Constraint characterization and degree of freedom counting in Lagrangian field theory

We present a Lagrangian approach to counting degrees of freedom in first-order field theories. The emphasis is on the systematic attainment of a complete set of constraints. In particular, we provide the first comprehensive procedure to ensure the functional independence of all constraints and discuss in detail the possible closures of the constraint algorithm. We argue degrees of freedom can but need not correspond to physical modes. The appendix comprises fully worked out, physically relevant examples of varying complexity.


Introduction
In most Lagrangian field theories, there exists a mismatch between the number of a priori independent field variables and the number of degrees of freedom N DoF being propagated.The determination of N DoF is of crucial importance, as it directly affects the physics.More often than not, this calls for non-trivial analyses.
All approaches to the degree of freedom count developed thus far revolve around constraint algorithms, whose origin dates back to the work of Dirac first and Bergmann shortly afterwards.These authors considered a coordinate-dependent approach for autonomous systems within the Hamiltonian formalism.A nice review of the initial proposal can be found in [13].It was only sensibly later that the geometrization of the procedure was carried out, yielding the celebrated Presymplectic Constraint Algorithm (PCA) [14].
A different yet equivalent geometric algorithm was put forward in [15].
The PCA was adapted into the Lagrangian formalism stepwise [16][17][18].The relation between constraints emerging in the Lagrangian and Hamiltonian algorithms was studied in a coordinate-dependent manner in [19].The said relation is based on the so-called (temporal) evolution operator, whose geometric definition and properties were clarified in [20].
As a remark, we note that even more general geometric constraint algorithms exist.Prominent examples include an algorithm for non-autonomous systems in the Lagrangian formalism [21] and algorithms for dynamical systems described implicitly via differential equations [22][23][24].Unfortunately, there does not seem to exist a complete review of constraint algorithms and relations between them.
In spite of the vast and sophisticated bodywork on geometric constraint algorithms, such approaches are intrinsically covariant and thus demanding for application in physical theories, where time plays a distinct role.The need to construct multiple auxiliary objects prior to implementation adds to the challenge.This is especially true for elaborate settings, including modified gravity proposals.The said and kindred hindrances presumably explain (at least in part) the extensive use of the Dirac-Bergmann algorithm, often supplemented by the Arnowitt-Deser-Misner (ADM) foliation [25], by the gravitational physics community.
In synergy with such fruitful background, few but noteworthy Lagrangian algorithms have been put forward, either providing or developed as a coordinate-dependent prescription [26,[29][30][31].Invariably, they aim to facilitate implementation in exigent theories and abridge the obtention of physically relevant quantities, such as the number of degrees of freedom N DoF .The present work delves into this line of research.

Organization
In the main section 2, we consider first-order Lagrangian field theories and present a coordinate-dependent constraint algorithm for them.Supplemented with information on local (gauge-like) symmetries, the algorithm yields the significant number N DoF in the theory.Most remarkably, a thorough procedure for the verification of functional independence among constraints is given in sections 2.2 and 2.3.This essential feature had only received modest attention thus far.Distinct closures of the algorithm are expounded in section 2.4.For clarity, appendix A consists of minute implementations of the method in diverse physical theories.In section 3, we reflect on the relation between N DoF and physics.We draw our conclusions in the final section 4.

Method
Let Q A " Q A px µ q be a finite set of real field variables.A " 1, 2, . . ., N is a collective index, running over all a priori independent components of possibly multiple fields of different types.Consider a first-order classical field theory, defined by the action Notice we do not consider Lagrangians with explicit spacetime dependence.The action (2.1) may but need not be invariant under local field transformations of the form with P P N 0 .Here, θ M denotes arbitrary functions of spacetime labeled by a collective index M , while R M A refers to fixed functions of pQ A , B µ Q A q.All pθ M , R M A q are taken to be smooth.
Gauge, Lorentz and diffeomorphism transformations comprise the physically most relevant examples of (2.2).Discrete, global and conformal (including Weyl) symmetries of the action do not affect the degree of freedom count and hence are not discussed.Typically, Lagrangians are postulated on the basis of a specific field content Q A entertaining certain symmetries, if any.Accordingly and when pertinent, we assume a priori knowledge of (2.2).Given a Lagrangian whose symmetries of the relevant form (2.2) are unknown a priori, there exist systematic procedures to their disclosure [29,32,33].See also the earlier studies [34,35].
The number of degrees of freedom propagated in the theory can be calculated as [26,29] with l, g, e P N 0 .g is the number of distinct θ M functions in (2.2), while e ě g is the number of distinct θ M functions plus their successive time derivatives p 9 θ M , : θ M , . ..q in (2.2).The focus henceforth is on the number l of functionally independent Lagrangian constraints, iteratively determined as l " l 1 `l2 `. . ., (2.4) where l 1 ě l 2 ě . . .count primary, secondary, etc. functionally independent Lagrangian constraints.We stress that (2.3) counts degrees of freedom exclusively in terms of Lagrangian parameters.In particular, it does not require the classification of constraints into first and second class.Nonetheless, such information is not lost, as (2.3) follows from the map between Hamiltonian and Lagragian parameters [26][27][28]] where pN 1 , N 2 q denote the number of first and second class constraints in Dirac's canonical formalism, respectively.Accordingly, (2.3) applies to both point particle systems and classical field theories, with the latter counting degrees of freedom per point in spacetime.An intrinsically Lagrangian count of degrees of freedom is an important result that was obtained geometrically in [29] -see equations ( 2), ( 3) and ( 16) therein.We remark that (2.3) was derived for first-order theories, but has also been employed in higher-order settings [36,37].

Initialization
Consider the (primary) equations of motion of the theory, in the form The (primary) Hessian captures the linear dependence on the generalized accelerations : Q A , while In the above, we have introduced the short-hands (2.9) We further rewrite the (primary) equations of motion as where pE p1q , : Q, U p1q q are N -dimensional column vectors and W p1q is an N ˆN square matrix.

First iteration
The (primary) equations of motion (2.10) may but need not encode second order differential equations (SODEs) in time for all Q A 's.The number of functionally independent such SODEs is given by the (row) rank of the (primary) Hessian W p1q .The theory may have up to `N ´rankW p1q ˘primary Lagrangian constraints.

Step I. Rank of the Hessian
In full generality, the determination of the (row) rank of the (primary) Hessian W p1q is a challenging task.
A conceptually neat and algebraically convenient manner to do so is as follows.Assume W p1q admits left null vectors If no non-trivial solution to (2.11) exists, then the (row) rank of the (primary) Hessian is N and the theory is said to be regular.In this case, the theory possesses no (primary) Lagrangian constraints l " l 1 " 0. The constraint determining algorithm thus terminates.
Else, let V p1q itself denote a maximal set of M 1 P r1, N q linearly independent solutions to (2.11), normalized as per convenience.In this case, the (row) rank of the (primary) Hessian is N ´M1 , the theory is said to be singular and up to M 1 primary Lagrangian constraints may exist, which we proceed to unveil.

Step II. Lagrangian constraints
Consider the left contraction of the M 1 null vectors V p1q with the (primary) equations of motion (2.10) φ p1q :" V p1q ¨Ep1q " V p1q ¨Up1q " 0. (2.12) By definition, φ p1q is a set of M 1 relations that do not depend on the generalized accelerations : Q A .Any maximal (sub)set of functionally independent relations among (2.12) can be regarded as the primary Lagrangian constraints in the theory.Hence, l 1 P r0, M 1 s.
It may happen that all relations in (2.12) identically vanish.If so, there exist no (primary) Lagrangian constraints l " l 1 " 0 and the constraint algorithm thus terminates.
In the absence of such trivialization, the distillation of a maximal (sub)set of functionally independent relations from (2.12) is generally demanding.A systematic way around the hurdle is similar to the determination of the (row) rank of the (primary) Hessian before.Assume the set of relations in (2.12), written as an M 1 -dimensional column vector φ p1q , admits solutions to Γ p1q ¨φp1q " 0. (2.13) A generic ansatz for such a vector is Notice that every component of Γ p1q includes spatial derivative operators up to second order.In this manner, both algebraic and spatial derivative dependences among the relations in (2.12) can be identified.
If no non-trivial solution to (2.13) exists, then all relations in (2.12) are functionally independent, implying l 1 " M 1 .Moreover, the relations (2.12) themselves can be interpreted as the primary Lagrangian constraints in the theory.
Else, let Γ p1q itself denote a maximal set of m 1 P r1, M 1 q linearly independent solutions to (2.13), normalized as per convenience.In this case, there exist l 1 " M 1 ´m1 primary Lagrangian constraints in the theory, which can be parametrized as where `Γp1q 0 ˘K stands for a maximal set of linearly independent row vectors orthogonal to normalized as per convenience1 .
To sum up, for l 1 ‰ 0, let (2.17) parametrize the primary Lagrangian constraints in the theory.For later convenience, let Φ p1q denote those same primary Lagrangian constraints, written as an l 1 -dimensional column vector.

Step III. Stability of the Lagrangian constraints
Let R " 1, 2, . . ., l 1 label the primary Lagrangian constraints (2.17).Self-consistency of the theory under time evolution implies must hold true, where It is convenient to rewrite the above as where pE p2q , U p2q q are l 1 -dimensional column vectors and W p2q is an l 1 ˆN rectangular matrix.
The l 1 conditions (2.18), viewed as the secondary equations of motion for the system (2.20), may give rise to up to l 1 secondary Lagrangian constraints, which are to be unveiled in a second iteration of the constraint algorithm.Indeed, a conceptually simple repetition of the just described procedure yields the secondary Lagrangian constraints in the theory, if any.The only formal subtlety amounts to ensuring that only functionally independent secondary Lagrangian constraints are considered in the degree of freedom count (2.3).To this aim, let encompass the primary (2.10) and secondary (2.20) equations of motion, in the form Here, E p2qÓ and U p2qÓ are pN `l1 q-dimensional column vectors, while W p2qÓ is an pN `l1 q ˆN rectangular matrix.The second iteration in the constraint algorithm takes E p2qÓ as a starting point, as opposed to merely E p2q .

Generic iteration n ě 2
For m ă n, let Φ pmq , denote the m-stage Lagrangian constraints, written as an l m -dimensional column vector.Then, let denote the ordered collection of all Lagrangian constraints unveiled thus far.We stress that Φ pn´1qÓ is an pN `´N q-dimensional column vector whose components have already been proven functionally independent, where On the other hand, let denote the n-stage equations of motion, written as an pl n´1 q-dimensional column vector.Notice that the n-stage Hessian W pnq is an pl n´1 ˆN q rectangular matrix.Further, let denote the ordered collection of all equations of motion up to and including the n-stage, in the form Here, pE pnqÓ , U pnqÓ q are N `-dimensional column vectors, while W pnqÓ is an N `ˆN rectangular matrix.

Step I. Rank of the Hessian
First, the row rank of W pnqÓ is to be determined.To this aim, assume it admits left null vectors (2.28) A generic ansatz for such a null vector is2 By construction, there indeed exist solutions to (2.28): they trivially extend the left null vector(s) V pn´1qÓ found in the immediately previous stage through V pnq " 0. Such solutions do not carry new information.Consequently, they are to be dismissed.If no left null vector to W pnqÓ exists such that V pnq ‰ 0, then rrank ´WpnqÓ ¯" rrank ´Wpn´1qÓ ¯`l n´1 , ( where rrank stands for row rank.It follows that l n " 0 and the algorithm thus terminates.In this case, l " N `´N .Else, let V pnqÓ itself denote a maximal set of M n P r1, l n´1 s linearly independent left null vectors to W pnqÓ such that V pnq ‰ 0, normalized as per convenience.In this case, rrank ´WpnqÓ ¯" rrank ´Wpn´1qÓ ¯`l n´1 ´Mn , (2.31) implying that up to M n n-stage Lagrangian constraints may exist.Algebraic ease dictates that the disclosure of the n-stage Lagrangian constraints, if any, is carried out in two steps.The first step guarantees functional independence within the nth iteration.The second step guarantees functional independence with respect to previous iterations.

Step II. Lagrangian constraints Substep IIA. Functional independence within the stage
Consider the set of M n relations φ pnq :" V pnqÓ ¨EpnqÓ " V pnqÓ ¨UpnqÓ " 0. (2.32) If the above relations trivially vanish, there exist no n-stage Lagrangian constraints l n " 0. The constraint algorithm thus terminates, yielding l " N `´N .Else, a maximal (sub)set of functionally independent relations among (2.32) is to be extracted.To this aim, let φ pnq denote the relations (2.32), written as an M n -dimensional column vector.Assume φ pnq admits solutions to Γ pnq ¨φpnq " 0. (2.33) A generic ansatz for such a vector is If no non-trivial solution to (2.33) exists, then all relations in (2.32) are functionally independent among themselves.
Else, let Γ pnq itself denote a maximal set of m n P r1, M n q linearly independent solutions to (2.33), normalized as per convenience.In this case, a maximal subset of M n ´mn functionally independent relations among (2.32) is where `Γpnq 0 ˘K stands for a maximal set of linearly independent row vectors orthogonal to normalized as per convenience.

Substep IIB. Functional independence with respect to previous stages
In an n ě 2 iteration of the constraint algorithm, the disclosed maximal (sub)set of functionally independent relations (2.37) cannot be immediately regarded as parametrizing the n-stage Lagrangian constraints.This is because (2.37) is not necessarily functionally independent from the Lagrangian constraints unveiled in previous iterations.We proceed to ensure such retroactive functional independence.Let denote the ordered collection of all previous stages' Lagrangian constraints Φ pn´1qÓ in (2.23) and the relations ϕ pnq in (2.37).Recall that, by construction, both distinct sets Φ pn´1qÓ and ϕ pnq comprise only functionally independent relations.As a result, upon joint consideration, l n " rrank ´ΨpnqÓ ¯´rrank ´Φpn´1qÓ ¯P r0, M n s. (2.39) In order to determine l n , assume Ψ pnqÓ admits solutions to Υ pnqÓ ¨ΨpnqÓ " 0. (2.40) A generic ansatz for such a vector is3 If a maximal set of M n linearly independent solutions to (2.40) exists, then all relations (2.37) are functionally dependent with respect to Φ pn´1qÓ .In this case, (2.38) has the minimal row rank rrank ´ΨpnqÓ ¯" rrank ´Φpn´1qÓ ¯" N `´N (2.42) and therefore l n " 0. The constraint algorithm thus terminates, yielding l " N `´N .Else, start by considering the diametrically opposite instance.If no non-trivial solution to (2.40) exists, then the relations (2.37) are functionally independent with respect to Φ pn´1qÓ .In this case, l n " M n .Moreover, the relations (2.37) can be regarded as the n-stage Lagrangian constraints in the theory.
Next, consider all intermediate instances.Let Υ pnqÓ itself denote a maximal set of m n P r1, M n q linearly independent solutions to (2.40), normalized as per convenience.In this case, l n " M n ´mn and a maximal subset of functionally independent relations among (2.37) is where `ΥpnqÓ 0 ˘K stands for a maximal set of linearly independent row vectors orthogonal to Step III.Stability of the Lagrangian constraints.
Self-consistency demands that the n-stage Lagrangian constraints are preserved under time evolution: 9 Φ pnq " 0, for all l n relations in (2.45).This condition may yield up to l n Lagrangian constraints in a subsequent iteration of the constraint algorithm

Closure
In order to unequivocally establish the number of degrees of freedom N DoF in a given theory, it is imperative to pursue any constraint algorithm to its closure.Unfortunately and especially within coordinate-dependent Lagrangian approaches, persistence to termination is not always the case, as alerted against in [31,38].In the method just advocated, there exist 3 distinct manners in which the constraint algorithm may close.
Let n f denote the (finite) final iteration, wherein l n f " 0. As per (2.25), let E pn f q and W pn f q denote the associated n f -stage equations of motions and Hessian, respectively.l n f " 0 is a direct consequence of one of the following instances.
1. W pn f q has maximal row rank (2.30).In this case, consistency under time evolution of the pn f ´1q-stage Lagrangian constraints is ensured dynamically, through second-order (in time) differential equations of the variables Q A .Examples of this closure can be found in appendices A.2 and A.4-A.6.
2. W pn f q does not have maximal row rank, but all contractions of its chosen left null vectors with E pn f q identically vanish.Namely, (2.32) is identically satisfied.This closure is the trivial expression of the functional dependence of the would-be Lagrangian constraints arising at the n f stage on the previous stages' Lagrangian constraints.An example of this closure is provided in appendix A.3.

Equation (2.32
) is not identically satisfied, but it exclusively comprises relations that are functionally dependent on the previous stages' Lagrangian constraints.Namely, (2.42) is fulfilled.This constitutes the non-trivial counterpart to the previously described closure.An example is given in appendix A.7.
We are not aware of any physical example within the scope of this work where constraint algorithms fail to close at a finite number of iterations.

Remarks
Lagrangian constraints are not uniquely defined, only the space they span is.At every iteration, we have advocated for the most convenient choice.Such choice is model-dependent.
For all iterations in the constraint algorithm, it has been implicitly assumed that the row rank of the relevant Hessian remains constant.Presumably, the distinct dynamical behavior of the field configuration(s) for which there is a change in the rank of one or more of the Hessians remains encoded in the final stage's stack of equations of motion E pn f qÓ -and hence in the later discussed dynamical problem (3.2).
The method readily applies to higher-order field theories whose equations of motion are linear in the generalized accelerations : Q A , as in (2.10).In this case, the (primary) Hessian and remaining terms have a more complicated, order-dependent relation to the Lagrangian than (2.7) and (2.8), but the algorithm per se remains unaltered.

Dynamical versus physical modes
Degrees of freedom are a foundational subject in physics.As such, they drive sustained investigations on and around themselves.Prominent questions under survey include their very definition, the well-posedness and solvability of their associated dynamical equations and their relation to physical observables.In this section, we briefly ponder on such open-ended problems.
It is customary to view N DoF as counting the (pairs of) initial conditions needed to define the dynamical problem of a given theory.We proceed to elucidate the previous assertion within the scope of section 2. This allows us to confront dynamical and physical modes.
Consider a theory of the form (2.1) for which the constraint algorithm has been successfully pursued until closure l n f " 0. As per (2.26) and (2.27), let E pn f qÓ and W pn f qÓ denote the exhaustive stack of associated equations of motions and Hessians, respectively.
By construction and for singular theories, W pn f qÓ is an N f ˆN matrix with non-maximal row rank (Recall that M n denotes the maximal number of linearly independent left null vectors to W pn f qÓ that necessarily and at most involve the n-stage Hessian W pnq .)It follows that W pn f qÓ admits a maximal set of M tot linearly independent left null vectors V pn f qÓ , normalized as per convenience.Let pV pn f qÓ q K denote a maximal set of ̺ linearly independent row vectors orthogonal to V pn f qÓ , normalized as per convenience.Consider E :" pV pn f qÓ q K ¨Epn f qÓ " 0. (3.2) The above comprises ̺ functionally independent second order differential equations (SODEs) in time for (some of) the variables Q A .Supplemented by 2N DoF initial values for pQ A , 9 Q A q, (3.2) defines the dynamic problem of the theory (2.1).Conversely, N DoF counts the pairs pQ A , 9 Q A q whose time dependence is encoded in the just described dynamical problem.In other words, N DoF counts dynamical modes.If any, variables Q A present in the Lagrangian (2.1) but not determined by the dynamical problem comprise pure gauge modes.In this regard, the interested reader is gladly referred to [39].
The dynamical problem may but need not fall within the scope of the Cauchy-Kovalevskaya (CK) theorem.When it does, a unique analytical solution is guaranteed to exist.When it does not, existence (let alone uniqueness) of analytical solutions cannot be generically ascertained.Physics-driven extensions to the CK theorem thus comprise an enticing line of mathematical research.
Consider a generic Lagrangian field theory, possibly beyond the subclass in section 2. Presume a fortunate case in which one or more analytical solutions to its dynamical problem can be found.Even then, the obtained dynamical modes should not be immediately identified with physical modes, in the sense that dynamical modes may exhibit a behavior that is incompatible with well-established physical principles and/or observation.Reasons are plentiful.
First, consider stability criteria.For instance, solutions could be perturbatively unstable, in the sense of lacking robustness against small deviations in the initial data and/or free parameters.Whenever in conflict with observation, such solutions are to be disregarded.A lucid introduction to the most frequent perturbative instabilities in gravitational settings is [40].Numerical examples of critical values for free parameters which dramatically destabilize a theory can be found in [41].Moreover, it has been known for a while that instabilities could also appear only at the non-perturbative level [42].For an enlightening recent review apropos, see [43].
Causality is yet another essential requirement for a dynamical mode to be deemed physical.As a remarkable example, we note [44], where causality was employed to constrain dynamical modes for certain massive gravity theories.Overall, the quest remains for necessary and sufficient conditions that guarantee physicality of dynamical modes, even within the theoretical realm.
More generally, it is worth considering physicality of a (classical) field theory as a whole.From a philosophical perspective, degrees of freedom might help to address the question of physical equivalence between theories, which enjoys a long tradition in the philosophy of science, see e.g.[45][46][47].While several notions of equivalence are discussed in the recent literature [48,49], there exists widespread consensus that dynamical equivalence of two theories is a necessary condition for their empirical equivalence [50].In this context, we regard a match in the number of degrees of freedom as a prefatory necessary (but not sufficient) condition for physical equivalence between two theories.

Conclusions
We have presented a Lagrangian method to count degrees of freedom in first-order classical field theories.The emphasis is two-fold.First, a systematic and algebraically convenient procedure to establish the functional independence of the constraints that may be present in such theories.Second, a detailed discussion on the possible closures of the associated constraint algorithm.Both are consequential aspects that are rarely explicitly addressed in akin Lagrangian approaches.Non-exhaustive counterexamples to the former omission can be found in [29,31].The latter was painstakingly discussed in [31] and, for a certain family of massive electrodynamics theories, in [38].
We stress that functional independence among constraints is essential to the postulation of self-consistent theories.This is particularly relevant when a given constraint structure is being pursued, i.e. fixed values for the number of primary, secondary, etc. (Lagrangian) constraints.Against this background, we note that a suitable (row) rank reduction of an n-stage Hessian does not ensure the desired number of n-stage constraints are generated.While such (row) rank reduction is a necessary condition for the sought constraint structure, it is premature to regard it as sufficient on its own [38].
Failure to close a constraint algorithm may yield an incorrect number of degrees of freedom N DoF .It could happen that N DoF is overestimated, via the overlooking of overconstrained systems.Contrastively, N DoF may be underestimated, via the misidentification of functionally dependent relations of the type in (2.32) as (Lagrangian) constraints.
Last but not least, we have discussed several non-trivial conditions that propagating degrees of freedom must fulfill before they can be regarded as physical modes.

A Appendix: Examples
We begin in section A.1, by providing simple yet illuminating examples for the obtention of functionally independent constraints from a given out-of-context set.The remainder of the appendix is devoted to the explicit application of the method presented in section 2 to count degrees of freedom in various examples of physical relevance.The appendix thus serves to amply illustrate the use of the method, at various levels of algebraic intricacy.Notation.Brackets denoting symmetrization and antisymmetrization of indices are defined as T pµνq " pT µν `Tνµ q{2 and T rµνs " pT µν ´Tνµ q{2, respectively.For the 2-dimensional examples in sections A.1, A.2, and A.6, we use the short hand T 1 " B 1 T .Natural units are employed throughout.

A.1 Detection and avoidance of functional dependence among ad hoc constraints
Toy model I Consider a theory of the form (2.1) in 2-dimensional Minkowski spacetime.Further consider the set of M n relations in (2.12) for n " 1 or (2.32) for n ě 2. Suppose M n " 2 has been obtained, with the relations arranged into a column vector of the form (For simplicity, we omit indices indicating the iteration.)In view of the spatial derivatives' order difference between the two relations, a generic ansatz to (2.13) for n " 1 or to (2.33) for n ě 2 is particularly simple in this case: which readily yields a single linearly independent solution m n " 1 parametrized by Γ 0 " 0 and Γ 1 " ´r Γ 0 .We choose a convenient normalization for the solution, look into its algebraic subspace and choose a convenient normalization for the M n ´mn " 1 linearly independent orthogonal vector: The left contraction of the latter with (A.1) yields a functionally independent relation: φ ˚" Γ K 0 ¨φ " F .For n " 1, φ ˚can be regarded as the primary Lagrangian constraint.For n ě 2, functional independence of φ ˚with respect to Lagrangian constraints unveiled in previous iterations must be ensured before regarding φ ˚as the n-stage Lagrangian constraint.

Toy model II
Consider a theory of the form (2.1) in 3-dimensional Minkowski spacetime.Further, consider the following set of M n " 3 relations in (2.12) for n " 1 or (2.32) for n ě 2, arranged into a column vector where pF, Gq denote obviously functionally independent relations; for instance F " F pQ A , p Q B , B µ Q A q and G " GpQ A , B µ Q A q, with the hat denoting a specific coordinate within the set Q A that is not present.Observation of the relative difference in the order of spatial derivatives between the relations leads us to postulate a generic ansatz to (2.13) for n " 1 or to (2.33) for n ě 2 of the form (A.5) The above readily yields a single linearly independent solution m n " 1 parametrized by r Γ x 1 " ´Γ0 and p Γ xy 2 " Γ 0 {2, with all other free functions set to zero.We choose a convenient normalization for this solution, look into its algebraic subspace and choose a convenient normalization for the M n ´mn " 2 linearly independent orthogonal vectors: The left contraction of the last two with (A.4) yields two functionally independent relations For n " 1, (A.7) can be readily regarded as the primary Lagrangian constraints.For n ě 2, functional independence of (A.7) with respect to Lagrangian constraints unveiled in previous iterations must be ensured before reaching such a conclusion.Notice that the simplest choice of orthogonal vectors pΓ K 0 , r Γ K 0 q does not yield the obviously simplest span of the constraint space, given by tF, Gu.Toy model II thus illustrates our first remark in section (2.5).

A.2 Floreanini-Jackiw chiral boson
The Lagrangian for the 2-dimensional theory of a chiral boson due to Floreanini and Jackiw [51] is Here, the scalar field φ " φpx 0 , x 1 q is the only a priori independent field variable Q A and so N " 1.As is well-known, this theory possesses no local symmetries -neither of the relevant form (2.2) nor otherwise -in its original formulation (A.8).Therefore, g, e " 0. It is worth mentioning that a manifestly Lorentz invariant action for the Floreanini-Jackiw chiral boson exists, which has been further generalized into the so-called 2k-form electrodynamics family of higher-dimensional theories [52].The (primary) equations of motion following from (A.8) are of the form (2.10), with Obviously, the (row) rank of the (primary) Hessian is zero.A convenient left null vector for it is simply V p1q " 1.As per (2.12) and since no identical vanishing happens, φ p1q :" 9 φ 1 ´φ2 " 0 (A.10) can be readily regarded as the only primary Lagrangian constraint in the theory l 1 " 1.
The (primary) equations of motion, together with the demand for stability under time evolution of the primary Lagrangian constraint, conform the starting point of the second iteration (2.21), where Clearly, W p2qÓ only admits left null vectors of the form V p2qÓ " pV p1q , 0q. (A.12) According to the discussion below (2.29), there exists no secondary Lagrangian constraint in the theory l 2 " 0. The constraint algorithm thus terminates, by means of closure 1.Using (2.3) and (2.4), we reproduce the renowned result that the theory propagates N DoF " 1{2 degrees of freedom.

A.3 Maxwell electrodynamics
The Lagrangian for standard electrodynamics in d ě 2 dimensions is where the components of the vector field A µ " A µ px 0 , x i q conform the a priori independent field variables Q A and thus N " d.The theory enjoys a manifest U p1q gauge invariance, under the transformation which is of the relevant form (2.2).It follows that g " 1 and e " 2. The (primary) equations of motion for A µ following from (A.13) are of the form (2.10), with It is easy to see that the (row) rank of the (primary) Hessian is d ´1.A convenient left null vector for it is V p1q " δ 0 µ .As per (2.12) and since no identical vanishing happens, φ p1q :" B i F i0 " 0 (A.16) can be identified with the primary Lagrangian constraint in the theory l 1 " 1.In fact, this is Gauss's Law for the electric field.The (primary) equations of motion, together with the demand for stability under time evolution of the primary Lagrangian constraint, conform the starting point of the second iteration.They can be written as (2.21), where It is easy to see that, up to normalization, there exists only one linearly independent left null vector to W p2qÓ that does not trivially extend V p1q .We choose it as V p2qÓ " pδ i µ B i , 1q.We remark that the above is a particular instance of the general form prescribed in (2.29).As per (2.32), Hence, no secondary Lagrangian constraint exists l 2 " 0. The constraint algorithm thus terminates, by means of closure 2. Using (2.3) and (2.4), we obtain the familiar result N DoF " d ´2.

A.5 Podolsky electrodynamics
Podolsky's proposal for a generalized electrodynamics theory [53] L Po " ´1 4 is arguably the best-known higher-order field theory.So as to remain within the scope of section 2, we consider its first-order formulation [54] in d ě 2 dimensions The components of the vector fields A µ " A µ px 0 , x i q, B µ " B µ px 0 , x i q are the a priori independent field variables Q A and hence N " 2d.The Lagrangian (A.26) inherits the symmetry of Maxwell electrodynamics and is gauge invariant under the field transformations which are of the relevant form (2.2).It follows that g " 1 and e " 2, as in the Maxwell case earlier on.The (primary) equations of motion following from (A.26) are of the form (2.10), with where pW µν , A µ q were introduced in (A.15) and B µ stands for the same quantity as A µ , but in terms of B µ instead of A µ .It is rather obvious that the (row) rank of the (primary) Hessian is 2pd ´1q.A convenient choice for the two linearly independent left null vectors is V p1q 1 " pδ 0 µ , 0q and V p1q 2 " p0, δ 0 µ q.Using (2.12), since no identical vanishing happens and taking into consideration the manifest functional independence, can be identified with the two primary Lagrangian constraints in the theory l 1 " 2.
In order to ensure the stability of the primary Lagrangian constraints, we calculate the theory's secondary equations of motions (2.20).We find and consider them together with the (primary) equations of motion, as described in (2.21) and (2.22).Beyond trivial extensions of V p1q 1 and V p1q 2 before and up to normalization, there exists another linearly independent left null vector V p2qÓ to W p2qÓ .It can be found as prescribed around (2.28).Explicitly, we postulate In (A.35), the components of the vector field A µ " A µ px 0 , x 1 q are the a priori independent field variables Q A and so N " 2. By definition, theories within the (E)PN class(es) do not possess any local symmetry.Therefore, g, e " 0. The (primary) equations of motion following from (A.35) are of the form (2.10), with where we have introduced the short-hands and the second (total) derivative of α has been defined for later convenience.It is easy to see that the (row) rank of the (primary) Hessian is 1.Following [58,59], we choose a left null vector to W p1q as As per (2.12) and since no identical vanishing happens, can be readily regarded as the only primary Lagrangian constraint in the theory l 1 " 1.Following (2.20), at the second iteration we find where we have introduced As per (2.21) and (2.22), we proceed to the joint consideration of the primary and secondary equations of motion.In particular, we inspect the row rank of W p2qÓ via a conveniently normalized generic left null vector of the form (2.29) Moreover, straightforward examination unequivocally indicates that the above determinant is functionally independent from both the primary (A.41) and the secondary (A.50) Lagrangian constraints.This proves there exists a maximal row rank minor within W p3qÓ .Hence, W p3qÓ itself has maximal row rank and does not admit left null vectors that non-trivially extend V p1q and V p2qÓ above.Using (2.3) and (2.4), we independently reproduce the recent result in [59] that the Minimal Model propagates N DoF " 1 degree of freedom, as a massive electrodynamics theory in 2 dimensions must do.

A.7 2-dimensional Palatini
Consider Einstein-Hilbert theory of gravity in 2 dimensions.A popular first-order reformulation attributed to Palatini [60] is where h µν " h νµ and G λ µν " G λ νµ denote tensors proportional to the metric and the connection, respectively.Their independent components conform the a priori independent field variables where we have introduced multiple renamings for notational simplicity.It readily follows that N " 9.The theory is invariant under the field transformations [61,62] which are of the relevant form (2.2).Here, θ µν " θ νµ , so it has g " 3 independent components.It is easy to see that e " 6.
As a direct consequence of the exclusively linear dependence of the Lagrangian (A.52) on the generalized velocities 9 Q A , the (primary) Hessian vanishes W p1q " 0 and its linearly independent left null vectors can be chosen as pV p1q I q A " δ A I , I " 1, 2, . . ., M 1 " 9. (A.55) Then, the components of the U p1q vector coincide with the relations (2.12): U p1q " φ p1q , where `h1 G 1 11 q " 0, φ 5 :" 2 `9 h 1 `hG 1 ´h11 G 1  11 ˘" 0, φ 6 :" 9 h 11 `2ph 1 G 1 `h11 G 1 1 q " 0, φ 7 :" B 1 h `2phG 1 `h1 G 11 q " 0, φ 8 :" 2 `B1 h 1 ´hG `h11 G 11 ˘" 0, φ 9 :" B 1 h 11 ´2ph 1 G `h11 G 1 q " 0. (A.56) (For simplicity, we omit indices indicating the iteration.)Observe that the first six relations depend on a distinct generalized velocity each, while the latter three do not depend on generalized velocities.Namely, functional independence is manifest for tφ 1 , . . ., φ 6 u.Functional (in)dependence among φ red " tφ 7 , φ 8 , φ 9 u is nonobvious and so we proceed to test it via (2.13).Postulating Γ red " `Γ0 `Γ1 B 1 , r Γ 0 `r Γ 1 B 1 , p Γ 0 `p Γ 1 B 1 ˘, (A.57) the reduced left null vector condition Γ red ¨φred " 0 yields no non-trivial solution, without much algebraic effort.It follows that φ red indeed comprises functionally independent relations.Consequently, l 1 " 9 and the relations (A.56) themselves can be regarded as the primary Lagrangian constraints in the theory.We proceed to the second iteration.As per (2.18), the secondary equations of motion are given by W p2q " ¨0 ´w 0 w 0 0 0 0 0 ‹ ‚, w " diagp1, 2, 1q, U p2q " ˜U 9 φ red ¸(A.58) where the components of U are It is evident that rrank `W p2q ˘" 6.Following (2.28) and leaving aside trivial extensions of the primary left null vectors (A.52), we find M 2 " 3 additional linearly independent left null vectors, up to normalization.We conveniently choose them as " `0, 0, . . ., 0 loooomoooon Following the prescription in (2.33), the search for functional (in)dependence within the secondary stage involves an ansatz for a vector Γ p2q with up to two spatial derivatives.Nonetheless, due to the relative structure of the first term in each relation, it is easy to see that Γ p2q ¨φp2q " 0 only admits the trivial solution.As a result, all relations in (A.61) are functionally independent among themselves.