JOURNAL OF GEOMETRIC MECHANICS doi:10.3934/jgm.2014.6.167 c American Institute of Mathematical Sciences Volume 6, Number 2, June 2014 pp. 167{236 AN EXTENSION OF THE DIRAC AND GOTAY-NESTER THEORIES OF CONSTRAINTS FOR DIRAC DYNAMICAL SYSTEMS Hernan´ Cendra Departamento de Matem´atica Universidad Nacional del Sur, Av. Alem 1253 8000 Bah´ıaBlanca and CONICET, Argentina Mar´ıa Etchechoury Departamento de Matem´atica, Facultad de Ciencias Exactas Universidad Nacional de La Plata CC 172, 1900 La Plata, Argentina Sebastian´ J. Ferraro Departamento de Matem´aticaand Instituto de Matem´aticaBah´ıaBlanca Universidad Nacional del Sur, Av. Alem 1253 8000 Bah´ıaBlanca and CONICET, Argentina (Communicated by Juan-Pablo Ortega) Abstract. 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 cases considered usually in the literature correspond to a trivial foliation, with 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 adapted total energy. The interesting example of LC circuits is developed in detail. The paper emphasizes the point of view that Dirac and Gotay-Nester theories are, in a certain sense, 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. Contents 1. Introduction 168 2. A brief review of the Dirac and Gotay-Nester theories 172 2.1. A brief review of Dirac's theory 174 2.2. A brief review of the Gotay-Nester algorithm 176 3. First and second class constraints, constraint submanifolds and equations of motion 177 3.1. Linear symplectic geometry 177 2010 Mathematics Subject Classification. Primary: 70H45, 70G45, 70H03, 70H05; Secondary: 34A09. Key words and phrases. Constrained systems, Dirac structures, implicit differential equations, nonholonomic mechanics, Poisson geometry. 167 168 HERNAN´ CENDRA, MAR´IA ETCHECHOURY AND SEBASTIAN´ J. FERRARO 3.2. Poisson-algebraic and geometric study of constraints and constraint submanifolds. 179 3.3. Second class subbundles 184 3.4. Equations of motion 191 4. Dirac structures 196 4.1. Dirac dynamical systems 199 5. The constraint algorithm for Dirac dynamical systems (CAD) 199 6. Examples 204 7. An extension of the Dirac theory of constraints 222 7.1. Dirac brackets adapted to foliated constraint submanifolds 223 8. Conclusions and future work 230 Acknowledgments 230 Appendix A. A canonical embedding of the Pontryagin bundle 230 REFERENCES 232 1. Introduction. The main purpose of this paper is to study an extension of the Dirac theory of constraints [54, 55, 56] and the Gotay-Nester theory [63, 58] for the case in which the equation of motion is a Dirac dynamical system on a manifold M, (x; x_) ⊕ dE(x) 2 Dx; (1) where D ⊆ TM ⊕ T ∗M is a given integrable Dirac structure on M, which gives a foliation of M by presymplectic leaves, and E is a given function on M, called the energy. The seminal article [39] by Courant and Weinstein contains the definition of a Dirac structure and addresses the Dirac theory of constraints as one of the motivations for introducing this notion. Another extension of the Dirac theory of constraints, inspired in the notion of Dirac structures, is given in the notes [16] by Bursztyn. Succinctly, for this extension one considers a Poisson manifold and a given primary constraint submanifold, rather than a symplectic manifold and a primary constraint submanifold as in the Dirac theory. This differs from the problem addressed in the present paper, since we start with a foliated submanifold of a symplectic manifold. See for example [8] as an introduction to the extensive literature on Poisson manifolds and algebras. For our purposes, we will first review only those aspects of the Gotay-Nester and the Dirac works which are relevant for our work and later on we will describe our extension. We will give detailed proofs under precise regularity conditions, at the cost of making the paper a little longer, but self contained. One of the features of systems like (1) is that they can represent some constrained systems where the constraints appear even if the Lagrangian is nonsingular, as it happens in the case of integrable nonholonomic systems. We will work under explicit hypotheses about regularity, locality and integrability, although some of our results can be applied in more general cases, as indicated in the paragraph Hypotheses, below in this section. In the Gotay-Nester theory the starting point is an equation of the form !(x)(x; _ ) = dE(x) (2) where ! is a presymplectic form on M. In fact, in the Gotay-Nester theory the more general case where dE(x) is replaced by an arbitrary closed 1-form α is considered, but this will not be relevant here. Example. Euler-Lagrange and Hamilton's equations. Let L: TQ ! R be a Lagrangian, degenerate or not. Let E(q; v; p) = pv − L(q; v) and let ! 2 THEORY OF CONSTRAINTS FOR DIRAC SYSTEMS 169 2 i Ω (TQ ⊕ T ∗Q) be the presymplectic form ! = dq ^ dpi on the Pontryagin bundle M = TQ ⊕ T ∗Q. Then the Euler-Lagrange equations are written equivalently in the form of equation (2) with x = (q; v; p). In fact, we have, in local coordinates, i i i i(_q;v;_ p_)dq ^ dpi =q _ dpi − p_idq (3) @E @E @E E = i + + i (4) d i dq dpi i dv @q @pi @v @L @L = − (q; v)dqi + vidp + p − dvi (5) @qi i i @vi Using equations (3){(5) we can easily see that (2) becomes q_i = vi @L p_ = (q; v) i @qi @L 0 = p − (q; v); i @vi which is clearly equivalent to the Euler-Lagrange equations. Note that in this approach the second-order conditionq _ = v appears automatically as a consequence of the formalism. This does not happen if we consider the equation i(_q;v_)!L(q; v) = dEL(q; v), where !L is the pullback of the canonical symplectic form on T ∗Q by the Legendre transformation and EL(q; v) = @L=@v (q; v) − L(q; v). Conditions for equations of this type to be second order have been studied in [59]. The case E(q; v; p) = H(q; p), where H is a given Hamiltonian on T ∗Q gives Hamilton's equations. In section6 we will give more examples showing the unifying power of Dirac systems. The idea of using the Pontryagin bundle to write such fundamental equations of physics appears in [29, 78, 97, 98, 99, 110, 111]. Equation (2) is equivalent to equation (1) in the case in which the Dirac structure on M is the one naturally associated to !, denoted D!, in which case the foliation of M has obviously only one presymplectic leaf. In general, equation (1) may be considered as a collection of equations of the type (2), one for each presymplectic leaf of the Dirac structure. However, in order to study the initial condition problem this approach might not be appropriate, because different initial conditions might belong to different presymplectic leaves and therefore correspond to different equations of the type (2), which is not the usual situation. Similarly, if D = Dπ is the Dirac structure associated to a Poisson bivector π the presymplectic leaves of Dπ are the symplectic leaves of π and equation (1) is equivalent to f_ = ff; Eg. The algorithm of the Gotay-Nester theory generates a finite sequence of secondary constraint submanifolds M ⊇ M1 ⊇ · · · ⊇ Mc. The final constraint submanifold Mc has the property that every solution curve of (2) is contained in it. In fact, Mc can be defined as being the smallest submanifold having that property, a non constructive definition but not involving any algorithm. Equations of motion are given by restricting the variable (x; x_) in equation (2) to TMc, and existence of solution curves x(t) for a given initial condition x0 2 Mc is guaranteed, under the condition that the kernel of [ ! (x)jTxMc : TxMc ! Tx∗M 170 HERNAN´ CENDRA, MAR´IA ETCHECHOURY AND SEBASTIAN´ J. FERRARO has locally constant rank. An important point in the Gotay-Nester approach is that the equations defining the secondary constraint submanifolds are written in terms of ! (see formula (12)), which makes the whole algorithm invariant under a group of transformations preserving !. In order to solve (1), we will develop a constraint algorithm that we call CAD (constraint algorithm for Dirac dynamical systems). The CAD extends the Gotay- Nester algorithm and gives a sequence of secondary foliated constraint submanifolds. The secondary foliated constraint submanifolds can be written in terms of the Dirac structure using formula (45), which generalizes (12). Let M ⊆ N be an embedding of M into a symplectic manifold (N; Ω), in such a way that the presymplectic leaves of M are presymplectic submanifolds of N and let, by slight abuse of notation, E : N ! R denote an arbitrary extension of the given energy on M. This kind of framework can be naturally constructed in many examples. Then one can consider, at least locally, that M is a primary foliated constraint submanifold defined by equating certain constraints to 0, and each leaf is defined by equating some other constraints to a constant depending on the leaf, which generalizes the situation of the Dirac theory of constraints. One of the results of this paper establishes that there exists a Dirac bracket whose symplectic leaves are adapted to the foliations of the primary and final foliated constraint submanifolds.