector field X H T = X qaL ( s ) ∂∂ q a + (cid:2) X qaL N ab (cid:3) (cid:12)(cid:12) s ∂∂ p b − ∂H C ∂q b ∂∂ p b + N X m =1 u m ( s ) P ( m ) , (91)in Flow H T when expressed in terms of local coordinates. Here, N ab = ∂ L∂v a ∂q b , (92)and P ( m ) = L ∗ [ P ( m ) ] for a choice { [ P ( m ) ] , m = 1 , . . . , N } of basis for ker Ω L ( u ) / G .With the canonical two-form ω = d q a ∧ d p a on T ∗ s P C , we have the collection of one-forms,Prim ♭ := (cid:8) π ∈ Λ ( L ( P L )) \ π = ω ♭ P ∀ P ∈ Prim (cid:9) . (93)which gives the primary constraints, andFlow ♭H T := (cid:8) α ∈ Λ ( L ( P L )) \ α = ω ♭ X H T ∀ X H T ∈ Flow H T (cid:9) , (94)which gives the set of total Hamiltonians. Prim and the Primary Hamiltonian Constraints Using the kernel of the pullback map,ker L ∗ := { φ ∈ Λ ( P C ) \ L ∗ φ = 0 } , (95)in this section we construct from ker Ω L ( u ) / G the primary Hamiltonian constraints.27 emma 9 For any one-form σ ∈ Λ ( L ( P L )) , σ ∈ ker L ∗ iff σ ∈ Prim ♭ . Proof. Suppose first that σ ∈ Prim ♭ . Then there exists a [ P ] ∈ ker Ω L ( u ) / G such that σ = i L ∗ [ P ] ω . As L ∗ ω = Ω L , L ∗ σ = i P Ω L = 0 , and it follows that σ ∈ ker L ∗ .Next suppose that σ ∈ ker L ∗ . Let X be the unique vector in T s P C such that i X ω = σ .Then L ∗ [ i X ω ] = 0 . But as both i X ω and ω are differential forms, their pullbacks are well-defined and there must then be a X ∈ T u P L such that L ∗ X = X . It then follows that i X Ω L = 0 , and thus σ ∈ Prim ♭ . (A coordinate-based proof of this lemma can also be given.)Consider now the Pfaff system of exterior equations, P f ( Prim ♭ ) := (cid:8) π ( n ) = 0 , n = 1 , . . . , N (cid:9) . (96)and the integral manifold ( P L , L ) of P f ( Prim ♭ ) [49]. As N = dim (cid:16) ker Ω L ( u ) / G (cid:17) = dim Prim = dim Prim ♭ , rank P f ( Prim ♭ ) = N . Of particular interest is the ideal [49] of P f ( Prim ♭ ) I [ P f ( Prim ♭ )] := ( N X n =1 ξ n ∧ π ( n ) \ ξ n ∈ Λ ( P C ) , π ( n ) ∈ P f ( Prim ♭ ) ) . (97) Lemma 10 I [ P f ( Prim ♭ )] = ker L ∗ . Proof. If σ ∈ I [ P f ( Prim ♭ )] , then σ = N X n =1 ξ n ∧ π ( n ) . (98) From Lemma 9 , L ∗ σ = N X n =1 L ∗ ξ n ∧ L ∗ π ( n ) = 0 , (99) so that I [ P f ( Prim ♭ )] ⊆ ker L ∗ .Next, choose a basis θ ( n ) , n = 1 , . . . , D of Λ ( L ( P L )) such that θ ( n ) = π ( n ) for n =1 , . . . , N . Let σ ∈ ker L ∗ be the p -form, σ ( s ) := 1 p ! D X n ,...n p =1 σ n ...n p ( s ) θ ( n ) ∧ · · · ∧ θ ( n p ) . (100) Then as L ∗ σ = 0 , p ! D X n ,...,n p =1 σ n ...n p ( L ( u )) L ∗ θ ( n ) ∧ · · · ∧ L ∗ θ ( n p ) , (101)28 nd from Lemma 9 we conclude that σ n ...n p ( L ( u )) = 0 for n s > N , s = 1 , . . . , p . Thusthere exists forms ξ ( n ) such that σ = N X n =1 ξ ( n ) ∧ π ( n ) , (102) so that ker L ∗ ⊆ I [ P f ( Prim ♭ )] as well. The construction of the primary Hamiltonian constraints is now trivial.Consider a π ∈ P f ( Prim ♭ ) . As L ∗ dπ = d L ∗ π = 0 , dπ ∈ ker L ∗ , and from Lemma 10 , dπ ∈ I [ P f ( Prim ♭ )] . There are then one-forms ξ ( n ) , n = 1 , . . . , N such that dπ = N X n =1 ξ ( n ) ∧ π ( n ) . (103)Then dπ ∧ π (1) ∧ · · · ∧ π ( N ) = 0 , and thus P f ( Prim ♭ ) is closed [49]. It follows fromthe Frobenius theorem that P f ( Prim ♭ ) is completely integrable. There are then N firstintegrals γ [0] n of P f ( Prim ♭ ) such that in a neighborhood about each generic point u ∈ P C , { π ( n ) = 0 } ∼ { d γ [0] n = 0 } ; these forms may be chosen such that π ( n ) = f n ( s ) d γ [0] n where f n is a C ∞ function on P C . The functions γ [0] n are the primary Hamiltonian constraints while P [0] C := (cid:8) s ∈ P C \ γ [0] n ( s ) = 0 , n = 1 , · · · , N (cid:9) , (104)is the primary constraint submanifold . Connections between the primary constraintsand vectors in ker Ω L ( u ) have been found previously by using the time-evolution operator K [24]. Such analyses make use of pullbacks of the primary Hamiltonian constraints, however,while the approach here is constructive. Lemma 11 L ∗ (cid:0) T u P L / G (cid:1) = T s = L ( u ) P [0] C . Proof. Let [ X ] ∈ T u P L / G . As [ X ] is projectable, h d γ [0] n |L ∗ [ X ] i = hL ∗ d γ [0] n | [ X ] i = 0 since d γ [0] n ∈ ker L ∗ , and it follows that L ∗ (cid:0) T u P L / G (cid:1) ⊆ T s = L ( u ) P [0] C . But as dim T u P L / G ] =2 D − N = dim T s = L ( u ) P [0] C , L ∗ (cid:0) T u P L / G (cid:1) = T s = L ( u ) P [0] C follows. The converse of Theorem 8 then follows. Importantly, because L ∗ (cid:0) S ol (cid:1) ⊂ T s = L ( u ) P [0] C , theintegral flow fields of SOELVFs lie on P [0] C . 29 . S ol and the total Hamiltonian On P [1] L , β [ X E ] = 0 , and the energy equation may be written as d E − i X L L ∗ ω . Itfollows that i L ∗ X L ω = d H C , (105)from which we conclude that if X C is the Hamiltonian flow field for H C , then X C = L ∗ X L .The image of the pushforward of Eq. (66) gives the vector field X H T := L ∗ X EL ∈ Flow H T , X H T = X C + N X m =1 u m ( s ) P ( m ) , (106)that is everywhere tangent to P [0] C . Correspondingly, a general one form in Flow ♭H T is i X HT ω = d H C + N X m =1 u m ( s ) f m ( s ) dγ [0] m , (107)which gives the total Hamiltonian, H T = H C + N X m =1 u m ( s ) f n ( s ) γ [0] m , (108)for the dynamical system. This leads to the sequence of maps: S ol L ∗ −−−−−−→ Flow H T ω ♭ −−−−−−→ Flow ♭H T L ∗ −−−−−−→ E, (109)and to each X EL ∈ S ol there is a corresponding total Hamiltonian H T ∈ Flow ♭H T . 3. The Equivalence of the Constraint Algorithm for Lagrangians and the Stability Analysis ofCanonical Hamiltonians It is well known that the integral flow generated by X H T need not be confined to P [0] C eventhough its initial data is chosen to be on this submanifold. This difficulty is resolved througha stability analysis [26] where { H T , γ [0] n } = 0 is imposed on the primary constraints, andwhen necessary, successively on the secondary, tertiary, and higher-level Hamiltonian con-straints. While this process is traditionally applied to the canonical Hamiltonian, SectionIII G describes a constraint algorithm for SOELVFs. We show here that this constraintalgorithm is equivalent to the stability analysis of the canonical Hamiltonian.30hoose a X [1] EL ∈ S ol, where we follow the notation established in Eq. (68) . There is thena corresponding X [1] H T = L ∗ X [1] EL , and total Hamiltonian H [1] T . The stability analysis of theprimary constraints under H [1] T then results in f n dγ [0] n dt = (cid:10) π ( n ) | X C (cid:11) + N X m =1 u m (cid:10) π ( n ) | P ( m ) (cid:11) , (110)after using Eq. (106) . But (cid:10) π ( n ) | P ( m ) (cid:11) = h ω |L ∗ ( P ( n ) ) ⊗ L ∗ ( P ( m ) ) i = h Ω L | P ( n ) ⊗ P ( m ) i = 0 ,while h π ( n ) | X C i = h ω |L ∗ ( P ( n ) ) ⊗ L ∗ (cid:0) X L (cid:1) i = h Ω L | P ( n ) ⊗ X L i = −h d E | P ( n ) i since we are onthe β [ X E ] = 0 surface. As h d E | P ( n ) i = γ [1] n , f n dγ [0] n dt = − γ [1] n ( s ) . (111)The projection of first-order constraints automatically gives the secondary Hamiltonian con-straints. It follows that { H T , γ [0] n } = 0 is automatically satisfied through the Lagrangianconstraint condition γ [1] n ( u ) = 0 .The stability analysis must now be applied to the secondary constraints: L X [1] HT γ [1] n ( s ) = 0 .But as X [1] H T = L ∗ X [1] EL , this requirement is equivalent to imposing the constraint condition: L X [1] EL γ [1] n ( u ) = 0 . From Section III G doing so results in the SOELVF X [2] EL , and thus givesa corresponding Hamiltonian flow field X [2] H T and total Hamiltonian H [2] T . If second-orderLagrangian constraints are introduced at this step, their projection will give the tertiaryHamiltonian constraints.This procession continues with the stability analysis of the n th -level Hamiltonian con-straints giving a X [ n ] EL , and thus a corresponding Hamiltonian flow field X [ n ] H T = L ∗ X [ n ] EL andtotal Hamiltonian H [ n ] T . If ( n + 1) th -level Hamiltonian constraints are introduced, they arethe projection of the n th -order Lagrangian constraints. The analysis stops when the La-grangian constraint algorithm ends: at the n F -step. The end result X [ n F ] EL of the constraintalgorithm gives a X [ n F ] H T with integral flows that lie on the Hamiltonian constraint subman-ifolds. Correspondingly, there is a H [ n F ] T that agrees with the end result of the stabilityanalysis of the total Hamiltonian. The Lagrangian constraint algorithm applied to X EL isthus equivalent to the stability analysis of the canonical Hamiltonian.31 . EXAMPLES OF ALMOST REGULAR LAGRANGIANS In this section we present three examples of dynamical systems with almost regular La-grangians. The first example describes a single particle interacting with an external poten-tial. It illustrates the role G plays, and the tight relationship between Gr S ym, the symmetriesof the Euler-Lagrange equations of motion, and the gauge symmetries of the Lagrangian.Moreover, it explicitly shows that G is not the generator of the local gauge symmetry, asis sometimes asserted in the literature. The second example consists of two interactingparticles with a Lagrangian that has a local conformal symmetry. It is an example of adynamical system for which only a subset of vectors in ker Ω L ( u ) generate the symmetrygroup. The third example consists of a particle with both local conformal symmetry andtime-reparametization invariance. It is an example of a fully constrained dynamical sys-tem—as such, S ol = ker Ω L ( u ) —that has two gauge symmetries. The analysis of all threesystems are done using the techniques and tools presented above. A. A Lagrangian With and Without a Local Gauge Symmetry Whether the action S := Z " m (cid:18) d b qdt (cid:19) − V ( q a ) dt, (112)with | q | = √ q a q a and b q a := q a / | q | , a = 1 , . . . , D , has a local gauge symmetry depends on thechoice of potential V ( q ) . With one choice both the Lagrangian and the equations of motionhave a local gauge symmetry; with another choice the equations of motion has a symmetrywhile the Lagrangian does not have a local gauge symmetry; and with a third choice, neitherhas a symmetry. Interestingly, L is singular irrespective the choice of V ( q ) , showing thatnot all singular Lagrangians need have a symmetry.With Π ab ( q ) := δ ab − b q a b q b , Ω M = m | q | Π ab ( q ) d q a ∧ d v b , Ω F = m | q | ( b q · d q ) ∧ ( v · Π( q ) · d q ) , (113)and C and G are spanned by U q (1) = b q · ∂ / ∂ q and U v (1) = b q · ∂ / ∂ v , respectively, whileker Ω L ( u ) is spanned by U q (1) and P (1) = b q · ∂∂ q + 1 | q | v · ∂∂ v . (114)32he energy is E = 12 m | q | v · Π( q ) · v + V ( q ) , (115)and there is only one first-order, Lagrangian constraint, γ [1] = U q (1) V, (116)with β [ X E ] = γ [1] Θ (1) q , Θ (1) q = b q · d q . As expected, L G γ [1] = 0 .We may choose X L = v · ∂∂ q + 2 ( b q · v ) | q | v · ∂∂ v − | q | m ∂V∂q · ∂∂ v . (117)As [ X L , U v (1) ] ∼ − P (1) , a symmetry transformation of X L does not result in a SOLVF.Instead, X L = v · Π( q ) · ∂∂ q + ( b q · v ) | q | v · ∂ ∂v − | q | m ∂V∂q · Π( q ) · ∂∂ v , (118)is constructed, and a general SOELVF is X EL = X L + u ( u ) (cid:2) P (1) (cid:3) , where u ( u ) ∈ F . Because L P (1) β = d h U q (1) V i − | q | b q · ∂∂q (cid:18) Π ba ( q ) ∂V∂ b q b (cid:19) d q a , (119)whether or not S ym is empty depends on the symmetries of V ( q ) . As the constraint algo-rithm gives L X EL γ [1] = v · Π · ∂γ [1] ∂q + u ( u ) U q (1) γ [1] , (120)whether or not u ( u ) is determined also depends on the symmetries of V ( q ) . There arethree possibilities, none of which will require the introduction of higher-order Lagrangianconstraints. The symmetric potential For P (1) to generate a symmetry, | q | b q · ∂∂q (cid:18) Π ba ( q ) ∂V∂ b q b (cid:19) , (121)and it follows that ∂V∂ b q a = ∂V AS ( b q a ) ∂ b q a , (122)where V AS is a function of b q a only. Then P (1) generates a symmetry iff V ( q, b q q ) = V Sph ( q ) + V AS ( b q a ) , where V Sph is a function of q only. The group S ym is one-dimensional, and spannedby P (1) . 33he constraint condition Eq. (120) for this potential reduces to u ( u ) d V Sph ( q ) d | q | , (123)which must be satisfied on P L . There are two cases: Case 1: d V Sph d | q | = 0 .Then V Sph = aq + b , but since γ [1] = dV Sph d | q | = a, (124)the condition γ [1] = 0 requires a = 0 . As we may choose b = 0 , V ( q ) = V AS ( b q a ) only.The Lagrangian is invariant under the local conformal transformation q a → αq a , where α is an arbitrary, nonvanishing function on P L . The function u ( u ) is not determined, andcorrespondingly, the dynamics of the particle is determined only up to an arbitrary function. Case 2: d V Sph d | q | = 0 .In this case u ( u ) = 0 , and the dynamics of the particle is completely determined by its initialdata. The Lagrangian does not have a local gauge symmetry. The first-order, Lagrangianconstraint γ [1] = 0 defines a surface on P L , and for dynamics to be possible the set ofsolutions ( R i ∈ R \ dV Sph d | q | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R i = 0 ) , (125)must be non-empty. Dynamics are on the surfaces | q | − R i = 0 , and on them the potentialreduces to V ( q ) = V Sph ( R i ) + V AS ( b q a ) . This reduced potential has the same symmetry as thepotential V AS ( b q a ) in Case 1 , leading to equations of motion that have the same generalizedLie symmetry. The Lagrangian for the two cases, however, do not have the same invariances,resulting in one case to dynamics that are determined up to an arbitrary u ( u ) while in theother case to u ( u ) = 0 and dynamics that are completely determined by the choice of initialdata.A specific example of this type of potential is the Mexican hat potential: V ( q ) = − λ | q | / β | q | / . Then γ [1] = − λ | q | + β | q | . (126)As | q | 6 = 0 , dynamics are thus on the surface | q | = ( β/λ ) / for β/λ > . This breaks thelocal conformal symmetry while preserving rotational symmetry.34 he asymmetric potential For a general V , the second term in Eq. (119) does not vanish, P (1) does not generate asymmetry of the equations of motion, S ym = {∅} , and Eq. (120) gives u = − v · Π · ∂γ [1] ∂q U q (1) γ [1] . (127)The dynamics of the particle is uniquely determined by its initial data.The passage to Hamiltonian mechanics is straightforward. With the canonical momen-tum, p a = m Π ab ( q ) v b /q , Eq. (115) gives H C = q p / m + V ( q ) , while γ [1] does not changeunder L . The projection of P (1) is P (1) = 1 | q | (cid:18) q · ∂∂ q − p · ∂∂ p (cid:19) , (128)giving π = d ( q · p ) / | q | , and the primary constraint γ [0] = q · p .The projection of X L gives the Hamiltonian flow X C = | q | m p · ∂∂ q − | q | p m b q · ∂∂ p − ∂V∂q · Π · ∂∂ p , (129)and the total Hamiltonian H T = H C + uγ [0] ( s ) . The projection of Eq. (120) is | q | m p · ∂γ [1] ∂q + u U q (1) γ [1] . (130)For each of the three possible choices of V ( q ) outlined above the total Hamiltonian obtainedhere agrees with the one obtained using constrained Hamiltonian mechanics. B. A Lagrangian with Local Conformal Symmetry The action, S := Z ( m (cid:18) d b q dt (cid:19) + 12 m (cid:18) d b q dt (cid:19) + λ (cid:20) q a q ddt (cid:18) q a | q | (cid:19) − q a q ddt (cid:18) q a | q | (cid:19)(cid:21) ) dt, (131)where a = 1 , . . . , d , D = 2 d , describes an interacting, two particle system that is invariantunder the local conformal transformation q a → α ( u ) q a and q a → α ( u ) q a .With 35 M = m | q | Π ab ( q ) d q a ∧ d v b + m | q | Π ab ( q ) d q a ∧ d v b , and Ω F = m | q | ( b q · d q ) ∧ ( v · Π( q ) · d q ) + m | q | ( b q · d q ) ∧ ( v · Π( q ) · d q ) − λ | q || q | [ d q a ∧ (Π( q ) · d q ) a + (Π( q ) · d q ) a ∧ d q a − (Π( q ) · d q ) a ∧ (Π( q ) · d q ) a ] − λ | q | ( b q · d q ) ∧ ( b q · Π( q ) · d q ) + λ | q | ( b q · d q ) ∧ ( b q · Π( q ) · d q ) , (132) C and G are two-dimensional and are spanned by U q (1) = b q · ∂∂ q , U q (2) = b q · ∂∂ q , and U v (1) = b q · ∂∂ v , U v (2) = b q · ∂∂ v , (133)respectively. The reduced ¯ F = 0 , and ker Ω L ( u ) is spanned by U v (1) , U v (2) , and P (1 , = b q , · ∂∂ q , + v , | q , | · Π( q , ) · ∂∂ v , +( − , λm (cid:20)b q , · Π( q , ) · ∂∂ v , + | q , || q , | b q , · Π( q , ) · ∂∂ v , (cid:21) , (134)The energy is E = 12 m | q | v · Π( q ) · v + 12 m | q | v · Π( q ) · v . (135)Although C is two-dimensional, γ [1](1) = − λγ [1] / | q | and γ [1](2) = λγ [1] / | q | , and the two first-order Lagrangian constraints reduce to one γ [1] = b q · Π( q ) · v | q | + b q · Π( q ) · v | q | , (136)with β [ X E ] = − λγ [1] Θ (1) q | q | − Θ (2) q | q | ! . (137)As expected, G γ [1] = 0 for any G ∈ G . We may choose X L = v · ∂∂ q + v · ∂∂ q + (cid:20) b q · v ) | q | v + λm (cid:18) | q || q | v − ( b q · v ) b q (cid:19)(cid:21) · ∂∂ v + (cid:20) b q · v ) | q | v − λm (cid:18) | q || q | v − ( b q · v ) b q (cid:19)(cid:21) · ∂∂ v . (138)As [ X L , U v (1 , ] ∼ − P (1 , /q , the action on X L by Gr S ym does not give a SOLVF. Instead,we construct X L = v · Π( q ) · ∂∂ q + v · Π( q ) · ∂∂ q + (cid:18) b q · v | q | (cid:19) v · Π( q ) · ∂∂ v + (cid:18) b q · v | q | (cid:19) v · Π( q ) · ∂∂ v + λm (cid:18) | q || q | v · Π( q ) · Π( q ) · ∂∂ v − | q || q | v · Π( q ) · Π( q ) · ∂∂ v (cid:19) . (139)36 general SOELVF is then X EL = X L + u ( − ) ( u ) (cid:2) P ( − ) (cid:3) + u (+) ( u ) (cid:2) P (+) (cid:3) , where u ( ± ) ( u ) ∈ F and P ( ± ) = | q | P (1) ± | q | P (2) . One of the arbitrary functions u ( − ) ( u ) = m λ i X L d γ [1] [1 − ( b q · b q )] , (140)is determined through the constraint algorithm with i X L d γ [1] = − b q · b q ) Em + 2 | q || q | v · Π( q ) · Π( q ) · v − λm ( b q · b q ) [ v · Π( q ) · b q − v · Π( q ) · b q ] . (141)The other one, u (+) ( u ) , is not.We find that L P (+) β = 0 , while L P ( − ) β = − λm (cid:2) − ( b q · b q ) (cid:3) . (142)Then S ym is one-dimensional, and spanned by P (+) .With the canonical momenta, p a := m | q | Π ab ( q ) v b − λ τ ab ( q ) | q | b q b , p a := m | q | Π ab ( q ) v b + λ τ ab ( q ) | q | b q b , (143)where τ ab := δ ab + b q a b q b , the passage to Hamiltonian mechanics is straightforward. Equation (135) gives H C = | q | L / m + | q | L / m with L a := | q | p a + λ τ ac ( q ) b q c , L a := | q | p a − λ τ ac ( q ) b q c , (144)and the projection of the first-order Lagrangian constraint is γ [1] = [ b q · L + b q · L ] /m .The projection of P ( ± ) is P (+) = b q · ∂∂ q + b q · ∂∂ q − p · ∂∂ p − p · ∂∂ p , P ( − ) = b q · ∂∂ q − b q · ∂∂ q − p · ∂∂ p + p · ∂∂ p − λ (cid:20) b q | q | · Π( q ) · ∂∂ p + b q | q | · Π( q ) · ∂∂ p (cid:21) , (145)giving π (+) = d ( q · p + q · p ) , and π ( − ) = d ( q · p − q · p + λ b q · b q ) . The primaryHamiltonian constraints are γ [0](+) := q · p + q · p and γ [0]( − ) := q · p − q · p + λ b q · b q .The projection of X L gives 37 H C = | q | m L · ∂∂ q + | q | m L · ∂∂ q − λ m | q | [( b q · b q ) L − L ] · ∂∂ p − [ L + λ ( b q · L )] m | q | b q · ∂∂ p + λ m | q | [( b q · b q ) L − L ] · ∂∂ p − [ L + λ ( b q · L )] m | q | b q · ∂∂ p . (146)Then, H T = H C + u ( − ) γ [0]( − ) ( s ) + u (+) γ [0](+) ( s ) , where after using Eq. (143) in Eqs. (140) and (141) , u ( − ) = 12 λ (cid:20) L · L /m + λ ( b q · b q ) b q · L − ( b q · b q ) H C − ( b q · b q ) (cid:21) , (147)while u (+) remains undetermined. C. A Lagrangian with Local Conformal and Time-reparametization Invariance The action S := sm Z " s (cid:18) d b qdt (cid:19) / dt, (148)where s = ± , is invariant under both the local conformal transformations, q a → α ( u ) q a ,and the reparametization t → τ ( t ) , where τ is a monotonically increasing function of t (seealso [50] and [51, 52] for systems with Lagrangians linear in the velocities). This action isa generalization of that for the relativistic particle, with the additional requirement that ithave a local conformal invariance.For this action, Ω L = m | q | P ab ( u ) p sv · Π( q ) · v d q a ∧ d v b , (149)and Ω F = 0 . Here, a = 1 . . . . , D , u a = Π ab ( q ) v b p sv · Π( q ) · v , (150)so that u = s , while P ab ( u ) = Π ab ( q ) − su a u b . Then ker Ω L ( u ) = ker Ω M ( u ) , and both C and G are two-dimensional. They are spanned by U q (1) = b q · ∂∂ q , U q (2) = u · ∂∂ q , and U v (1) = b q · ∂∂ v , U v (2) = u · ∂∂ v , (151)respectively. 38ecause this system is fully constrained, E = 0 . As Ω F = 0 as well, there are noLagrangian constraints. We may choose X L = v · ∂ / ∂ q . As [ X L , U v (1 , ] ∼ − U q (1 , , theaction of X L by Gr S ym does not give a SOLVF. The vector field X L can be constructed,and as expected for a fully constrained system, X L = 0 . A general SOELVF is then X EL = u ( u ) h U q (1) i + u ( u ) h U q (2) i , with u n ( u ) ∈ F for n = 1 , . The Lie algebra S ym itself is twodimensional, and spanned by U q (1) and U q (2) .For the passage to Hamiltonian mechanics, H C = 0 as E = 0 . With the canonicalmomentum p a = mu a / | q | , the projection of P (1 , is P (1) = b q · ∂∂ q − | q | p · ∂∂ p , P (2) = qm p · ∂∂ q − p m c | q | · ∂∂ p , (152)and π (1) = d ( q · p ) / | q | , π (2) = d ( | q | p ) / m | q | . The primary Hamiltonian constraints are γ [0]1 := q · p and γ [0]2 := | q | p − sm . As X L = 0 , X C = 0 , and we find that H T = u (1) | q | γ [0]1 ( s ) + u (2) m | q | γ [0]2 ( s ) . (153) VI. CONCLUDING REMARKS With the benefit of hindsight, the many roles that G plays in determining both thegeometric structure of P L for singular Lagrangians, and the connection between these struc-tures and dynamics become readily apparent. What also becomes clear are the reasons whySOELVFs and their dynamical structures are projectable.Because G is involutive, it gives a foliation of P L . There is then a neighborhood U abouteach point u ∈ P L on which we can define the equivalence relation u ∼ u iff u − u = g ,where g is a point on the leaves F ol ( G ) of the foliation. This leads to the quotient space P L / F ol ( G ) , which has dimension D − N . Importantly, P L / F ol ( G ) is projectable, and L ( P L / F ol ( G )) = L ( P L ) = P [0] C . This structure, and the role that G plays in its construction,is well known in the literature [4, 6].Next, because G ⊂ T u P L and is involutive, it is natural to follow the construction of P L / F ol ( G ) and consider the set of vector fields in T u P L for which G is an ideal. This leadsus to T u P L , and the quotient space T u P L / G . As dim T u P L / G = 2 D − N , it is expected that T u P L / G = T p [ P L / F ol ( G )] for p ∈ P L / F ol ( G ) . That T u P L / G is projectable is then readilyapparent. 39inally, for singular Lagrangians the acceleration is not determined uniquely by the Euler-Lagrange equations of motion, an ambiguity due to the generalized Lie symmetry. Thissymmetry is generated by vectors that must lie in the kernel of Ω L , and yet cannot be in G , leading naturally first to the construction of ker Ω ( u ) / G , and then to the constructionof S ol. Both ker Ω ( u ) / G ⊂ T u P L / G and S ol ⊂ T u P L / G , and thus the evolution of thedynamical system is confined to the tangent bundle T [ P L / F ol ( G )] . The projectability of T [ P L / F ol ( G )] ensures that all of the dynamical structures needed to describe the evolutionof dynamical systems on the Lagrangian phase space is projectable, and agrees with thoseobtained through constrained Hamiltonian mechanics.While G does play an important role in determining the general Lie symmetry group, ititself is not the generator of this group. This can be readily seen in the first example in Section V where the Lagrangian may or may not have a local gauge symmetry dependingon the choice of potential. Nevertheless, G is present and plays its usual role in determin-ing P L / F ol ( G ) . It is instead vectors in ker Ω L ( u ) / G —with G removed—that generate thegeneralized Lie symmetry. We emphasize here that while this symmetry plays an importantand guiding role, this role is nevertheless supportive in the construction of the algebraic-geometric structures on P L needed in determining both the geometric structure of P L , andthe connection between these structures and the evolution of the dynamical system.The application of these algebraic-geometric structures go beyond showing the equiva-lence of the Lagrangian and Hamiltonian formulations of mechanics for singular Lagrangians,however. While the primary Hamiltonian constraints play a critical role in the Hamiltonianconstraint analysis, the constraints themselves have traditionally been found by inspection;the expectation is that this inspection is able to both determine their form and to ensurethat all of the constraints has been found for the system at hand. As a result of the La-grangian phase space analysis presented here we are able to determine the number of primaryconstraints for any dynamical system, and the constraints themselves can be calculated bysolving a first-order, quasi-linear differential equation. In addition, while the end result ofthe Lagrangian constraint algorithm is a SOELVF defined in terms of a certain numberof arbitrary functions, and the end result of the Hamiltonian constraint analysis is a totalHamiltonian with the same number of arbitrary functions, how many arbitrary functions areneeded, and their relationship to the original symmetries of the action is not known. Withdirect access to the Lagrangian and its symmetries, these questions can now be addressed40n the Lagrangian phase space formulation. ACKNOWLEDGMENTS This paper would not have been possible without the contributions by John Garrison,who provided most of the essential mathematics in Section III . 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