An extension of the Dirac and Gotay-Nester theories of constraints for Dirac dynamical systems

This paper extends the Gotay-Nester and the Dirac theories of constrained systems in order to deal with Dirac dynamical systems in the integrable case. Integrable Dirac dynamical systems are viewed as constrained systems where the constraint submanifolds are foliated, the case considered in Gotay-Nester theory being the particular case where the foliation has only one leaf. A Constraint Algorithm for Dirac dynamical systems (CAD), which extends the Gotay-Nester algorithm, is developed. Evolution equations are written using a Dirac bracket adapted to the foliations and an abridged total energy which coincides with the total Hamiltonian in the particular case considered by Dirac. The interesting example of LC circuits is developed in detail. The paper emphasizes the point of view that Dirac and Gotay-Nester theories are dual and that using a combination of results from both theories may have advantages in dealing with a given example, rather than using systematically one or the other.


Introduction
In this brief introduction we give a sketch of the present work. References, preliminaries and a description of the main works related to the present paper will be given in section 2. The Dirac theory of constrained Hamiltonian systems was written by Dirac in terms of the canonical Poisson brackets in the space of classical observables (functions on the phase space) avoiding the notion of constraint submanifolds, which, on the other hand, is naturally present in the theory. For instance, instead of introducing the final constraint submanifold the notion of weak equality of functions is preferred. This indicates a deliberate decision to study the Poisson-algebraic aspect of constrained Hamiltonian systems, and for good reasons, in view of the success of the theory.
However, from the point of view of classical mechanics the states are points in phase space and the constraint submanifolds also play an important role in understanding the geometry of equations of motion and solution curves. This point of view also suggests, for instance, that one can view a constrained Hamiltonian system as an IDE (Implicit Differential Equation). Then the Dirac algorithm, as well as several questions about dynamics like existence of solutions for the initial condition problem, has a meaning also in the context of IDEs, which is an active field of study.
In many papers after Dirac's work, for instance the work of Gotay and Nester, cited in the next section, the geometric side of his theory has been developed and proved very useful. The geometric side is, in a sense, dual to the algebraic side and this duality is apparent in the commutative diagrams of subsection 3.2 where geometric diagrams have a Poisson-algebraic counterpart. This algebro-geometric possibility of approaching questions is present throughout the paper. Even though there are many beautiful works emphasizing the Poisson-algebraic or the geometric aspects our main references will be the works of Dirac and those of Gotay and Nester, respectively.
The notion of Dirac structure (Courant and Weinstein [1988]; Courant [1990]; Bursztyn and Crainic [2009]) is originated in part in Dirac's work and gives a new possibility to understand and extend the theory. The present paper should be followed soon by a generalization in the realm of Dirac geometry.
In section 2 we review the Gotay-Nester and the Dirac algorithms and define the notion of secondary constraint submanifolds, in particular, the final constraint submanifold. Both the primary and the final constraint submanifolds are important for writing equations of motion.
In subsection 3.2 we perform a careful study of the primary and final constraint submanifolds and various quotient manifolds and commutative diagrams, which helps to understand important aspects of the dynamics, for instance, the question of uniqueness of solutions. The latter is related to the notion of physical variables. We also study the dual point of view using the Poisson algebra of first class constraints and various quotients and commutative diagrams, which shows the duality between the geometric and the Poisson-algebraic points of view. In subsection 3.3 we show that the second class constraint submanifolds are submanifolds of the phase space tangent to a second class vector subbundle along the final constraint submanifold, which may lead to a classification, at least in some examples, of second class constraint submanifolds modulo tangency. The second class vector subbundle carries enough information to write the Dirac bracket at points of the final constraint submanifold. We write equations of motion in terms of the Dirac bracket in subsection 3.4.
Another feature of our work is that notions such as second class constraint submanifolds and Dirac brackets are defined globally (in the sense explained in theorem 3.20), and notions such as the standard Dirac bracket appear as coordinate expressions of a global object.
Along this paper we assume strong regularity conditions that lead for instance, to the fact that the Dirac bracket is locally constant, which can be established using the Weinstein splitting theorem.

Constraint Algorithms.
Implicit Differential Equations We now briefly review some basic results concerning general IDEs and constraint algorithms. Let M be a given differentiable manifold. An IDE on M , written as ϕ(x,ẋ) = 0, (2.1) of which ODEsẋ − f (x) = 0 or algebraic equations φ(x) = 0 are considered trivial particular cases, appear naturally in science and technology. A solution of (2.1) at a point x is a vector (x,ẋ) ∈ T x M satisfying (2.1). A solution curve, say x(t), t ∈ (a, b), must satisfy, by definition, that (x(t),ẋ(t)) is a solution at x(t) for all t ∈ (a, b). In the local case, M is an open set of R n and the IDE can be written equivalently in the formẋ = u 0 = ϕ(x, u), that is, A(y) ·ẏ = f (y), where y = (x, u) and f (y) = (u, ϕ(x, u)). This is by definition a quasilinear IDE. Basic questions such as existence, uniqueness or extension of solutions are not completely answered yet, although many partial results in this direction have been established. One of the common features of those results is that they show, at least under certain regularity conditions, how to transform, using a certain constraint algorithm, a given IDE into an equivalent parameter dependent ODE on a certain final constraint manifold.
Very briefly, the idea consists in finding a decreasing sequence of constraint submanifolds M ⊇ M 1 ⊇ . . . ⊇ M c , which appears naturally by imposing the condition of existence of a solution (x,ẋ) ∈ T M k to the given IDE at a each point x ∈ M k+1 . Under certain additional conditions, like locally constant rank conditions, the original IDE is reduced to an equivalent ODE depending on parameters on the final constraint submanifold M c , which, by construction, has the fundamental property that it must contain all solutions curves of the given IDE.
In spite of the simplicity of the general algorithm, there are fundamental examples where extra meaningful structures are used to build the submanifolds M k and to write the equations of motion. Moreover, in the Dirac approach the sequence of submanifolds M k is not emphasized and the methods of Poisson geometry are used, for good reasons.
In the present paper we will concentrate on the Dirac and the Gotay-Nester points of view (which represent the algebraic and the geometric side), see Dirac [1950Dirac [ , 1958Dirac [ , 1964 and Gotay, Nester, and Hinds [1978]. One may say that some aspects of Dirac's idea have been nicely formalized and generalized in Gotay, Nester, and Hinds [1978] in the context of presymplectic geometry on reflexive Banach manifolds. Both the algebraic and the geometric aspects of Dirac's theory have been treated by many people, cited below, with different ideas. The Dirac algorithm is not the same as the Gotay-Nester algorithm although the two methods are essentially equivalent in fundamental examples, like degenerate Lagrangian systems, as shown in Gotay, Nester, and Hinds [1978]. The Dirac algorithm provides explicit equations of motion written in terms of the canonical bracket of the ambient symplectic manifold and a total Hamiltonian depending on parameters. Besides, the Dirac approach yields the Dirac bracket defined in a neighborhood of the final constraint submanifold. Equations of motion written in terms of the Dirac bracket are specially simple and elegant, as we will see. On the other hand, the IDE obtained on the final constraints submanifold by the Gotay-Nester algorithm does not depend on any parameters or an embedding in a symplectic manifold.
General assumptions. All manifolds involved will be finite-dimensional smooth manifolds and all maps will be smooth, unless otherwise specified. Several arguments in this paper are of a local character, but they can be regarded as coordinate versions of global results. For instance, this is the case for the notion of second class constraints, which represent a second class constraint submanifold.
The Gotay-Nester and the Dirac constraint algorithms considered in this paper can be considered as particular ways of writing the general constraint algorithm for (quasilinear) IDEs mentioned above, using the special structure available in each case (presymplectic structures, symplectic and Poisson structures, respectively). Therefore, the sequence of secondary constraints M k , k = 1, . . . is the same for all these algorithms. It is important to have a criterion to ensure that this sequence stops. We will assume that each M k+1 is a closed submanifold of M k defined by equations. Also, we assume that for each k and each x ∈ M k+1 , if dim x M k = dim x M k+1 , then x ∈ M k+p and dim x M k = dim x M k+p for all p ∈ N. This implies that the sequence stabilizes, that is there is some c such that M c = M c+p , for all p ∈ N.
For each M k one has the corresponding k-th IDE, which can be written as A point x will be in M k+1 , by definition, iff this equation has a solution (x,ẋ). We will assume throughout the paper that the rank of A k (x) is locally constant on M k+1 . This implies that the rank of A c (x) is locally constant on M c , and on each point of M c there is at least one solution that is tangent to M c . The main property of M c from the dynamical point of view is that every solution curve to the original system must lie on M c . Since we are assuming the locally constant rank condition, the final system on M c can be converted, at least locally, into a parameter-dependent family of ODEs.
We will recall some essential aspects of the Dirac theory of constraints, following Dirac [1964], but using a more modern language, adapted to our purposes, and assuming explicitly certain regularity conditions. Dirac's theory starts with a given singular Lagrangian system L : T Q → R, since in the case of a regular Lagrangian the theory becomes trivial. Then, in order to construct a Hamiltonian theory one must consider from the beginning the image of the Legendre transformation, which may be a very complicated subset M 0 ⊆ T * Q. By definition, solution curves to the Hamiltonian system must be exactly the image of solutions to Euler-Lagrange equations under the Legendre transformation. Natural questions such as existence of solution curves to the Euler-Lagrange equations for a given initial condition are not completely solved in general, to the best of our knowledge. In order to obtain this kind of results one would need to choose mathematically precise hypotheses, a topic not considered in Dirac's work.
In this paper we will assume a general hypothesis about regularity , under which this kind of problem is easier. Regularity means, among other hypotheses to be established along the paper as they are needed, that certain sets M 0 ⊇ M 1 . . . are submanifolds of T * Q defined regularly by equations φ are functions defined on T * Q constructed by the Dirac constraint algorithm and called constraints. The submanifolds M k are called the constraint submanifolds.
As usual, one assumes that the algorithm stops for k = c. One also assumes that the ranks of the matrices ({φ . . , c, are called secondary constraints. The main property of the final constraint submanifold M c is that any motion of the classical particle, that is, any solution (q(t), p(t)), must remain in M c , and Dirac shows how to write Poisson equations of motion in terms of position and momentum using the canonical Poisson bracket on T * Q and the total Hamiltonian H T . For given initial conditions belonging to M c , solutions are not necessarily unique and Dirac interprets this fact as being due to the nonphysical character of some of the variables. Of fundamental importance for Dirac's theory, especially for quantization, are the classification of constraints into first class and second class in terms of certain commutation relations, and the construction of a Poisson bracket { , } * called Dirac's bracket. An important result is that with respect to the Dirac bracket all final constraints φ (c) i appear to be first class constraints, in other words, {φ Dirac's procedure also shows how to deal with the nonphysical variables and find the correct notion of state of the system. One shows that there are physically meaningful variables in terms of which the evolution for a given initial state is determined. This is important from the classical and also from the quantum mechanics point of view. Now, we will be more precise. The image of the Legendre transformation F L : T Q → T * Q, that is, M 0 = F L(T Q), contained in the canonical symplectic manifold T * Q, is assumed to be defined by equations φ In the case in which the Legendre transformation is degenerate the Hamiltonian H : T * Q → R is not uniquely defined from the formulas pv − L(q, v) = H(q, p), p = ∂L(q, v)/∂v, but in Dirac's theory one assumes that such a function can be conveniently defined on M 0 (which can be done in examples using ideas akin to the Pontryagin maximum principle, like the fact that for each (q, p) ∈ M 0 the derivative of pv − L(q, v) with respect to v is 0) and then extended, more or less arbitrarily, to T * Q. Then one defines, following Dirac, the total Hamiltonian i , with arbitrary parameters λ i (0) to be determined. The Dirac constraint algorithm goes as follows. The preservation of the primary constraints is written {φ Then M 1 is defined by the condition that x ∈ M 1 if and only if there exist λ (0) = (λ 1 (0) , . . . , λ a 0 (0) ) such that the system of equations φ i , H T }(x) = 0, i = 1, . . . , a 0 , is satisfied. Clearly, M 0 ⊇ M 1 , and one assumes that M 1 is a submanifold regularly defined by equations, say, φ is a secondary constraint, by definition. By proceeding iteratively one obtains a sequence M 0 ⊇ M 1 ⊇ . . . , and we will assume that this sequence stops. Then there are final constraints, say φ For each x ∈ M c the space of solutions of the linear system of equations (2.2) in the unknowns λ j (0) is an affine subspace of R a 0 , called S (c) x whose dimension is a locally constant function d (c) j }(x)). One can locally choose d (c) (x) unknowns as being free parameters and the rest will depend affinely on them. Then the solutions of (2.2) form an affine bundle S (c) over M c . After replacing λ (0) ∈ S (c) in the expression of the total Hamiltonian, the corresponding Hamiltonian vector field, x ∈ M c , which will depend on the free unknowns, will be tangent to M c . Its integral curves, for an arbitrary choice of a time dependence of the free unknowns, will be solutions of the equations of motion, which is the main property of the final constraint submanifold M c from the point of view of classical mechanics. The lack of uniqueness of solution for a given initial condition in M c , given by the presence of free parameters, indicates, according to Dirac, the nonphysical character of some of the variables. In our context the physical variables can be defined on a quotient manifold.
Remark Dirac introduces the notion of weak equality for functions on T * Q. Two such functions are weakly equal , denoted f ≈ g, if f |M c = g|M c . Then, for instance φ i , for some functions ν i on T * Q and conversely.
Since we have introduced the notion of a constraint submanifold, in particular the final constraint submanifold, we prefer not to use the notation ≈. Now let us make some comments on the notions of first class and second class constraints. The rank of the matrix ({φ j }(x)), i, j = 1, . . . , a c , is necessarily even, say, 2s, and it is assumed to be constant. Then, using elementary properties of determinants (like adding to a row or column a linear combination of the other rows or columns) one can find, at least locally in a neighborhood of each point x ∈ M c , functions ψ i , i = 1, . . . , a c − 2s, with, χ j , j = 1, . . . , 2s, such that the equations ψ i = 0, χ j = 0, define M c regularly and, besides, . . , a c − 2s, j, j ′ = 1, . . . , 2s and x ∈ M c . In fact, we will assume that this is can be done globally, for simplicity. The φ (c) j are linear combinations with smooth coefficients of the χ j and ψ i , and conversely. The functions χ j , j = 1, . . . , 2s, are called second class constraints and the functions ψ i , i = 1, . . . , a c − 2s, are called first class constraints.
More generally, any function ρ on T * Q satisfying ρ|M c = 0, {ρ, ψ i }|M c = 0, {ρ, χ j }|M c = 0, is a first class constraint with respect to the submanifold M c , by definition. Any function g on T * Q satisfying {g, ψ i }|M c = 0, {g, χ j }|M c = 0, is a first class function, by definition. For instance, the total Hamiltonian H T is a first class function. Now define the Hamiltonian h c in terms of ψ i , χ j , i = 1, . . . , k, j = 1, . . . , 2s, as The preservation of the constraints for the evolution generated by h c can be rewritten as {ψ i , h c }(x) = 0, which is equivalent to {ψ i , H}(x) = 0 for all x ∈ M c , and {χ j , h c }(x) = 0, for all x ∈ M c . The latter is equivalent to for all x ∈ M c , which determines the µ j as well-defined functions on M c . Then the solutions (µ(x), λ) form an affine bundle with base M c and whose fiber, parametrized by the free parameters λ, has dimension a c − 2s. Any section (µ(x), λ(x)) of this bundle determines h c as a first class function. This means that X hc (x) ∈ T x M c , for each x ∈ M c , and therefore a solution curve of X hc is contained in M c provided that the initial condition belongs to M c . We will show in this section that there is a symplectic manifoldM c such that one can pass to the quotient M c →M c and also h c passes to the quotienth c in such a way that solution curves of h c become solutions curves of the Hamiltonianh c . Moreover, we will show that there is a manifoldM c and natural maps M c →M c →M c such that the Hamiltonian h c passes to the quotient to a functionh c onM c . The extended Hamiltonian defined by Dirac is related toh c . One can show thath c passes to the quotient viaM c →M c to the functionh c defined above.
Dirac defines an interesting bracket, now called the Dirac bracket, which is defined on an open set in T * Q containing M c , where c ij , which by definition is the inverse matrix of {χ i , χ j }, is defined. The Dirac bracket is a Poisson bracket and has the important property that, for any function F on T * Q, the condition {F, χ j } * = 0, j = 1, . . . , 2s, is satisfied on a neighborhood of M c , which implies thaṫ Because of this, one may say that, with respect to the Dirac bracket, all the constraints χ j , j = 1, . . . , 2s and ψ i , i = 1, . . . , k, are first class with respect to M c . This is important for purposes of quantization.

A brief Review of the Gotay-Nester Theory
In this section we recall some aspects of the Gotay-Nester theory Gotay, Nester, and Hinds [1978] which we need. This theory was developed in Gotay, Nester, and Hinds [1978] to deal geometrically with the Dirac-Bergman theory of constraints. The main equation studied is an IDE of the type where ω is a closed 2-form on a manifold M and α ∈ Ω 1 (M ) is a closed 1-form on M . As we have indicated before this kind of equation appears naturally in classical Lagrangian mechanics, in fact, we will show later that the Euler-Lagrange equations can be rewritten equivalently in the form which is clearly of the type (2.3).
Description of the Gotay-Nester Algorithm. As we mentioned in the Introduction, in order to deal with IDEs one can apply a basic idea which consists in building a sequence of constraint submanifolds. Let us first describe that basic approach for a system like (2.3) without using explicitly the presymplectic form, and later on we will briefly explain how the presymplectic form can be used to write equations for the constraints submanifolds explicitly. The latter is an important contribution of the Gotay-Nester algorithm. We will assume for simplicity that M is a vector space.
We want to find solution curves to (2.3). Let x(t) be such a solution curve; then for each t the linear algebraic system has at least one solution, namely, v(t) =ẋ(t). This implies that, for each t, x(t) must belong to the subset Assume, as in Gotay, Nester, and Hinds [1978], that M 1 is a submanifold of M . Since x(t) ∈ M 1 for all t we must have thatẋ(t) ∈ T x(t) M 1 for all t. This implies that, for each t, x(t) must belong to the subset We can continue in a similar way and define M k+1 recursively as This sequence stabilizes, under the General Assumptions described in section 1. Under the assumption that the map has locally constant rank on the final constraint manifold M c , existence of local solution curves to (2.3) for each initial condition in M c is guaranteed. For given local coordinates (x 1 , . . . , x m ) on M c and for a given initial condition x 0 ∈ M c one can fix some appropriate coordinates as functions of t, say x 1 (t), . . . , x r (t), where r = dim ker ω, in a neighborhood of x 0 and then solve (2.3) uniquely for x r+1 (t), . . . , x m (t). More precisely, in local coordinates our equation becomes (2.5) i = 1, . . . , n, j = 1, . . . , m. Since r = dim ker ω, in a neighborhood of a given point x 0 one can solve, after relabeling the coordinates if necessary, x r+k = x r+k (x 1 , . . . , x r ), k = 1, . . . , m − r. After choosing arbitrarily the curves x 1 (t), . . . , x r (t) and replacing in (2.5) one obtains a time dependent ODE in the remaining variables x r+1 , . . . , x m . We can also interpret the previous arguments by saying that the implicit differential equation The following geometric description will be useful later on. Since by assumption r does not depend on x ∈ M c , at least locally, then the equation on M c where X ∈ T M c , defines an affine distribution on M c of locally constant rank. More precisely, one has an affine bundle S (c) with base M c whose fiber S (c) x at a given point x ∈ M c is, by definition, is not locally constant we still have a distribution S (c) on M c , but it may be singular. The analysis of existence of solution curves in this case may be problematic, see Cendra and Etchechoury [2006], Pritchard [2003] and references therein. The algorithm developed in Cendra and Etchechoury [2006] for a general system of the type a(x) ·ẋ = f (x), with analytic data, represents an improvement of the previous basic ideas also in the sense that the final system obtained after applying the algorithm has always locally constant rank, and that singular cases are also studied using desingularization methods.
(b) In Gotay, Nester, and Hinds [1978] it is explained how solutions can be expressed using brackets, as in Dirac's work.
Example. Let L : T Q → R be a Lagrangian, degenerate or not. Since the problem is of a local nature we can use local coordinates. Let E(q, v, p) = pv − L(q, v) and let ω ∈ Ω 2 (T Q ⊕ T * Q) be the presymplectic form ω = dq i ∧ dp i on the Pontryagin bundle M = T Q ⊕ T * Q. Then Euler-Lagrange equations are written equivalently in the form of equation (2.4) with x = (q, v, p). In fact, we have Using equations (2.8)-(2.10) we can easily see that (2.4) is equivalent tȯ which is clearly equivalent to the Euler-Lagrange equations. The idea of using the Pontryagin bundle to write important equations of physics like Euler-Lagrange or Hamilton's equations appears in Livens [1919]; Skinner [1983]; Skinner and Rusk [1983a,b]; Cendra, Marsden, Pekarsky, and Ratiu [2003]; Yoshimura and Marsden [2006a,b].
Describing the Secondary Constraints Using ω Now we shall explain how the constraint manifolds M k defined by the algorithm can be described using the presymplectic form ω. This is a simple but important idea because, in particular, we show how each M k is defined by equations written in terms of ω. Depending on the nature of ω one may obtain analytic, smooth, linear, etc., equations, which may simplify matters in given examples. This idea is also important in the context of reflexive Banach manifolds, as remarked in Gotay, Nester, and Hinds [1978]. Besides, those equations will be obviously invariant under changes of coordinates preserving ω.
The idea is the following. The condition defining the subsets M k+1 , k = 0, 1, . . . , calling M 0 = M to uniformize the notation, namely,

Preliminaries
We will need the following results about linear symplectic geometry which are an essential part of many of the arguments in our treatment of Dirac and Gotay-Nester theories. This is because under our strong regularity assumptions those theories are, to a certain extent, linear.
Lemma 3.1. Let (E, Ω) be a symplectic vector space of dimension 2n, V ⊆ E a given subspace. For a given basis Then the rank of the matrix [α i (X j )] is even, say 2s, and X i , i = 1, . . . , r form a basis of V Ω . Moreover, the basis α i , i = 1, . . . , r can be chosen such that for all j = 1, . . . , r i . The first part of the lemma is easy to prove using this.
Since the first 2s columns of the matrix [α i (X j )] are linearly independent and the rest are zero, we must have λ i = 0, for 1 ≤ i ≤ 2s, and λ i , i = 2s + 1, . . . , r are arbitrary. This means that V ∩ V Ω is generated by X i , i = 2s + 1, . . . , r.
Let ω be the pullback of Ω to V via the inclusion. ω = j * Ω. Then (V, ω) is a presymplectic space. In what follows, the ♭ and ♯ operators are taken with respect to Ω unless specified otherwise.
Lemma 3.5. Let γ i , i = 1, . . . r be a given basis of V • and let Y i = γ ♯ i , i = 1, . . . r. Let β ∈ E * be given. Then the following conditions are equivalent.
. . , λ r ) satisfies (3.1). Then we have X ∈ V as we have just seen and we also have We have proven that X is a solution to (3.3).
To prove that every solution X to (3.3) can be written as before, we can reverse the previous argument. Using this, it is clear that if ω is symplectic then (3.1) has unique solution, in particular, we have that det Since Y j , j = 1, . . . , r is a basis of V Ω , using lemma 3.4 and the fact that V ω = {0} we get that λ j = 0 for j = 1, . . . , r.
Proof. ker ω = V ω , which has dimension r − 2s from corollary 3.3 and lemma 3.4. On the other hand the dimension of the subspace of λ satisfying (3.1) is clearly also r − 2s, since the coefficient matrix has rank 2s.

A Poisson-Algebraic and Geometric Study of the Primary and
Final Constraint Submanifolds.
The Dirac algorithm, briefly explained in the previous subsection, can be applied to any given constrained Hamiltonian system (P, Ω, H, M ) where (P, Ω) is a symplectic manifold, the primary constraint submanifold M is a given submanifold of P defined regularly by an equation φ = 0 and H is a Hamiltonian defined on P . This is because the particular cotangent bundle structure of the symplectic manifold T * Q is not essentially used in the Dirac algorithm. For instance, an interesting variant of the Dirac algorithm for a degenerate Lagrangian system is the following.
Consider the canonical symplectic manifold N = T * T Q with the canonical symplectic form Ω, and let the primary constraint be M = T Q ⊕ T * Q, canonically embedded in N via the map given in local coordinates (q, v, p, ν) Remark This embedding of M in N is globally defined, and it is a coisotropic embedding, see Appendix A. For a given presymplectic manifold (M, ω) one can always find a coisotropic embedding in a symplectic manifold (P, Ω), see Gotay and Sniatycki [1981/82]. The number pv is a well-defined function on M and can be naturally extended to a function on a chart with coordinates (q, v, p, ν), but this does not defines a function on N consistently. In any case, it can be extended to a smooth function on N and any such extension will give the same equations of motion. The Dirac theory of constraints is essentially a local theory. However, we will see a global version of the notion of Dirac bracket, in a sense, as well as its local descriptions.
Consider the function E : M → R given by E = pv − L(q, v). Using the fact that E can be extended naturally on a chart with coordinates (q, v, p, ν) and taking an appropriate partition of unity we can choose once and for all an extension to a smooth function E on N called the Energy. Then we can apply the Dirac algorithm to the constrained Hamiltonian system (N, Ω, E, M ).
There are some interesting features in this approach, as compared with the one described in subsection 2.1, where the symplectic manifold is T * Q and the primary constraint is the image of the Legendre transformation. For instance, the primary constraint M = T Q ⊕ T * Q is well defined in a natural way as a closed submanifold of the symplectic manifold N . Besides, the comparison with the Gotay-Nester approach becomes clear from the beginning and the Euler-Lagrange equations are derived quickly as a differential-algebraic equation (DAE). On the other hand, this approach may have the disadvantage of introducing the extra variable v, which may lead to longer calculations in some examples.
We shall start with the constrained Hamiltonian system (N, Ω, E, M ), and we will work locally, for simplicity. We will call φ i (0) , i = 1, . . . , r 0 , the primary constraints ν i defining M 0 ≡ M regularly by equations.
We will emphasize the Gotay-Nester point of view and we will see how it combines with the Dirac procedure.
Accordingly, we shall study the Dirac dynamical system on the manifold M = T Q ⊕ T * Q, already considered in the example at the end of section 2.2 where the Dirac structure is associated to a presymplectic form ω which is the pullback of the symplectic form Ω on N = T * T Q via the inclusion M ⊆ N . Then M is the primary constraint. Then the equation to be solved, according to the Gotay-Nester algorithm, is the equation where X ∈ T x M c and x ∈ M c , M c being the final constraint. Let ω c be the pullback of Ω via the inclusion of M c in N . Since ω c is presymplectic, ker ω c is an involutive distribution.
From now on we will assume the following.
Assumption K 1 . The distribution ker ω c has constant rank and defines a regular foliation K c , that is, the natural map p Kc : M c →M c , whereM c = M c /K c is a submersion.
Lemma 3.8. The following assertions hold: (a) There is a uniquely defined symplectic formω c onM c such that p * Kcω c = ω c . (b) LetX be a given vector field onM c . Then there is a vector field X on M c that is p Kc -related toX.
(c) Letf ∈ F(M c ). Then there exists a vector field X on M c such that X is p Kcrelated to Xf , and for any such vector field X the equality ω c (x)(X, Then one can choose the functionf ∈ F(M c ) and the vector field X in (c) in such a way that X(x 0 ) = X x 0 .
Proof. (a) By definition, the leaves of the foliation K c are connected submanifolds of M c , that is, each A, B). To prove that this is a good definition observe first that it is a consistent definition for fixed x, which is easy to prove, using the fact that ker ω c (x) = ker T x p Kc . Now choose a Darboux chart centered at x, say U × V , such that, in this chart, p Kc : U × V → U and ω c (x 1 , x 2 ) =ω c (x 1 ), where ω c (x 1 , x 2 ) andω c (x 1 ) are independent of (x 1 , x 2 ). This shows thatω c is well defined on the chart. Using this and the fact that one can cover the connected submanifold p −1 Kc (z) with charts as explained above, one can deduce by a simple argument thatω c (z) is well defined.
(b) Let g be a Riemannian metric on M c . Then for each x ∈ M c there is a uniquely determined X(x) ∈ T x M c such that X(x) is orthogonal to ker T x p Kc and T x p Kc X(x) =X(x), for all x ∈ M c . This defines a vector field X on M c which is p Kc -related toX.
(c) Givenf and using the result of (b) we see that there is a vector field X on M c that is p Kc -related to Xf . Then, for every x ∈ M c and every Y (d) One can proceed as in (b) and (c) and choosingf such that (df (p Kc (x 0 ))) ♯ = T x 0 p Kc X x 0 and, besides, the metric g such that X x 0 is perpendicular to ker T x 0 p Kc .
The space of first class functions is defined as In other words, R (c) is the largest subset of F(N ) satisfying i = 0, respectively. Dirac was interested in classical mechanics, where states are points in phase space, as well as in quantum mechanics where functions are observables and states are not points in phase space. In the present paper we focus mainly in classical mechanics, and therefore we need to concentrate on the constraint submanifolds. In particular, M and M c are the only ones that play an important role. The other secondary constraints submanifolds seem to be less important.
(b) The total Hamiltonian H T is a first class function, by construction.
. Moreover, for any vector field X on M c that is p Kc -related to a vector field Xf onM c there exists a function f ∈ R (c) such that f |M c = p * Kcf and X = X f . In particular, any vector field X on M c satisfying X(x) ∈ ker ω c (x) for all x ∈ M c is p Kc -related to the vector field 0 on the symplectic manifoldM c , which is associated to the functionf = 0, therefore there exists a function f ∈ R (c) , which satisfies f |M c = 0, such that X( . Then X f (x) and X g (x) are both tangent to M c at points x of M c which implies that −X {f,g} (x) = [X f , X g ](x) is also tangent to M c at points of x of M c . This shows that {f, g} ∈ R (c) . It is easy to see that any linear combination of f, g and also f g belong to R (c) .
(b) By definition ∆ R (c) ⊆ T M c . We need to show the converse inclusion. Let Choose the function f and the vector field X on M c as in lemma 3.8, (d). Choose any extension of p * Kcf to a function g on N . For each x ∈ M c , we can apply lemmas 3.5 and 3.6 with which can be naturally extended by 0 on N . Then it is easy to see, using the fact for each x ∈ M c , and in particular X 0 = X(x 0 ) = X f (x 0 ). Lemma 3.11. (a) Each function f ∈ R (c) is locally constant on the leaves of K c therefore, since they are connected, for each f ∈ R (c) there is a uniquely determined f ∈ F(M c ), called (p Kc ) * f , such that f |M c = p * Kcf . Moreover, the vector fields X f (x), x ∈ M c , on M c and Xf onM c are p Kc -related.
(b) For eachf ∈ F(M c ) there exists f ∈ R (c) such that f |M c = p * Kcf and the vector fields X f (x), x ∈ M c , on M c and Xf onM c are p Kc -related.
Proof. (a) Let f ∈ R (c) , we only need to show that f is constant on the leaves of K c , which is equivalent to showing that df (x)| ker ω c (x) = 0, for all x ∈ M c . For a given x ∈ M c , let v x ∈ ker ω c (x); then using lemma 3.10 (b) one sees that there is a function g ∈ R (c) such that v x = X g (x). Then we have Now we shall prove that X f and Xf are p Kc -related. For each x ∈ M c and each Using this we obtain which shows that Xf (p Kc (x)) = T x p Kc X f (x), becauseω c is symplectic and T x p Kc Y x represents an arbitrary element of T p Kc (x)M . (b) To find f we choose a vector field X that is p Kc -related to Xf according to lemma 3.8 and then use 3.10, (b).
Elements of I (c) are called first class constraints.
Proof. (a) Let f, g ∈ I (c) and h ∈ R (c) . Then it is immediate that f + g and hg belong to I (c) . For any h ∈ R (c) , we have {f, h}|M c = X h (f )|M c = 0.
(b) Follows from (a), using basic Poisson algebra arguments.
Lemma 3.14. The following conditions are equivalent for a function f ∈ R (c) . ( Proof. Assume (i). Then {f, h}|M c = 0 for all h ∈ R (c) , that is, df (x)X h (x)|M c = 0. By lemma 3.10, (b), we know that X h (x) represents any vector in T x M c . We can conclude that f |M c is locally constant, so (ii) holds. Now we will prove that (ii) implies (iii). Let f |M c be locally constant. Then for all g ∈ R (c) and all x ∈ M c , ).
Since, again by lemma 3.10, X g (x) represents any element of T x M c , we can conclude that X f (x) ∈ ker ω c (x), so (iii) holds true. Now we will prove that (iii) implies (i).
Assume that X f (x) ∈ ker ω c (x), x ∈ M c . Then for all g ∈ R (c) and all x ∈ M c , c) . Using this and the definitions, we see that f ∈ Z I (c) R (c) . Proof. Surjectivity of (p Kc ) * and the fact that its kernel is I (c) follows immediately from lemma 3.11 and the definitions. This implies that (p Kc ) * I (c) is an algebra isomorphism. Also, using the definitions, for f, g ∈ R (c) and any x ∈ M c we can prove easily that All the arrows are defined in a natural way and they are surjective Poisson algebra homomorphisms.

Equations of Motion and Physical Variables. It is immediate to see from the definitions that ker
for all x ∈ M c .
From now on we will assume the following.
Assumption K 2 . (a) ker ω(x) is a regular distribution, that is, it determines a regular foliation K and the natural projection p K : M →M , whereM = M/K, is a submersion.
(b) The distribution ker ω(x) ∩ T x M c is a distribution of constant rank.
Theorem 3.16. The distribution ker ω(x) ∩ T x M c is regular and has rank d (c) (x). Its integral manifolds are S ∩M c , where S is an integral manifold of ker ω. Moreover, these integral manifolds give a foliationK c of M c which is regular, that is, the natural map pK c : M c →M c , whereM c = M c /K c is a submersion. Besides, each leaf of the foliation K c is foliated by leaves ofK c , which gives a naturally defined submersion p KcKc :M c →M c . In other words, we obtain the commutative diagram where each arrow is a naturally defined submersion.
Proof. The first assertion, about the rank of the distribution ker ω(x)∩T x M c , is easy to prove. Let x 0 ∈ M c . Then there exists a uniquely determined integral manifold S of the distribution ker ω such that x 0 ∈ S. Using that, by assumption, ker ω(x) ∩ T x M c is a distribution of constant dimension and that dim (ker ω(x) ∩ T x M c ) = dim (T x S ∩ T x M c ) we can conclude that the intersection S ∩ M c coincides with the integral leaf of the integrable distribution of ker ω ∩T M c containing x 0 . So we obtain the foliationK c of M c . Using (3.5) we can deduce that each leaf of the foliation K c is foliated by leaves ofK c . The rest of the proof follows by standard arguments.
Lemma 3.17. (a) The following diagram is commutative where the arrows are defined as follows. The maps p K , p Kc , pK c and p KcKc are defined in Assumption K 1 , Assumption K 2 and theorem 3.16. By definition, the map f c is the inclusion. The mapf c is an embedding defined byf c (S ∩ M c ) = S, where S is a leaf of the foliation K. We will think off c as being an inclusion. The vector bundle TM |M c is the tangent bundle TM restricted toM c . In other words, sincef c is an inclusion, TM |M c is identified via some isomorphism, called ǫ c , with the pullback of TM byf c . We callF c the natural map associated to the pullback.
(b) The presymplectic form ω on M passes to the quotient via p K giving a uniquely defined symplectic formω onM , satisfying p * Kω = ω. The presymplectic form ω c , which, by definition is f * c ω, defines uniquely a presymplectic formω c oñ M c via pK c satisfying p * K cω c = ω c ,ω c =f * cω . The energy function E on M satisfies dE(x)| ker ω(x) = 0, for all x ∈ M c , therefore it defines uniquely a 1-form on TM |M c , called (F c • ǫ) * dE ∈ Γ (TM |M c ) * . Since E is constant on each leaf of K c , it also defines a functionẼ c onM c . Since TM c ⊆ TM c via the inclusion Tf c we have (F c • ǫ) * dE|TxM c = dẼ c (x), for allx ∈M c .
(c) Equation of motion (3.4) on M c passes to the quotientM c as , is a solution of (3.6). Therefore, a solution curve x(t) of (3.4) projects to a solution curvex(t) = pK c (x(t)) of (3.6) onM c . Equation (3.6) has unique solutionX(x) for eachx ∈M c . This solution also satisfies the equationω c (x)(X(x), ) = dẼ(x). (3.7) However solutions to equation (3.7) are not necessarily unique, since kerω c (x) is not necessarily 0. for all x ∈ M c , therefore there is a uniquely defined functionĒ c onM c such that p * KcĒ c = E|M c . The equationω has unique solutionX(x) forx ∈M c . IfX(x) is a solution of (3.6) thenX(x) = Txp KcKcX (x) is a solution of (3.8). Therefore, a solution curvex(t) of (3.6) projects to a solution curvex(t) = p KcKc (x(t)) of (3.8) onM c .
Proof. (a) The equality p Kc = p KcKc • pK c was proven in theorem 3.16. The equality τM •F c • ǫ c =f c • τM results immediately from the definitions. The equality p K • f c =f c • pK c results by applying the definitions and showing that, for given where S x is the only leaf of K containing x. (b) Existence and uniqueness ofω andω is a direct consequence of the definitions and standard arguments on passing to quotients. For any x ∈ M c we know that there exists a solution X of equation (3.4), from which it follows immediately that dE(x)| ker ω(x) = 0. The rest of the proof of this item consists of standard arguments on passing to quotients.
(c) We shall omit the proof of this item which is a direct consequence of the definitions and standard arguments on passing to quotients.
for all x ∈ M c . The rest of the proof is a consequence of standard arguments on passing to quotients.
Remark. Recall that the locally constant function d (c) (x) on M c is the dimension of the distribution ker ω(x) ∩ T x M c on M c and also the dimension of the fiber of the bundle S (c) (x). If d (c) (x) is nonzero then there is no uniqueness of solution to equation of motion (3.4), since solution curves to that equation satisfy, by definition, where (x,ẋ) ∈ T x M c . Passing to the quotient manifoldM c eliminates this indeterminacy and uniqueness of solution is recovered. This is related to the notion of physical variables mentioned by Dirac.

First Class and Second Class Constraints and Constraint Sub-
manifolds. The Tangent Bundle V Ω to a Second Class Submanifold.
As we have said before, an important topic in the theory of constraints as developed by Dirac is the distinction between first class and second class constraints.
His treatment is intended to solve systems with constraints coming from degeneracies in the Lagrangian from both the classical and the quantum mechanics point of view. The Poisson algebra structure of functions on a symplectic manifold is the context in which this theory is developed and it is not very geometric and almost no attention is paid to the constraint submanifolds defined by the several equations involved. Among several interesting references we citeŚniatycki [1974] which has several points of contact with our work.
In this paragraph we shall give a geometric framework and describe its close relationship to the Poisson algebra point of view to deal with the notions of first class and second class constraints and functions and also first class and second class submanifolds. These notions only depend on the final constraint submanifold M c and the ambient symplectic manifold N and do not depend on the primary constraint M 0 = M or the Hamiltonian H : N → R. Accordingly, in this paragraph we will adopt an abstract setting, where only an ambient symplectic manifold and a submanifold are given. This kind of abstract setting was studied inŚniatycki [1974], in particular the notion of second class constraint submanifold and its connection with the Dirac bracket.
Then we will go back to equations of motion in the next subsection, where the role of both the final and the primary constraint is essential.
The definitions given at the beginning of this section inspire the following one.
Definition 3.18. Let (P, Ω) be a symplectic manifold and S ⊆ P a given submanifold. Then, by definition, Elements of R (S,P ) are called first class functions. Elements of I (S,P ) are called first class constraints. The submanifold S is called a first class constraint submanifold if for all f ∈ F(P ) the condition f |S = 0 implies f ∈ I (S,P ) , that is, I (S,P ) ⊆ I (S,P ) , where I (S,P ) is the ideal of the ring F(P ) of all functions vanishing on S. Obviously, using the notation introduced before in the present section, R (Mc,N ) = R (c) and I (Mc,N ) = I (c) . All the properties proven for R (c) and I (c) hold in general for R (S,P ) and I (S,P ) . For instance, T x S is the set of all X f (x), f ∈ R (S,P ) . Every function f ∈ R (S,P ) satisfies df (x)(X x ) = 0, for all X x ∈ ker(Ω(x)|T x S) and ker(Ω(x)|T x S) is the set of all X f (x), f ∈ I (S,P ) .
The following lemma 3.19 is one of our main results. It studies the vector subbundles V ⊆ T P |S which classify all second class submanifolds S V containing S at a linear level, that is, V Ω is tangent to the second class submanifold. For such a second class submanifold, which is a symplectic submanifold, the Dirac bracket of two functions F and G at points x ∈ S can be calculated, by definition, as the canonical bracket of the restrictions of F and G. This has a global character.
A careful study of the global existence of a bracket defined on sufficiently small open sets U ⊆ P containing S which coincides with the previous one on the second class submanifold will not be considered in this paper. However, to write global equations of motion on the final constraint submanifold one only needs to know the vector bundle V Ω , which carries a natural fiberwise symplectic form.
Local descriptions of second class constraint submanifolds on sufficiently small open sets give the usual expression of the Dirac bracket.
All these are fundamental properties of second class constraints and constraint submanifolds, and theorem 3.20 collects the essential part of them.
(b) The vector subbundle V χ ⊆ T P |S with base S and fiber x ∈ S. (c) There is a neighborhood U of S such that the equations χ 1 = 0, . . . , χ 2s = 0, on U define a symplectic submanifold S χ such that S ⊆ S χ and for x ∈ S χ , where we have extended the definition of V χ x for x ∈ S χ using the expression V χ x = span (X χ 1 (x), . . . , X χ 2s (x)) ⊆ T x P, for x ∈ S χ . The submanifold S χ has the property I (S,S χ ) ⊆ I (S,S χ ) , that is, S is a first class constraint submanifold of S χ , defined regularly by ψ i |S χ = 0, i = 1, . . . , r − 2s, and ψ i |S χ ∈ I (S,S χ ) , i = 1, . . . , r − 2s. Moreover, it has the only possible dimension, which is dim S χ = dim S + dim ker ω = 2n − 2s, for symplectic submanifolds having that property. It is also a minimal object in the set of all symplectic submanifolds P 1 ⊆ P , ordered by inclusion, satisfying S ⊆ P 1 .
(d) Let V be any vector subbundle of T P |S such that (3.12) or equivalently, Then dim V x = 2s, for x ∈ S. Let S V be a submanifold of P such that T x S V = V Ω x , for each x ∈ S. Then S is a submanifold of S V . Such a submanifold S V always exists. Moreover, for such a submanifold there is an open set U ⊆ P containing S such that S V ∩ U is a symplectic submanifold of P .
Letx ∈ S and let χ ′ 1 = 0, . . . , χ ′ 2s = 0 be equations defining S V ∩ U ′ for some open neighborhood U ′ ⊆ P and satisfying that dχ ′ All the properties established in (a), (b), (c) for χ 1 , . . . , χ 2s on S hold in an entirely similar way for Let ω χ be the pullback of Ω to S χ and {, } χ the corresponding bracket. For which is the famous bracket introduced by Dirac, called Dirac bracket, and it is defined for x in the neighborhood U where c χ ij (x) has an inverse c ij χ (x). Then, for any x ∈ S χ , {F χ , χ i }(x) = 0 for i = 1, . . . , 2s, and also If we denote X (χ),F the Hamiltonian vector field associated to the function F ∈ F(P ), with respect to the Dirac bracket {, } (χ) then the previous equalities are equivalent to The Jacobi identity is satisfied for the Dirac bracket {F, G} (χ) on S χ , that is, (f ) Let U be an open neighborhood of S such that c χ ij (x) is invertible for x ∈ U . For each C = (C 1 , . . . , C 2s ) ∈ R 2s let χ C i = χ i − C i and define S χ C by the equations χ C i (x) = 0, i = 1, . . . , 2s, x ∈ U . For any C in a sufficiently small neighborhood of 0, S χ C is a nonempty symplectic submanifold of P . Define the matrix c χ C ij (x) = {χ C i , χ C j }(x), and also c ij χ C (x) as being its inverse, x ∈ U . Then, the equalities

14)
and also, are satisfied for all x ∈ U . All the definitions and properties proved in (e) for the case C = 0 hold in general for any C in a neighborhood of 0 small enough to ensure that S χ C is nonempty. In particular, the equalities and hold for x ∈ S χ C , and any C in such a neighborhood. The Dirac bracket {F, G} (χ) satisfies the Jacobi identity for F, G ∈ F(U ) and the symplectic submanifolds S χ C are the symplectic leaves of the Poisson manifold (F(U ), { , } (χ) ). By shrinking, if necessary, the open set U and for C in a sufficiently small neighborhood of 0 ∈ R 2s , the equations ψ k |S χ C = 0, k = 1, . . . , r − 2s, define regularly a first class constraint submanifold S C ⊆ S χ C ⊆ U , and the functions ψ k |S χ C ∈ R (S C ,S χ C ) ⊆ F(S χ C ) are first class constraints, that is, ψ k |S χ C ∈ I (S C ,S χ C ) k = 1, . . . , r − 2s. We have that dim S χ C = dim S C + dim ker ω C , where ω C is the pullback of Ω to S C . One has dim S C = dim S and dim S χ C = dim S χ , therefore dim ker ω = dim ker ω C .
(c) Since dχ 1 (x), . . . , dχ 2s (x) are linearly independent for x ∈ S they are also linearly independent for x in a certain neighborhood U of S. Then χ 1 (x) = 0, . . . , χ 2s (x) = 0 define a submanifold S χ of U containing S. To see that it is a symplectic submanifold choose x ∈ S χ and apply corollary 3.7 with E := T x P ; V := T x S χ ; β := 0; γ i := dχ i , i = 1, . . . , 2s; ω := Ω(x)|T x S χ . We can conclude that dim(ker Ω(x)|T x S χ ) = 0. Now let us prove that T x S χ = (V χ x ) Ω , namely, To prove that S ⊆ S χ is a first class constraint submanifold defined by first class constraints ψ i |S χ , i = 1, . . . , r − 2s, on S χ , we observe first that it is immediate that ψ i |S χ = 0, i = 1, . . . , r − 2s, define S regularly. It remains to show that X ψ i |S χ (x) ∈ T x S, i = 1, . . . , r − 2s, x ∈ S, where X ψ i |S χ is the Hamiltonian vector field associated to the function ψ i |S χ with respect to the symplectic form ω χ . This is equivalent to showing that for x ∈ S. We know that ω χ is the pullback of Ω to S χ and ψ i |S χ is the pullback of ψ i , via the inclusion S χ ⊆ U , then we have for x ∈ S, since ψ i are first class constraints, i = 1, . . . , r − 2s. Finally, using the definitions we can easily see that dim S χ = 2n − 2s and that dim S = 2n − r and from (a) we know that dim ker ω = r − 2s. We can conclude that 2n − 2s = dim S + dim ker ω.
(d) We know that, for x ∈ S, dim T x S = 2n − r, and dim ker ω(x) = r − 2s; then using (3.12) we obtain dim V x = 2s. Also from (3.12) we immediately deduce applying Ω to both sides, V Ω ∩ (ker ω) Ω = T S, in particular T S ⊆ V Ω . Let g be a given Riemannian metric on P and let W x be the g-orthogonal complement of T By choosing τ (x) appropriately one can ensure that S V is a submanifold and, moreover, it is easy to see from the definition of x , for each x ∈ S. We leave for later the proof that S V ∩ U is a symplectic submanifold of P , for an appropriate choice of the open set U , which amounts to choosing τ (x) appropriately. Then using (3.13), we see that dχ ′ 1 (x), . . . , dχ ′ 2s (x), dψ 1 (x), . . . , dψ r−2s (x) are linearly independent and span On the other hand, since ϕ|S = 0, then {ϕ, ψ j }(x) = 0, j = 1, . . . , r − 2s. We can conclude that ϕ ∈ I (S,U ′ ) and then X ϕ (x) ∈ ker ω(x), in particular, . . , dψ r−2s (x). It follows from which precedes that by replacing χ i by χ ′ i , i = 1, . . . , 2s and S by S ∩ U ′ all the properties stated in (a), (b) and (c) are satisfied. In particular, S ∩ U ′ = S χ ′ and S ∩ U ′ is symplectic. It is now clear that by covering S with open subsets like the U ′ we can define U as being the union of all such open subsets and one obtains that S ∩ U is a symplectic submanifold.
(e) Let x ∈ S χ . Then, since Using this we obtain For any F ∈ F(P ), x ∈ S χ and k = 1, . . . , 2s, we have X Fχ (x)χ k = {χ k , F χ }(x) = 0, so X Fχ (x) ∈ T x S χ . Therefore, for any Y x ∈ T x S χ , which shows that X Fχ (x) = X F |S χ (x), where both Hamiltonian vector fields are calculated with the symplectic form ω χ . Using this, for any G ∈ F(P ) and any x ∈ S χ , one obtains The equality X (χ),F (x) = X Fχ (x) = X F |S χ (x) is an immediate consequence of the previous ones. The Jacobi identity for the bracket { , } (χ) follows using the previous formulas, namely, for x ∈ S χ , one obtains where the bracket in the last term is the canonical bracket on the symplectic manifold S χ , for which the Jacobi identity is well known to be satisfied.
(f ) The equalities (3.14) and (3.15) are proven in a straightforward way. The equations (3.16) and (3.17) follow easily using a technique similar to the one used in (e). Using all this, the proof of the Jacobi identity for the bracket { , } (χ) on U goes as follows. Let x ∈ U and let C be such that x ∈ S χ C . For F, G, H ∈ F(U ) using (e) we know that the Jacobi identity holds for { , } (χ C ) on S χ C . But then, according to (3.15) it also holds for { , } (χ) for all x ∈ S χ C . Now we will prove that S χ C are the symplectic leaves. Since they are defined by equations χ C i = 0, i = 1, . . . , 2s on U we need to prove that {F, χ C i } χ (x) = 0, x ∈ S C , for all F ∈ F(U ), i = 1, . . . , 2s. Using (3.15) and (3.16) we see that {F, To finish the proof, observe first that, since χ i = 0, ψ i = 0, i = 1, . . . , 2s, j = 1, . . . , r − 2s define regularly the submanifold S ⊆ U , by shrinking U if necessary and for all C sufficiently small, we have that χ C i = 0, ψ i = 0, i = 1, . . . , 2s, j = 1, . . . , r − 2s define regularly a submanifold S C ⊆ U and therefore ψ j |S χ C = 0, j = 1, . . . , r −2s, define regularly S C as a submanifold of S χ C . To prove that it is a first class constraint submanifold and that ψ j |S χ C , j = 1, . . . , r − 2s, are first class constraints, that is ψ j |S χ C ∈ I (S C ,S χ C ) , j = 1, . . . , r − 2s, we proceed in a similar fashion as we did in (c), replacing χ by χ C . The fact that ψ j |S χ C , j = 1, . . . , r − 2s, are first class constraints defining S C implies that dim S χ C = dim S C + dim ker ω C . From the definitions one can deduce that dim S C = dim S and dim S χ C = dim S χ , therefore dim ker ω = dim ker ω C .
The following theorem summarizes the essential part of the previous lemma.
Theorem 3.20. Let (P, Ω) be a symplectic manifold, S ⊆ P and let ω be the pullback of Ω to S. Assume that ker ω has constant rank. Let V be a vector subbundle of T P |S such that V ⊕ ker ω = (T S) Ω . Then there is a symplectic submanifold S V containing S of dimension dim S + dim ker ω and the condition T x S V = V Ω x , for all x ∈ S holds. The vector bundle V Ω is called the second class subbundle tangent to the second class submanifold S V . For given functions F , G on P , one defines the Poisson bracket {F, G} V (x) := {F |S V , G|S V }(x), x ∈ S. We call this the V -Dirac bracket on S.
Remark (a) Since we are interested mainly in describing equations of motion we will not consider the definition of a global Poison bracket on a neighborhood of S such that one of its symplectic leaves coincides with S V . By choosing a local regular description χ k = 0 of S V one obtains the usual expression for the Dirac bracket, as it was shown in lemma 3.19. Under our strong regularity conditions the symplectic leaves of the Dirac bracket give a (local) regular foliation of a neighborhood of the final constraint submanifold S. This implies by the Weinstein splitting theorem (Weinstein [1983]) that there are local charts where the Dirac bracket is constant.
(b) The tangent second class subbundle V Ω in a sense (modulo tangency) classifies all the possible second class constraint submanifolds containing a given submanifold S ⊆ P . It carries enough information to write the Dirac brackets along the final constraint submanifold S and therefore also equations of motion, as we show in subsection 3.4.

Equations of motion
We are going to describe equations of motion in the abstract setting of the paragraph First Class and Second Class Constraints and Constraint submanifolds of subsection 3.2, that is, a symplectic manifold (P, Ω) and a submanifold S ⊆ P , defined regularly by equations φ i = 0, i = 1, . . . , a. We are going to assume all the results, notation and regularity conditions of that paragraph. We need to introduce in this abstract setting, by definition, the notions of primary constraints, primary constraint submanifold and the Hamiltonian.
The primary constraint submanifold is a given submanifold S ′ ⊆ P containing S, and in this context, S will be called the final constraint. We will assume without loss of generality that S ′ is defined regularly by the equations φ i = 0, i = 1, . . . , a ′ , with a ′ ≤ a, where each φ i , i = 1, . . . , a ′ will be called a primary constraint while each φ i , i = a ′ + 1, . . . , a will be called a secondary constraint, for obvious reasons. In this abstract setting the Hamiltonian is by definition a given function H ∈ F(P ).
The equations of motion can be written in the Gotay-Nester form, where (x,ẋ) ∈ T x S, for all δx ∈ T x S ′ . Now we will transform this equation into an equivalent Poisson equation using the Dirac bracket.
The condition {H, ψ}(x) = 0, for all x ∈ S and all first class constraints ψ will appear later as a necessary condition for existence of solutions for any given initial condition in S, so we will assume it from now on.
For each section of Λ one has a Hamiltonian H T (x) = H(x) + λ i (x)φ i (x), x ∈ P , and an equation of motion Equations of motion can be described nicely using the Dirac bracket as we will see in a moment. Choose first class and second class constraints (ψ 1 , . . . , ψ a−2s , χ 1 , . . . , χ 2s ) as in lemma 3.19, then since {H T , χ i }(x) = 0, for any function F ∈ F(P ) and any x ∈ S we obtain Then we can describe equations of motion in terms of the Dirac bracket as We want a more precise description of equations of motion. The total Hamiltonian has the equivalent expression for all x ∈ S, while χ ′ i ∈ I (S ′ ,P ) , i = 1, . . . , s ′ , are such that can be chosen as a set of primary constraints which justifies the expression (3.19) for the total Hamiltonian. One can see that the rank of the matrix {χ j , χ ′ i }(x), j = 1, . . . , 2s, i = 1, . . . , s ′ , x ∈ S is s ′ . Now the conditions (3.18) are equivalent to {H T , ψ j }(x) = 0, j = 1, . . . , a − 2s, which gives {H, ψ j }(x) = 0, j = 1, . . . , a − 2s and {H T , χ i }(x) = 0, i = 1, . . . , 2s, for all x ∈ S which gives {H, χ i }(x) + µ ′j (x){χ ′ j , χ i }(x) = 0, i = 1, . . . , 2s, x ∈ S, from which we obtain µ ′j , j = 1, . . . , s ′ , as a well-defined function (on a neighborhood of S, then we can extend it arbitrarily to P ). One can write χ ′ j = a k j ψ k + b l j χ l with a k j , b l j ∈ F(P ) uniquely defined on S, for j = 1, . . . , s ′ , k = 1, . . . , a − 2s, l = 1, . . . , 2s. Then the total Hamiltonian can be written where ψ (S,S ′ ,H) = µ ′j a k j ψ k is a first class constraint, ψ (S,S ′ ,H) ∈ I (S,P ) . We can conclude, using the fact that {χ k , F } (χ) (x) = 0, for all x ∈ S and k = 1, . . . , 2s and (3.18), that for any function F ∈ F(P ) and any x ∈ S. We shall call H (χ,S,S ′ ) := H + ψ (S,S ′ ,H) + λ ′i ψ ′ (S,S ′ ,i) the Hamiltonian of the system with respect to the Dirac bracket { , } (χ) , where we have denoted ψ ′ i by ψ ′ (S,S ′ ,i) to emphasize that these functions depend on S and S ′ . We have proven the following theorem.
Theorem 3.21. Let (P, Ω) be a symplectic manifold and S ⊆ S ′ ⊆ P given regularly defined submanifolds. In the situation described above, equations of motion on S can be written in the following equivalent ways: = X (χ),H (x) + X (χ),ψ (S,S ′ ,H) (x) + λ ′i X (χ),ψ ′ i (x) for all x ∈ S.
Remark. (a) The family of Hamiltonian vector fields (3.20), each one of them tangent to S, depends on a finite number of arbitrary parameters λ ′i ∈ R, i = 1, . . . , a ′ − s ′ , and it generates an affine distribution on S. This should be compared with the affine bundle in equation (2.7). Any vector field, even time-dependent, X t on P whose restriction to S is a section of that distribution gives equations of motion. Note that, since ψ ′ i |S = 0, then for any choice of functions, even time-dependent, λ ′i t ∈ F(P ), i = 1, . . . , a ′ − s ′ we have λ ′i t X (χ),ψ ′ i (x) = X (χ),λ ′i for all x ∈ S.
(b) The equations of motion can be globalized, using the bracket { , } V , as far as one can find a global ψ (S,S ′ ,H) .

Future work
The present work should be followed immediately by an extension of the Dirac theory, and also the dual Gotay-Nester theory for the case of a Dirac dynamical system (x,ẋ) ⊕ dE(x) ∈ D x . This will expand the field of applications, for instance, one will have a theory of constraints for LC circuits and holonomic systems, if the Dirac structure D is integrable.
The reduction theory for the constraint algorithm will be also the purpose of future work. This can be approached using the category of Dirac anchored vector bundles. This will extend part of the results in Cendra, Marsden, and Ratiu [2001].
Singular cases, where the strong regularity assumptions made in the present paper are weakened in several ways are also important and will be the purpose of future work.

A Appendix
Lemma A.1. There is a canonical inclusion ϕ : T Q ⊕ T * Q → T * T Q. In addition, consider the canonical two-forms ω T * Q and ω T * T Q on T * Q and T * T Q respectively, the canonical projection pr T * Q : T Q⊕T * Q → T * Q, and define the presymplectic twoform ω = pr * T * Q ω T * Q on T Q⊕T * Q. Then the inclusion preserves the corresponding two-forms, that is, ω = ϕ * ω T * T Q .
Proof. If τ Q : T Q → Q and τ T Q : T T Q → T Q are the tangent projections, we can consider the dual tangent rhombic for w vq ∈ T vq T Q.
Here v q ⊕ α q denotes an element in the Pontryagin bundle over the point q ∈ Q. Note that the following diagram commutes.
Let us see that ϕ is an injective vector bundle map from the bundle pr T Q : T Q ⊕ T * Q → T Q to the cotangent bundle π T Q : T * T Q → T Q, over the identity of T Q. The last part of this assertion follows from the commutative diagram above.
First, if ϕ(v q ⊕ α q ) = ϕ(v ′ q ′ ⊕ α ′ q ′ ) then both sides are in the same fiber T * vq T Q = T * v ′ q ′ T Q, so v q = v ′ q ′ . Also, for all w vq ∈ T * vq T Q we have ϕ(v q ⊕ α q ) · w vq = ϕ(v q ⊕ α ′ q ) · w vq or α q · T τ Q (w vq ) = α ′ q · T τ Q (w vq ). Since τ Q : T Q → Q is a submersion, we have α q = α ′ q and ϕ is injective. Second, ϕ is linear on each fiber, since For the second part of the lemma, let us recall the definition of the canonical one-form on θ T * Q ∈ Ω 1 (T * Q). For α q ∈ T * Q, θ T * Q (α q ) is an element of T * αq T * Q such that for w αq ∈ T αq T * Q, θ T * Q (α q ) · w αq = α q (T π Q (w αq )), where π Q : T * Q → Q is the cotangent bundle projection.
However, the following diagram commutes