Constrained dynamics: generalized Lie symmetries, singular Lagrangians, and the passage to Hamiltonian mechanics

Guided by the symmetries of the Euler–Lagrange equations of motion, a study of the constrained dynamics of singular Lagrangians is presented. We find that these equations of motion admit a generalized Lie symmetry, and on the Lagrangian phase space the generators of this symmetry lie in the 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 always possible to construct from them a SOELVF that is. We find that all SOELVFs are projectable to the Hamiltonian phase space, as are all the dynamical structures in the Lagrangian phase space needed for their evolution. In particular, the primary Hamiltonian constraints can be constructed from vectors that lie in the kernel of the Lagrangian two-form, and with this construction, we show that the Lagrangian constraint algorithm for the SOELVF is equivalent to the stability analysis of the total Hamiltonian. Importantly, the end result of this stability analysis gives a Hamiltonian vector field that is the projection of the SOELVF obtained from the Lagrangian constraint algorithm. The Lagrangian and Hamiltonian formulations of mechanics for singular Lagrangians are in this way equivalent.


I. INTRODUCTION
The Lagrangian phase space formulation of mechanics [1][2][3][4], with its roots in differential geometry, provides an especially fruitful framework with which to analyze dynamical systems of singular Lagrangians L. Instead of trajectories q(t) = (q 1 (t), . . . , q D (t)) on a D-dimensional configuration space Q that are solutions of the Euler-Lagrange equations of motion, trajectories in the Lagrangian phase space formulation u(t) = (q 1 (t), . . . , q D (t), v 1 (t), . . . , v D (t)), are on a 2D-dimensional Lagrangian (or velocity) phase space P L = TQ embodied with a Lagrangian two-form Ω L . They are determined by vector fields X E ∈ T u P L that are solutions of the energy equation with E being the energy of the system. For regular Lagrangians, Ω L is symplectic. The solution to Eq. (2) is unique, and is a second-order, Lagrangian vector field (SOLVF) [1] (also called a second-order dynamical equation in the literature). For singular Lagrangians, on the other hand, Ω L is presymplectic [5]. The solution to Eq. (2) is not unique, need not be 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 physical reasons: the conditionq a = v a , for a = 1, . . . , D, immediately follows for trajectories determined by such fields. This focus on SOLVFs has consequences, however.
The presence of a singular Lagrangian often predicates the existence of a Lagrangian constraint submanifold of P L , and for solutions of Eq.
(2) to exist, trajectories of these dynamical systems must lie on this submanifold. Algorithms-called a constraint algorithm or a stability condition-for constructing such solutions have been developed [6,[8][9][10][11][12][13][14]. However, irrespective of the one used, the end result of these algorithms is a vector field that has a number of troubling attributes. First, this vector field need not be a SOLVF, even though physical 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 result of these algorithms be a SOLVF is very restrictive, and additional conditions may need to be imposed [16]. Second, the fibre derivative (Legendre transform) L for singular Lagrangians is singular, and thus the rank of the Hessian of L is not maximal. Because of this the passage from Lagrangian to Hamiltonian mechanics is problematic. The ability to map dynamical structures from the Lagrangian to the Hamiltonian phase space has long been studied for SOLVFs [4,8,11,12,[16][17][18][19][20][21][22][23][24]. It is found that a general SOLVF-even after the application of the constraint algorithm, and even under the weak projectability condition [11]-is not projectable [11,16]. (Examples of systems for which a SOLVF is projectable, and for which it is not are given in [11].) One immediate consequence of this non-projectability is dynamical systems for which the Hamiltonian flow field determined through constrained Hamiltonian mechanics as described in [25,26] (see [27,28] for more modern approaches)-even after its restriction to the primary constraint submanifold-need have little relation to the SOLVF obtained from the Lagrangian constraint algorithm. Third, it is known that the Lagrangian constraints obtained while a constraint algorithm is being imposed on a SOLVF also need not be projectable [4,11,16,18,19,22,29]. Combined, this means that dynamics on the Lagrangian phase space and dynamics on the Hamiltonian phase space can take place on two inequivalent submanifolds, be determined by two inequivalent vector fields, resulting in two different families of trajectories on the configuration space Q for the same dynamical system with the same initial data.
We take a different starting point in our analysis of singular Lagrangians, one that is rooted in the generalized Lie symmetries of the Euler-Lagrange equations of motion. The analysis is guided by the observation that if a symmetry of the dynamical system has been determined through Lagrangian mechanics, it must be present in the Lagrangian and Hamiltonian phase space descriptions of motion as well. We emphasize, however, that while these symmetries play an important role, this role is nevertheless supportive. One of the main goals of this work is the construction of algebraic-geometric structures within differential geometry that will then be used to implement these symmetries; to characterize the relevant geometry of the Lagrangian phase space; to determine the structures needed to decribe dynamics on this phase space; and to show the equivalence of the Lagrangian and Hamiltonian phase space 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, are projectable to the Hamiltonian phase space, and lie on Lagrangian constraint submanifolds that also are projectable. Importantly, the projection of the SOELVF is the Hamiltonian flow field of the total Hamiltonian obtained from constrained Hamiltonian mechanics. (We follow the terminology in [26], and call the result of augmenting the canonical Hamiltonian with the primary constraint functions the total Hamiltonian.) The generalized Lie symmetry [30] of the Euler-Lagrange equations of motion is generated by second-order prolongation vectors in the tangent space TM (2) of the second-order jet space M (2) . This symmetry is reflected in the Lagrangian phase space description of motion, and the projection of TM (2) to TP 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 Sym is not a symmetry group for SOLVFs; action on a SOLVF by Gr Sym results in a vector field that is no longer a SOLVF, nor need it even be a solution of Eq. (2). It is, however, always possible to construct from a SOLVF vector fields that do have Gr Sym as a symmetry group, and are solutions to Eq. (2). These vector fields are the SOELVFs, and they resolve the issues listed above for the SOLVF.
That a SOELVF is projectable is a natural consequence of having Gr Sym as a symmetry group. Moreover, all the Lagrangian constraints-both those due to the energy equation and those introduced through the application of the constraint algorithm to the SOELVF-are projectable as well. We find also that there is a choice of a basis for the kernel of Ω L that is projectable, and that the primary Hamiltonian constraints can be constructed from their image. Indeed, all the dynamical structures needed to describe evolution on the Lagrangian phase space are projectable, and their image corresponds to the dynamical structures needed to describe evolution on the Hamiltonian phase space obtained through constrained Hamiltonian mechanics [14,[25][26][27][28][31][32][33]. In this way the Lagrangian and Hamiltonian fomulations of mechanics are equivalent even for singular Lagrangians.
Importantly, the great majority of these analyses have been done using first-order prolongations on first-order jet bundles with a focus on the Lie symmetries of first-order evolution equations. Our interest is in the symmetries of the Euler-Lagrange equations of motion-a system of second-order differential equations-that come from singular Lagrangians. This naturally 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 before. (Although the framework for k th -order prolongations on k th -order jet bundles have been introduced before [10,40,53,54], they have not been applied to the Euler-Lagrange equations of motion.) In this paper we only consider autonomous Lagrangians for which the rank of Ω L is constant on P L . We also require such Lagrangians to have a fibre derivative that is a submersive map of P L to the Hamiltonian phase space P C ; the rank of the Hessian of such Lagrangians 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). These Lagrangians are called almost regular Lagrangians in the literature [2,4,8,16,24], and we also use this terminology.
Some of the results on the equivalence of singular Lagrangian and Hamiltonian mechanics presented in this paper have been presented elsewhere. However, the approaches used previously often rely on such dynamical structures from constrained Hamiltonian mechanics as the primary Hamiltonian constraints; pullbacks of their derivatives to the Lagrangian 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 the Lagrangian phase space formulation, and, with restrictions imposed by Gr Sym , is one which shows that the dynamical structures on the Hamiltonian phase space necessary to describe the dynamical system can be obtained directly from those on the Lagrangian phase space.
The importance of establishing the equivalence between the Lagrangian and Hamiltonian formulations of mechanics for singular Lagrangians can been seen from the starting point of any physically relevant system: the action, and through it, its Lagrangian. For systems with local gauge or diffeomorphism symmetry, the Lagrangian is singular, and dynamics have traditionally been analysed using Hamiltonian constraints [26]. This is done by first using the fibre derivative to construction the Hamiltonian from the Lagrangian, and then using Hamiltonian stability analysis to determine both the total Hamiltonian and the Hamiltonian constraint surfaces. However, as the Lagrangian was the starting point, and as the fibre derivative is not invertable for such Lagrangians, a natural question to ask is whether the dynamics described by the total Hamiltonian has any relation to the original dynamics given by the Lagrangian. With the equivalence between the Lagrangian and the Hamiltonian phase space formulations of dynamics demonstrated, we have shown that they are. This equivalence is even more important for the path integral formulation of quantum mechanics and quantum field theory. Both are based on the action, and integration is over paths on the configuration space. For systems with local gauge or diffeomorphism symmetries these integrals must be restricted, which for non-abelian gauge theories leads to the use of BRST symmetries. These symmetries have traditionally been constructed using the Hamiltonian and Hamiltonian constraint analysis (see [26]).
Although we freely use the tools and language of differential geometry, we are aware that interest 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. To ensure that the tools and methodologies given in this paper can be so applied to the analysis of quantum field-theoretic systems, we have also written a number of the expressions given in this paper in terms of local coordinates using a notation that is both familiar and useful for calculations. In particular, the general solution to i K Ω L = 0 given in Section III B is given in terms of local coordinates as are the construction of second-order Lagrangian and Euler-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 existence conditions for the generators of this symmetry. In Section III the vectors that lie in the kernel of Ω L are found, and the role they play in generating Gr Sym is determined.
Physically relevant solutions of the energy equation are characterized, and the SOELVF is defined and constructed. First-order Lagrangian constraints are also constructed, and a constraint algorithm for SOELVFs is presented. In Section IV focus is on the passage from 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 TP L is presented.
The dynamical structures needed to describe evolution with SOELVFs are shown to be projectable, and the primary Hamiltonian constraints are constructed. The equivalence of the constraint algorithm presented in Section III with the usual Hamiltonian stability analysis is shown. In Section V application of the analysis given here to three different dynamical systems with singular Lagrangians is presented. Concluding remarks can be found in Section VI, with the crucial role that the vertical vector fields in the kernel of Ω L play summarized.

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 generators of this symmetry will be established.
The Euler-Lagrange equations of motion are the system of D second-order differential where is the energy, while M ab (q,q) := ∂ 2 L (q,q) ∂q a ∂q b , and F ab (q,q) := Here, Einstein's summation convention is used. Lie symmetry [30].
Following Olver [30], we define which reduces to Eq. (3) on the surfaces ∆ a (q,q,q) = 0. The set is the family of solutions to Eq. (3) that evolve from u 0 . Consider two such solutions q a (t) and Q a (t). From Eq. (3) there exists a z a (u) ∈ ker M(u) such thatQ a −q a = z a (u). As z a depends on both q a andq a , we are led to consider generalized Lie symmetry groups generated by with a ρ(u) that does not depend explicitly on time. This gives the total time derivative withρ := dρ/dt.
With this g, the second-order prolongation vector, on the second-order jet space M (2) = {(t, q,q,q)} with pr g ∈ TM (2) can be constructed.
Its action on the Euler-Lagrange equations of motion gives 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 trajectories in O(u 0 ), 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 abρ b for some function b a (u) whereḃ a = 0. However, because all the solutions in O(u 0 ) have the same initial data, ρ a (u 0 ) = 0 andρ a (u 0 ) = 0, and thus b a = 0. This leads to the following new result.
We denote by g the set of all vector fields g that satisfy Lemma 1, 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 well known [30]. However, because our Lagrangians are singular, three additional conditions must be imposed: 1. While ρ a = 0 andρ a = z a for any z ∈ ker M ab (u) is a solution of Eq. (12), we require thatρ a = dρ a /dt, and they must be removed.
2. Ifρ a is a solution of Eq. (12), thenρ a + z a is a solution of Eq. (12) as well. Theρ a are not unique, and this, along with the first condition, leads us to generators that are constructed from equivalence classes of prolongations.
3. For any z a ∈ ker M ab (u), Eq. (6) gives, on the solution surface ∆ a (q,q,q) = 0. If Eq. (13) does not hold identically, it must be imposed, leading to the well-known, first-order Lagrangian constraints. As each q(t) ∈ O(u 0 ) must lie on this constraint submanifold, any symmetry transformation of q(t) generated by pr g must give a path Q(t) that also lies on the constraint submanifold.
Not all vectors in prg will be generators of the generalized Lie symmetry group for O(u 0 ).
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 Then for pr g A , pr g B , ∈ pr g, 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 is involutive. Importantly, dim pr g = dim K = 2N 0 .

III. THE LAGRANGIAN PHASE SPACE
In this section we determine the generators of the generalized Lie symmetry found in Section 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 energy equation, and with them, the constraint algorithm and the Lagrangian constraint submanifold. 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 notation and 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, trajectories t → q(t) ∈ Q are replaced by integral flows t → u(t) ∈ P L [1], which for the initial data u 0 = (q 0 , v 0 ) are given by solutions to where X is a smooth vector field in the tangent space TP L . As P L = TQ, we have the bundle projections τ Q : TQ → Q and τ TQ : T(TQ) → TQ with τ Q • τ TQ : T(TQ) → Q (see [6] and [1]). It is also possible to construct Tτ Q , the prolongation of τ Q to T(TQ), which is a second projection map Tτ Q : T(TQ) → TQ which is determined by requiring that τ Q • τ TQ and τ Q • Tτ Q map a point in T(TQ) to the same point in Q. The vertical subbundle which in terms of local coordinates are, In particular, a second order Lagrangian vector field X L is the solution of Eq.
(2) such that Tτ Q • X L is the identiy on TQ (see [1]). In terms of local coordinates 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 a vector field X ∈ T u P L , we define the dual prolongation map T * τ Q by Here, we adapt Dirac's bra and ket notation to denote the action of a k-form ω(x) by where Y j ∈ T u P L . The k-form bundle is Λ k (P L ), while the exterior algebra of forms is In terms of local coordinates ϕ q := ϕ qa dq a and ϕ v := ϕ va dv a .
There are a number approaches [1] used in the literature to obtain the Lagrange twoform 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 for all X, Y ∈ T u P L ; this is a horizontal two-form of Ω L . Then ; this is a mixed two-form of Ω L . In terms of local coordinates, while If u(t) is to describe the evolution of the dynamical system given by L, then the vector field X must be chosen so that its integral flows on P L faithfully represent their trajectories on Q. For regular Lagrangians this is guaranteed by setting X = X L , and is a unique ker 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 that lie in it.
The two-form Ω L gives the lowering map Ω ♭ L : As while Moreover, because the rank of M ab (u) is constant on P L there exists a basis, for ker M ab (u) at each u ∈ P L . This in turn gives the bases 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) To determine the vectors in ker Ω L (u), choose a K ∈ ker Ω L (u). Then Solutions of these equations are found with the use of the following theorem from linear algebra 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 * of rank r, the inhomogeneous linear equation, has solutions if and only if where This theorem is applied to Eq. (31) by setting To make the connection with the results of Section II clear, this application is done locally. The condition that X q ∈ A is Ω q♭ Using the action of Ω v♭ M and Ω q♭ M in Eq. (31), we find that M ab K qa = 0, and thus K q ∈ C. The existence condition for solutions of Eq. (31) is then, With the basis given in Eq. (29), we may express and Eq. (35) becomes, is the reduced matrix of F ab . Then K q is restricted to the subspace, Theorem 3 The vectors K = K q + K v ∈ ker Ω L are given by, where, C ∈ C, G ∈ G, and C ∈ G ⊥ is the unique solution of Proof. The horizontal component, K q = C, of K satisfies Eq. (35), and is a solution of for e, f = N 0 + 1, . . . , D, and M is nonsingular. In this basis Eq.
The nonvanishing components of C and R are vectors with (D − N 0 )-components that satisfy Thus there is a unique solution for C that belongs to G ⊥ .
The constant rank assumption for M ab (u) together with the definition of G shows that there are N 0 free choices for G at each u ∈ P L . According to Theorem 3 the C-term is uniquely specified by the choice of C ∈ C. In general,F is not determined by M ab , and thus the constant rank of M ab does not guarantee that the rank ofF is constant. We then find The results of Theorem 3 are more general than we need. Because its proof is local and algebraic, even though our focus is on Ω L with constant rank the theorem would nevertheless still hold if the rank was not. It would only have to be applied to each region of P L on which the rank of Ω L is constant, resulting in aD that takes on different values on the Lagrangian phase space.
Consider a region U Sol ∈ M (2) on which solutions of the Euler-Lagrange equations of motion exist, and a point (t, q,q,q) ∈ U Sol . Then under the isomorphism (t, q,q,q) → Although this conclusion can only be reached on U Sol , the rank of Ω L is constant on P L , and thusD = N 0 on all of P L . As such C = C, and for any K ∈ ker Ω L (u), K = C +Ċ + G where nowĊ =Ċ a ∂/∂v a . (This expression for K can also be established directly using Eq. (31).) This result is also new.

D. First-order Lagrangian constraints
For singular Lagrangians solutions of the energy equation X E are not unique. They also do not, in general, exist throughout P L , but are rather confined to a submanifold of the space given by Lagrangian constraints. The first-order constraints, those that come directly from the energy equation, are the focus of this section. Although most of this analysis is done for a SOLVF, we show later that our results do not depend on this choice.
With X L = X q L + X v L , the energy equation can be expressed in terms of a one-form Ψ as The existence condition for solutions to Eq. (44) is given again by Theorem 2 by identifying (34) with the action of Ω q♭ M on vector fields yields A = C; consequently Eq. (33) requires that Ψ| C = 0 ∀ C ∈ C or, after using the basis U q The condition γ [1] n = 0 imposes relations on the coordinates q and v, and defines a set of submanifolds of P L . These γ [1] n are called the first-order constraint functions. (Because they are obtained through the energy equation, they are also called dynamical constraints in the literature [11,16].) While the number of first-order constraint functions in C [1] L := γ is equal to the dimension of C, these functions need not be mutually independent. Let I [1] be the number of independent functions in C [1] L . Then I [1] = rank dγ [1] n ≤ N 0 , and P [1] L := u ∈ P L \ γ [1] n (u) = 0 , n = 1, . . . , N 0 -called the first-order Lagrangian constraint submanifold-has dim P [1] L = 2D − I [1] . We will assume that P [1] L is not empty, i.e. that the first-order constraint functions are consistent. Otherwise, there is no SOLVF, and the integral flows that give the evolution of the dynamical system would not exist.
While the energy equation is usually written as dE = i X E Ω L , in doing so we have implicitly restricted ourselves to P [1] L . This is too restrictive for our purposes, and in this paper we introduce the constraint one-form (see also the approach in [16]). The condition β[X E ] = 0 then gives both solutions of the energy equation and the submanifold P and where , and the construction of γ [1] n does not depend on our use of X L .

E. The Generalized Lie Symmetry Group
Our construction of the generalized Lie symmetry group for O(u 0 ) is guided by the three conditions listed in Section II, and makes use of the projection of K to ker Ω L (u) in Section III C. It begins with the vector space, along with the following collection of functions on P L , The following result will be used a number of times in our analysis.
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).
Lemma 5 There exists a choice of basis for ker Ω L (u) that is also a basis of ker Ω L (u). on P L such that

Proof. Given a basis {G
Let {C (n) , n = 1, . . . , N 0 } be another choice of basis of C where Requiring Tτ Q [G (m) , C (n) ] = 0 in turn requires that ω m n be a solution of This is a linear, first-order Cauchy problem [48]. A solution exists for a given set of boundary conditions for ω m n given on a surface S as long as G (l) is nowhere tangent to S [48]. As N 0 < D, and as we have complete freedom to choose both the boundary conditions and S, a solution of Eq. (53) can always be found. The collection of vector fields {P (n) = C (n) + C (n) , G (n) } is then a basis of both ker Ω L (u) and ker Ω L (u).
For the rest of this paper we will assume that this choice of basis for C and ker Ω L (u) has been made.
As G is an ideal of ker Ω L (u), we may define for any P 1 , P 2 ∈ ker Ω L (u) the equivalence relation: P 1 ∼ P 2 iff P 1 − P 2 ∈ G. The equivalence class, then follows, along with the quotient space ker Ω L (u)/G. This results in a collection of vectors that lie in the kernel of Ω L , but with the vectors in G removed. This ker Ω L (u)/G addresses the first two conditions listed at the end of Section II. We now turn our attention to the third condition.
Because the integral flow u X (t) of any solution X of the energy equation must lie on P [1] L , a symmetry transformation of u X (t) must result in an integral flow u Y (t) of another solution Y of the energy equation. This flow must also lie on P [1] L . Implementing this condition is done through β[X E ], and it is this one-form that singles out the vectors in ker Ω L (u) that generate the generalized Lie symmetry. We do this by looking at the action of a vector since γ [1] n = 0 on P [1] L . Given a P (n) ∈ ker Ω L (u) such that P (n) = U q (n) + U (n) + G ′ with G ′ ∈ G, it has been shown in [6,9] that γ [1] But G is an ideal of ker Ω L (u), while a straightforward calculation shows that GE = 0.
Thus Gγ [1] (n) = 0, and it follows that L G β = 0. The subspace is then well defined, and we find that P ∈ Sym iff i P dβ[X E ] = 0. A simple calculation shows that Sym is involutive. There is then a corresponding set of one-parameter subgroups σ P (ǫ, x) for any P ∈ Sym given by with σ P (0, u) = u for u ∈ P L . The collection of such subgroups gives the Lie group Gr Sym .
This Gr Sym is the generalized symmetry group we are looking for, as we see below.

F. Euler-Lagrange Solutions of the Energy Equation
The set of general solutions to the energy equation is Importantly, while a SOLVF X L ∈ Sol, the majority of vectors in Sol are not SOLVFs. This is 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 Sol whose projection onto Q corresponds to a trajectory q(t) that is a solution of the Euler-Lagrange equations of motion, then Gr Sym must map one of such flows into another. However, while 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 energy equation. By necessity, general solutions of the energy equation must be considered. Only a specific subset of such solutions are physically relevant, however. As for P ∈ ker Ω L (u) and X E ∈ Sol, in general L P X E / ∈ ker Ω L (u). The exception is when P ∈ Sym as well, which leads us to the subset of solutions Moreover, as i [G,X EL ] Ω L = −L G β = 0, from Lemma 4 [G, X EL ] ∈ G.
Proof. As P ∈ Sym, from Eq. (59) [X EL , P] ∈ ker Ω L (u). Next, for each G ∈ G, there is a G X ∈ G such that G X = [G, X EL ]. There is also a G P ∈ G such that G P = [G, P]. It then follows from the Jacobi identity that [[X EL , P], G] ∈ G, and [X EL , P] ∈ ker Ω L (u).
The vector fields in Sol generate the family of integral flows The physical significance of these flows can be seen from the following theorem. Proof. Let u X EL (t, u 0 ) ∈ O EL (u 0 ) be an integral flow generated by X EL , and let σ P (ǫ, u) ∈ Gr Sym be a one-parameter subgroup of Gr Sym generated by P ∈ Sym with σ P (0, u) = u.
The action of σ P on u X E gives u Y (t, u 0 ) := σ P (ǫ, u X EL (t, u 0 )), while the choice ǫ = 0 when t = t 0 ensures that u X EL and u Y have the same initial data. The tangent to this path is where σ * P is the pullback map of σ P . As σ P is also a mapping of P L into itself, for a suitably small neighborhood about u X EL we may expand Y about ǫ = 0 in the Lie series, .
While X L / ∈ Sol, it is possible to construct from X L a vector field X L that is. Choose a basis P (n) , G (n) , n = 1, . . . , N 0 of ker Ω L (u), and consider a vector field X L such that where G ∈ G, and f m (u) are functions on P L . For X L ∈ Sol as well, we must have [X L , G (n) ] ∈ G, and thus these functions must be solutions of This leads us to the second-order, Euler-Lagrange vector field (SOELVF), where {[P (n) ], n = 1, . . . , N 0 } is a choice of basis for ker Ω L (u)/G. By construction, X EL ∈ Sol. Conversely, if Y EL ∈ Sol, then Y EL − X L ∈ ker Ω L (u) and Y EL is a SOELVF. Thus,

G. A constraint algorithm for second-order, Euler-Lagrange vector fields
For most dynamical systems the flow fields in O EL (u 0 ) will not be confined to P [1] L , and yet this is the submanifold on which the solutions X EL ∈ Sol of the energy equations exist.
In these cases it is necessary to jointly choose a SOELVF X EL and a submanifold of P [1] L on which u X EL will be confined. Doing so requires that which is called the constraint condition. Implementing it involves imposing successive conditions on X EL . At each step additional constraints may be introduced, giving a succession 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, or when the possibility of dynamics on P L is exhausted. This process is called a constraint algorithm, and has been introduced often in the literature. While such an algorithm will also be presented here, its purpose is to show that the end result X EL of the algorithm is once again a SOELVF, and a second-order problem is avoided. Later, it will also be used to show that both this X EL and the Lagrangian constraints-whether first-order or introduced by the algorithm-are projectable.

A collection C
[n F ] L ⊂ F of constraint functions of order 1 to n F . , and

A second-order, Euler-Lagrange vector field
where the N 0 , have been uniquely 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. In addition, 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 space are 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, and often make use of pullbacks of structures on the Hamiltonian phase space in the construction of 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 constraints and vector fields on TP L , respectively. However, while primary Hamiltonian constraints play a central role in the construction of both operators, the existence of such constraints is presumed. Moreover, because of the reliance on primary constraints, a number of subtleties involving first-and second-class Hamiltonian constraints must be dealt with.
These subtleties and their conclusions, present for SOLVFs, are not present for SOELVFs.
The approach used here focuses on the symmetry group Gr Sym , and the geometric structures inherent to almost regular Lagrangians. The passage from Lagrangian to Hamiltonian mechanics 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 the function f c (s) := (f • L −1 )(s) = f (L −1 (s)) = f (u)| L(u)=s , on P C . The action of L on a vector field X ∈ T u P L is given by the pushforward map L * : T u P L → T L(u) P C , while its action on a 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 simply involves replacing s by L(u), the action of both L and L * involve solving for u in s = L(u).
which is a submanifold of P L . As such, the pullback of a function now results in the collection of functions while the pushforward L * X of X gives the collection of vectors This ambiguity for Eq. (81) can be avoided by focusing on functions that are constant on 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 condition for a function f to be projectable is for all G ∈ G.
For vector fields, the ambiguity Eq. (82) is avoided when the components of X C are constant on the preimage. Then and we say a vector field on T u P L is projectable when Eq. (85) holds. To determine which vectors are projectable, consider first the collection of vectors in T u P L for which G is an ideal, Applying the same arguments using Lemma 5 to T u P L as was applied to ker Ω L (u) gives similar results: dim T u P L = 2D, and T u P L is involutive. The equivalence relation X 1 ∼ X 2 iff X 1 − X 2 ∈ G then follows along with the quotient space T u P L /G.
Choose an open covering U of P L , and a point u in an open neighborhood U u ∈ U such that Gu A = 0, A = 1, . . . , 2D, for all G ∈ G. Choose also a X ∈ T u P L /G, and consider the path u X (t, u) given by with u X (0, u) = u. This open neighborhood can always be chosen small enough such that, on U u . Then as G is an ideal of T u P L , e −tX Ge tX ∈ G, and Gu A X (t, u) = 0 in U u . By applying Eq. (88) to a sequence of such open neighborhoods, we can extend this result to any connected region 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). The integral 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, as we show in the next section.

B. 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, GE = 0, and E is projectable with its image H C = (E • L −1 )(s) being the canonical Hamiltonian. With the exception of the energy, we avoid introducing new notation, and will represent the projection of any function f (u) ∈ F through its argument: f (s).
Both Sol and ker Ω L (u)/G are subsets of T u P L /G, and are projectable. Of particular interest are and In particular, the general X EL in Eq. (66) gives the general vector field in Flow H T when expressed in terms of local coordinates. Here, With the canonical two-form ω = dq a ∧dp a on T * s P C , we have the collection of one-forms, which gives the primary constraints, and which gives the set of total Hamiltonians.

Prim and the Primary Hamiltonian Constraints
Using the kernel of the pullback map, in this section we construct from ker Ω L (u)/G the primary Hamiltonian constraints.
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 welldefined 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, and the integral manifold (P L , L) of P f (Prim ♭ ) [49].
The construction of the primary Hamiltonian constraints is now trivial.
n ∈ ker L * , and it follows that L * T u P L /G ⊆ T s=L(u) P

Sol and the total Hamiltonian
On P [1] L , β[X E ] = 0, and the energy equation may be written as 0 = dE − i X L L * ω. It follows that 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 , that is everywhere tangent to P [0] C . Correspondingly, a general one form in Flow ♭ H T is which gives the total Hamiltonian, for the dynamical system. This leads to the sequence of maps: and to each X EL ∈ Sol there is a corresponding total Hamiltonian H T ∈ Flow ♭ H T .

The Equivalence of the Constraint Algorithm for Lagrangians and the Stability Analysis of Canonical Hamiltonians
It is well known that the integral flow generated by X H T need not be confined to P [0] C even though its initial data is chosen to be on this submanifold. This difficulty is resolved through a stability analysis [26] where {H T , γ n } = 0 is imposed on the primary constraints, and when necessary, successively on the secondary, tertiary, and higher-level Hamiltonian constraints. While this process is traditionally applied to the canonical Hamiltonian, Section III G describes a constraint algorithm for SOELVFs. We show here that this constraint algorithm is equivalent to the stability analysis of the canonical Hamiltonian.
The stability analysis must now be applied to the secondary constraints: L X [1] H T 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 gives a corresponding Hamiltonian flow field X [2] H T and total Hamiltonian H [2] T . If second-order Lagrangian constraints are introduced at this step, their projection will give the tertiary Hamiltonian constraints.
This procession continues with the stability analysis of the n th -level Hamiltonian constraints giving a X  T . If (n + 1) th -level Hamiltonian constraints are introduced, they are the projection of the n th -order Lagrangian constraints. The analysis stops when the Lagrangian constraint algorithm ends: at the n F -step. The end result X

V. EXAMPLES OF ALMOST REGULAR LAGRANGIANS
In this section we present three examples of dynamical systems with almost regular Lagrangians. The first example describes a single particle interacting with an external potential. It illustrates the role G plays, and the tight relationship between Gr Sym , the symmetries of 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, as is sometimes asserted in the literature. The second example consists of two interacting particles with a Lagrangian that has a local conformal symmetry. It is an example of a dynamical system for which only a subset of vectors in ker Ω L (u) generate the symmetry group. The third example consists of a particle with both local conformal symmetry and time-reparametization invariance. It is an example of a fully constrained dynamical system-as such, Sol = ker Ω L (u)-that has two gauge symmetries. The analysis of all three systems are done using the techniques and tools presented above.

A. A Lagrangian With and Without a Local Gauge Symmetry
Whether the action with |q| = √ q a q a and q a := q a /|q|, a = 1, . . . , D, has a local gauge symmetry depends on the With Π ab (q) := δ ab − q a q b , and C and G are spanned by U q (1) = q · ∂/∂q and U v (1) = q · ∂/∂v, respectively, while ker Ω L (u) is spanned by U q (1) and The energy is and there is only one first-order, Lagrangian constraint, q = q · dq. As expected, L G γ [1] = 0. We may choose As [X L , U v (1) ] ∼ −P (1) , a symmetry transformation of X L does not result in a SOLVF. Instead, is constructed, and a general SOELVF is whether or not Sym is empty depends on the symmetries of V (q). As the constraint algorithm gives L X EL γ [1] = v · Π · ∂γ [1] ∂q whether or not u(u) is determined also depends on the symmetries of V (q). There are three possibilities, none of which will require the introduction of higher-order Lagrangian constraints.

The symmetric potential
For P (1) to generate a symmetry, and it follows that where V AS is a function of q a only. Then P (1) generates a symmetry iff V (q, q q ) = V Sph (q) + V AS ( q a ), where V Sph is a function of q only. The group Sym is one-dimensional, and spanned by P (1) .
The constraint condition Eq. (120) for this potential reduces to which must be satisfied on P L . There are two cases: Case 1: Then V Sph = aq + b, but since the condition γ [1] = 0 requires a = 0. As we may choose b = 0, V (q) = V AS ( 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, and correspondingly, the dynamics of the particle is determined only up to an arbitrary function.
Case 2: In this case u(u) = 0, and the dynamics of the particle is completely determined by its initial data. The Lagrangian does not have a local gauge symmetry. The first-order, Lagrangian constraint γ [1] = 0 defines a surface on P L , and for dynamics to be possible the set of must be non-empty. Dynamics are on the surfaces |q| − R i = 0, and on them the potential reduces to V (q) = V Sph (R i ) + V AS ( q a ). This reduced potential has the same symmetry as the potential V AS ( q a ) in Case 1, leading to equations of motion that have the same generalized Lie 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 the other case to u(u) = 0 and dynamics that are completely determined by the choice of initial data.

The asymmetric potential
For a general V , the second term in Eq. (119) does not vanish, P (1) does not generate a symmetry of the equations of motion, Sym = {∅}, and Eq. (120) gives The dynamics of the particle is uniquely determined by its initial data.
For each of the three possible choices of V (q) outlined above the total Hamiltonian obtained here agrees with the one obtained using constrained Hamiltonian mechanics.
We find that L P (+) β = 0, while Then Sym is one-dimensional, and spanned by P (+) .
With the canonical momenta, where τ ab := δ ab + q a q b , the passage to Hamiltonian mechanics is straightforward. Equation and the projection of the first-order Lagrangian constraint is γ [1] = [ q 1 · L 2 + q 2 · L 1 ] /m.

C. A Lagrangian with Local Conformal and Time-reparametization Invariance
The action S 3 := sm s d q dt where s = ±1, 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 (see also [50] and [51,52] for systems with Lagrangians linear in the velocities). This action is a generalization of that for the relativistic particle, with the additional requirement that it have a local conformal invariance.
For this action, and Ω F = 0. Here, a = 1. . . . , D, so that u 2 = 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 respectively.
Because this system is fully constrained, E = 0.
As Ω F = 0 as well, there are no Lagrangian constraints. We may choose X L = v · ∂/∂q. As [X L , U v (1,2) ] ∼ −U q (1,2) , the action of X L by Gr Sym 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 1 (u) U q (1) + u 2 (u) U q (2) , with u n (u) ∈ F for n = 1, 2. The Lie algebra Sym itself is two dimensional, and spanned by U q (1) and U q (2) . For the passage to Hamiltonian mechanics, H C = 0 as E = 0. With the canonical momentum p a = mu a /|q|, the projection of P (1,2) is and π (1) = d(q · p)/|q|, π (2) = d(|q| 2 p 2 )/2m|q|. The primary Hamiltonian constraints are 2 := |q| 2 p 2 − sm 2 . As X L = 0, X C = 0, and we find that

VI. CONCLUDING REMARKS
With the benefit of hindsight, the many roles that G plays in determining both the geometric structure of P L for singular Lagrangians, and the connection between these structures and dynamics become readily apparent. What also becomes clear are the reasons why SOELVFs and their dynamical structures are projectable.
Because G is involutive, it gives a foliation of P L . There is then a neighborhood U about each point u ∈ P L on which we can define the equivalence relation u 1 ∼ u 2 iff u 1 − u 2 = g, where g is a point on the leaves Fol(G) of the foliation. This leads to the quotient space P L /Fol(G), which has dimension 2D − N 0 . Importantly, P L /Fol(G) is projectable, and L(P L /Fol(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 /Fol(G) and consider the set of vector fields in T u P L for which G is an ideal. This leads us to T u P L , and the quotient space T u P L /G. As dim T u P L /G = 2D − N 0 , it is expected that T u P L /G = T p [P L /Fol(G)] for p ∈ P L /Fol(G). That T u P L /G is projectable is then readily apparent.
Finally, for singular Lagrangians the acceleration is not determined uniquely by the Euler-Lagrange equations of motion, an ambiguity due to the generalized Lie symmetry. This symmetry 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 construction of Sol. Both ker Ω(u)/G ⊂ T u P L /G and Sol ⊂ T u P L /G, and thus the evolution of the dynamical system is confined to the tangent bundle T [P L /Fol(G)]. The projectability of T[P L /Fol(G)] ensures that all of the dynamical structures needed to describe the evolution of dynamical systems on the Lagrangian phase space is projectable, and agrees with those obtained through constrained Hamiltonian mechanics.
While G does play an important role in determining the general Lie symmetry group, it itself 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 depending on the choice of potential. Nevertheless, G is present and plays its usual role in determining P L /Fol(G). It is instead vectors in ker Ω L (u)/G-with G removed-that generate the generalized Lie symmetry. We emphasize here that while this symmetry plays an important and guiding role, this role is nevertheless supportive in the construction of the algebraicgeometric structures on P L needed in determining both the geometric structure of P L , and the connection between these structures and the evolution of the dynamical system.
The application of these algebraic-geometric structures go beyond showing the equivalence of the Lagrangian and Hamiltonian formulations of mechanics for singular Lagrangians, however. While the primary Hamiltonian constraints play a critical role in the Hamiltonian constraint 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 ensure that all of the constraints has been found for the system at hand. As a result of the Lagrangian phase space analysis presented here we are able to determine the number of primary constraints for any dynamical system, and the constraints themselves can be calculated by solving a first-order, quasi-linear differential equation. In addition, while the end result of