On Representations and Integrability of MathematicalStructures in Energy-Conserving Physical SystemsMorten Dalsmo1ABB Corporate ResearchDepartment of Information Technology and Control SystemsP.O. Box 90, N-1361 Billingstad, NorwayTel.: +47 66 84 34 21, Fax: +47 66 84 35 41E-mail: md@nocrc.abb.noArjan van der SchaftUniversity of TwenteSystems and Control Group, Department of Applied Mathematics7500 AE Enschede, The NetherlandsTel.: +31 53 489 3449, Fax: +31 53 434 0733E-mail: A.J.vanderSchaft@math.utwente.nlRevised manuscript (SICON ms # 0312031)November 25, 1997AbstractIn the present paper we elaborate on the underlying Hamiltonian structure of interconnectedenergy-conserving physical systems. It is shown that a power-conserving interconnection ofport-controlled generalized Hamiltonian systems leads to an implicit generalized Hamiltoniansystem, and a power-conserving partial interconnection to an implicit port-controlled Hamil-tonian system. The crucial concept is the notion of a (generalized) Dirac structure, de�ned onthe space of energy-variables or on the product of the space of energy-variables and the spaceof ow-variables in the port-controlled case. Three natural representations of generalized Diracstructures are treated. Necessary and su�cient conditions for closedness (or integrability) ofDirac structures in all three representations are obtained. The theory is applied to implicitport-controlled generalized Hamiltonian systems, and it is shown that the closedness conditionfor the Dirac structure leads to strong conditions on the input vector �elds.Key wordsHamiltonian systems, Dirac structures, implicit systems, external variables, integrability,actuated mechanical systems, kinematic constraints, interconnections.AMS subject classi�cations (1991)93C10, 93A30, 70F25, 58F05Abbreviated titleMathematical Structures in Physical Systems1Corresponding author 1

1 IntroductionMost of the current modelling and simulation approaches to (complex) physical systems (e.g. multi-body systems) are based on some sort of network representation, where the physical system underconsideration is seen as the interconnection of a (possible large) number of simple sub-systems.This way of modelling has several advantages. From a physical point of view it is usually naturalto regard the system as composed of sub-systems, possibly from di�erent domains (mechanical,electrical, ...). The knowledge about sub-systems can be stored in libraries, and is re-usable forlater occasions. Because of the modularity the modelling process can be performed in an \iterative"manner, gradually re�ning - if necessary - the model by adding other sub-systems. Further, theapproach is suited to general control design where the overall behavior of the system is sought tobe improved by the addition of other sub-systems or controlling devices. From a system-theoreticpoint of view this modular approach naturally emphasizes the need for models of systems withexternal variables, e.g. inputs and outputs.In this paper we concentrate on the mathematical description of network representations of(lumped-parameter) energy-conserving physical systems. In our previous work we have shown howenergy-conserving physical systems with independent energy variables can be naturally described asgeneralized Hamiltonian systems (with external variables). However, a general power-conserving in-terconnection of such systems will lead to a system described by di�erential and algebraic equations,that is an implicit dynamical system, which cannot be anymore directly described as an explicitgeneralized Hamiltonian system. This motivates the de�nition of implicit generalized Hamiltoniansystems, as introduced in [SM2, SM3]. The main ingredient in this de�nition is that of a (general-ized) Dirac structure. The relevance of Dirac structures in the Hamiltonian modelling of electricalLC-circuits with dependent storage elements (a clear example of interconnected energy-conservingsystems) was already recognized in [C2].The notion of Dirac structures was introduced by Courant and Weinstein [CW] and furtherinvestigated by Courant in [C1], as a generalization of Poisson and (pre-)symplectic structures.Dorfman [D1, D2] developed an algebraic theory of Dirac structures in the context of the study ofcompletely integrable systems of partial di�erential equations, with the aim of describing within aHamiltonian framework certain sets of p.d.e's which do not admit an easy Hamiltonian formulationin terms of Poisson or symplectic structures, due to non-locality of the involved operators. Theconceptual novelty in the approach initiated in [C2, SM2, SM3] is to use Dirac structures for thedirect Hamiltonian description of di�erential-algebraic equations resulting from the interconnec-tion of energy-conserving systems, including constrained systems. Although the terminology Diracstructure is derived from the "Dirac bracket" introduced by Dirac in his study of of constrainedHamiltonian systems arising from degenerate Lagrangians [D3], our use of Dirac structures deter-mining, together with the stored energy (Hamiltonian), the algebraic constraints as well as thedynamical equations of motion seems to be new. Furthermore, we stress the "physical" relevanceof Dirac structures as naturally capturing the geometric structure of the system as arising from theinterconnection of sub-systems (see e.g. Proposition 2.2).In Courant and Dorfman [C1, D2] the de�nition of a Dirac structure includes a closedness (orintegrability) condition generalizing the Jacobi-identity for Poisson brackets or the closedness oftwo-forms de�ning symplectic structures. This condition is naturally satis�ed for constant Diracstructures (as in the case of LC-circuits) and for Dirac structures arising from holonomic kinematicconstraints in mechanical systems, but not for the generalized Dirac structures arising from non-holonomic kinematic constraints [SM1, SM3] or from general kinematic pairs in multibody systems[M2].The structure of this paper is as follows. In Section 2 we will recall the de�nitions of a(generalized) Dirac structure and of an implicit Hamiltonian system, and we will show how thepower-conserving interconnection of port-controlled (explicit) Hamiltonian systems leads to suchan implicit Hamiltonian system. In Section 3 we will investigate various useful ways of representinggeneralized Dirac structures and consequently of representing implicit Hamiltonian systems, andwe will study their relationship. Then in Section 4 the closedness (or integrability) condition forDirac structures will be worked out for the three di�erent representations obtained. Both Sections3 and 4 use extensively techniques and results from the work of Courant and Dorfman, althoughthe emphasis is rather di�erent. The results of Section 3 and 4 are applied in Section 5 to Diracstructures as arising in implicit generalized Hamiltonian systems with external variables. In partic-ular it is shown that the closedness condition translates into strong conditions on the input vector2

�elds.A main motivation for the Hamiltonian modelling of interconnected energy-conserving physicalsystems is, apart from the clear motivation from a general modelling and simulation point of view,the generalization of the theory of \passivity-based control" to complex interconnected physicalsystems. Key concepts in this theory (see e.g. [TA, OS, S]) are the use of the internal energyas candidate Lyapunov function, the shaping of the internal energy via state feedback, and theinjection of \damping" in order to achieve asymptotic stability. This approach has shown tobe very powerful in the robust and/or adaptive control of physical systems described by Euler-Lagrange or Hamiltonian equations of motion (such as robot manipulators, mobile robots andelectrical machines), and can be expected to be equally powerful for interconnected physical systems.Although it is not the topic of the present paper to demonstrate this, we indicate at the end ofSection 4 how the usual stability theory of Hamiltonian systems based on the Hessian matrix of theHamiltonian can be naturally extended to implicit Hamiltonian systems. Moreover, at the end ofSection 5 we show the link between results in this paper and \passivity-based control" of actuatedmechanical systems with kinematic constraints.In the control design of interconnected physical systems also the system-theoretic properties(such as controllability and observability) of implicit port-controlled Hamiltonian systems will proveto be instrumental (e.g. in the analysis how much damping injection is needed for asymptoticstabilization). For explicit port-controlled generalized Hamiltonian systems some of these topicsalready have been studied in our previous work [SM2, MS1, MS2]. Section 5 only provides abasic framework for a study of these issues. Apart from \passivity-based control" also the furtherexploitation of the structure of symmetries and conservation laws has a great potential (see e.g.[BKMM] for related developments). All this is a large area for further research.2 Generalized Hamiltonian modelling of interconnected sys-temsIn our previous work ([MS1], [MS2], [MBS], [MSB1], [MSB2], [SM1], [SM2], [SM3]) we have arguedthat the basic dynamic building blocks in the network representation of energy-conserving physicalsystems are systems of the form _x = J(x)@H@x (x) + g(x)fe = gT (x)@H@x (x) (2.1)Here x = (x1; : : : ; xn) denotes the vector of (independent) energy variables, coordinatizing the statespace manifold X , H(x1; : : : ; xn) is the total stored energy in the system, with @H@x (x) denotingthe column-vector of partial derivatives of H, and the n�n skew-symmetric structure matrix J(x)is associated with the network topology of the system. The columns gj(x), j = 1; : : : ;m, of thematrix g(x) de�ne the (state modulated) transformers describing the in uence of the external owsources (or inputs) fj , j = 1; : : : ;m. The components ej of e are the corresponding conjugated(with respect to the power) e�orts (or outputs). Since the matrix J(x) is skew-symmetric weimmediately obtain the energy balance ddtH = eT f (2.2)expressing that the increase in energy equals the externally supplied power (ejfj is the power ofthe j-th source). Thus (2.1) describes an energy-conserving physical system with internal variablesx1; : : : ; xn (associated with energy storage) and external (or port) variables f1; : : : ; fm; e1; : : : ; em(associated with power), which can be regarded as input, respectively output, variables.The system (2.1) is called a port-controlled generalized Hamiltonian system because of thefollowing. We may de�ne a generalized Poisson bracket operation on the real functions on X asfF;Gg(x) = �@F@x (x)�T J(x) @G@x (x); F;G : X ! R (2.3)Clearly, this bracket is skew-symmetric and satis�es the Leibniz identityfF;G1G2g(x) = fF;G1g(x)G2(x) + G1(x)fF;G2g(x); for all F;G1; G2 : X ! R (2.4)3

and thus _x = J(x)@H@x (x) can be seen as the generalized Hamiltonian vector �eld corresponding toH and the generalized Poisson bracket f ; g. This generalized Poisson bracket is a true Poissonbracket if additionally the Jacobi-identity is satis�ed, that isfF; fG;Kgg+ fG; fK;Fgg+ fK; fF;Ggg= 0; 8F;G;K : X ! R (2.5)If (and only if) the Jacobi-identity holds there exists in a neighborhood of every point x0 2 X whereJ(x) has constant rank local canonical coordinates (q; p; r) = (q1; : : : ; qk; p1; : : : ; pk; r1; : : : ; rl) forX in which J(x) takes the form (see e.g. [O])J(q; p; r) = 24 0 Ik 0�Ik 0 00 0 035 (2.6)implying that the Hamiltonian vector �eld _x = J(x)@H@x (x) takes the form_q = @H@p (q; p; r)_p = �@H@q (q; p; r)_r = 0 (2.7)which are almost the standard Hamiltonian equations of motion except for the appearance of theconserved quantities r1; : : : ; rl. Although in many cases of interest the Jacobi-identity is satis�ed,there are clear examples where it is not satis�ed (e.g. mechanical systems with nonholonomickinematic constraints; see [SM1]).The overall energy-conserving physical system is now obtained by interconnecting the variousport-controlled generalized Hamiltonian sub-systems as above in a power-continuous fashion (e.g.by using Kirchho�'s laws). In general this will result in a mixed set of di�erential and algebraicequations, which nevertheless is expected to be again Hamiltonian in some sense. Indeed, it canbe seen that it is an implicit generalized Hamiltonian system, as de�ned in [SM2, SM3]. The keyconcept in the de�nition of an implicit generalized Hamiltonian system is the notion of a generalizedDirac structure, as introduced (in a rather di�erent context) in [C1, D2].First we concentrate on interconnected energy-conserving physical systems without any remain-ing external sources; see Section 5 for the general case. In this case the Dirac structure for theinterconnected system is de�ned solely on the space of energy-variables. Let X be an n-dimensionalmanifold with tangent bundle TX and cotangent bundle T �X . We de�ne TX �T �X as the smoothvector bundle over X with �ber at each x 2 X given by TxX � T �xX . Let X be a smooth vector�eld and � a smooth one-form on X respectively. Then we say that the pair (X;�) belongs to asmooth vector subbundle D � TX � T �X (denoted (X;�) 2 D) if (X(x); �(x)) 2 D(x) for everyx 2 X . Furthermore for a smooth vector subbundle D � TX � T �X we de�ne the smooth vectorsubbundle D? � TX � T �X asD? = f(X;�) 2 TX � T �X j h� j Xi + h� jXi = 0; 8(X; �) 2 Dg (2.8)with h j i denoting the natural pairing between a one-form and a vector �eld. In (2.8) andthroughout in the sequel the pairs (X;�); (X; �) are assumed to be pairs of smooth vector �eldsand smooth one-forms.De�nition 2.1 ([C1, D2]) A generalized Dirac structure on an n-dimensional manifold X is asmooth vector subbundle D � TX � T �X such that D? = D.If D satis�es an additional closedness (or integrability) condition then D de�nes a Dirac structure;see Section 4. Later on we will see that the dimension of the �bers of a generalized Dirac structureon an n-dimensional manifold is equal to n. By taking � = �, X = X in (2.8) we obtainh� jXi = 0; for all (X;�) 2 D (2.9)Conversely, if (2.9) holds then for every (X;�); (X; �) 2 D0 = h�+ � jX + Xi = h� jXi + h� j Xi+ h� jXi + h� j Xi= h� j Xi+ h� jXi (2.10)4

and thus D � D?. Hence a Dirac structure is a smooth vector subbundle of TX � T �X which ismaximal with respect to property (2.10) or (2.9).Let now X be an n-dimensional manifold with a generalized Dirac structure D, and let H :X ! R be a Hamiltonian (energy function). Then the implicit generalized Hamiltonian system onX corresponding to D and H is given by the speci�cation (see [SM2])� _x; @H@x (x)� 2 D(x) (2.11)By (2.9) we immediately obtain the energy conservation property dHdt = h@H@x (x) j _xi = 0. Notethat in general the speci�cation (2.11) puts algebraic constraints on X , since in general there willnot exist for every x 2 X a tangent vector _x 2 TxX such that (2.11) is satis�ed. Thus (2.11) isin general a set of DAE's (Di�erential Algebraic Equations). It can be seen that (2.11) generalizesthe notion of an (explicit) generalized Hamiltonian system_x = J(x)@H@x (x); J(x) = �JT (x) (2.12)by noting that D = f(X;�) 2 TX � T �X jX(x) = J(x)�(x); x 2 Xgde�nes a generalized Dirac structure. (If �T (x)J(x)�(x) + �T (x)X(x) = 0 for all �, then X(x) =J(x)�(x).)A special case of a Dirac structure is that of a constant Dirac structure on a linear space.De�nition 2.2 A constant Dirac structure on a linear n-dimensional space V is a linear subspaceD � V � V� with the property that D? = D, whereD? = f(v; v�) 2 V � V� j hv� j vi+ hv� j vi = 0 for all (v; v�) 2 Dgwhere h j i denotes the natural pairing between V and V�.It is straightforward to derive the following proposition.Proposition 2.1 Let V be an n-dimensional linear space. A linear subspace D � V � V� de�nesa constant Dirac structure if and only if dimD = n andhv� j vi = 0; for all (v; v�) 2 D (2.13)Proof: (Sketch; see [SM3] for details.) As in (2.9) and (2.10) we see that if D de�nes a constantDirac structure then (2.13) holds, while if (2.13) holds then equivalentlyhv� j vi+ hv� jvi = 0 for all (v; v�) 2 D (2.14)Furthermore, a subspace D of V � V� de�nes a Dirac structure if it is maximal with respect toproperty (2.14), which is equivalent (see [C1]) to the property dimD = n. �Now let us consider k port-controlled generalized Hamiltonian systems as in (2.1), i.e., fori = 1; : : : ; k _xi = Ji(xi)@Hi@xi (xi) + gi(xi)fiei = gTi (xi)@Hi@xi (xi)xi 2 Xi; fi 2 Fi := Rmi; ei 2 Ei := F�i = Rmi (2.15)with Xi an ni-dimensional state space. Consider a general power-conserving interconnection of thesesystems given by an (m1 + : : :+mk)-dimensional subspace (possibly parametrized by x1; : : : ; xk)I(x1; : : : ; xk) � F1 � : : :�Fk � E1 � : : :� Ek (2.16)with the property (f1; : : : ; fk; e1; : : : ; ek) 2 I(x1; : : : ; xk) ) kXi=1 eTi fi = 0 (2.17)5

Remark 2.1 By Proposition 2.1 it follows that I(x1; : : : ; xk) de�nes a constant Dirac structureon F1 � : : :� Fk, parameterized by (x1; : : : ; xk).Proposition 2.2 Consider k port-controlled generalized Hamiltonian systems (2.15) subject to aninterconnection (2.16) satisfying (2.17). Then the resulting interconnected system is an implicitgeneralized Hamiltonian system with state space X := X1� : : :�Xk, Hamiltonian H(x1; : : : ; xk) :=H1(x1) + : : :+Hk(xk), and generalized Dirac structure D on X given as(X;�) = (X1; : : : ; Xk; �1; : : : ; �k) 2 D ()8xi 2 Xi; i = 1; : : : ; k; 9(f1; : : : ; fk; e1; : : : ; ek) 2 I(x1; : : : ; xk) such thatXi(xi) = Ji(xi)�i(xi) + gi(xi)fiei = gTi (xi)�i(xi) (2.18)Proof: The main point is in proving that D given by (2.18) de�nes a generalized Dirac structure.Let (X;�) = (X1; : : : ; Xk; �1; : : : ; �k) be in D?, that is h� jXi + h� j Xi = 0, for all (X; �) =(X1; : : : ; Xk; �1; : : : ; �k) satisfying (2.18). This means0 = kXi=1 h�Ti (xi)Xi(xi) + �Ti (xi)Xi(xi)i= kXi=1 h�Ti (xi)Xi(xi) + �Ti (xi)Ji(xi)�i(xi) + �Ti (xi)gi(xi)fii= kXi=1 ��Ti (xi) [Xi(xi)� Ji(xi)�i(xi)] + �Ti (xi)gi(xi)fi� (2.19)for all �i, fi such that ei = gTi (xi)�i(xi) satis�es (f1; : : : ; fk; e1; : : : ; ek) 2 I(x1; : : : ; xk). Letting�rst fi = 0 and ei = 0, we obtainkXi=1 �Ti (xi) [Xi(xi)� Ji(xi)�i(xi)] = 0 (2.20)for all �i(xi) such that gTi (xi)�i(xi) = 0. This means that there exist vectors f1; : : : ; fk such thatXi(xi) = Ji(xi)�i(xi) + gi(xi)fi (2.21)Substitution into (2.19) yields0 = kXi=1 ��Ti (xi)gi(xi)fi + �Ti (xi)gi(xi)fi�= kXi=1 �eTi fi + eTi fi� (2.22)for all fi and ei = gTi (xi)�i(xi) satisfying (f1; : : : ; fk; e1; : : : ; ek) 2 I(x1; : : : ; xk). If gTi (xi) issurjective for all i = 1; : : : ; k, this means that (2.22) is satis�ed for all (f1; : : : ; fk; e1; : : : ; ek) 2I(x1; : : : ; xk), and by Proposition 2.1 and Remark 2.1 this implies that (f1; : : : ; fk; e1; : : : ; ek) 2I(x1; : : : ; xk) and thus (X;�) 2 D. In general we proceed as follows. De�ne the space of achievable ows and e�ortsC(x1; : : : ; xk) := f(f1; : : : ; fk; e1; : : : ; ek) j fi 2 Fi; ei 2 ImgTi (xi); i = 1; : : : ; kgThen (2.22) implies that(f1; : : : ; fk; e1; : : : ; ek) 2 (I(x1; : : : ; xk) \C(x1; : : : ; xk))? = I?(x1; : : : ; xk) + C?(x1; : : : ; xk)6

where ? denotes orthogonal complement with respect to property (2.22). By Proposition 2.1 itfollows that I?(x1; : : : ; xk) = I(x1; : : : ; xk), while C?(x1; : : : ; xk) is seen to be given asC?(x1; : : : ; xk) = f(f1; : : : ; fk; e1; : : : ; ek) j fi 2 ker gi(xi); ei = 0; i = 1; : : : ; kgThus there exist ow vectors f 01; : : : ; f 0k such that (f 01; : : : ; f 0k; e1; : : : ; ek) 2 I(x1; : : : ; xk), withXi(xi) = Ji(xi)�i(xi) + gi(xi)f 0i , ei = gTi (xi)�i(xi), showing that (X;�) 2 D. Hence D? � D.Since it is easily seen that D � D?, this shows that D de�nes a Dirac structure. �We note that the de�nition of a power-conserving interconnection is very general, and for ex-ample includes Kirchho�'s laws for electrical systems, the interconnection relations for generalizedvelocities and forces for interconnected mechanical systems (Newton's third law), as well as trans-formers in electrical circuits and kinematic pairs in multibody systems.From a classical control point of view an important example of a power-conserving interconnec-tion is the standard feedback interconnection:Example 2.1 Consider two input-state-output systems (\plant" and \controller")_xi = gi(xi; ui)yi = hi(xi); ui; yi 2 Rm; i = 1; 2 (2.23)and impose the (negative) feedback interconnectionu2 = y1u1 = �y2 (2.24)leading to the explicit system _x1 = g1(x1;�h2(x2))_x2 = g2(x2; h1(x1)) (2.25)If we equate the input vectors ui with ow vectors, and the output vectors yi with e�ort vectors,then (2.24) is a power-conserving interconnection. Proposition 2.2 applied to this particular casesays that if both systems in (2.23) are Hamiltonian, then also (2.25) is Hamiltonian. This can beregarded as a special instance of the Passivity theorem in input-output stability theory.3 Representations of generalized Dirac structures and im-plicit generalized Hamiltonian systemsThere are di�erent ways of representing generalized Dirac structures, and consequently of writingthe equations of an implicit generalized Hamiltonian system. These representations each have theirown advantages and are connected to di�erent but equivalent ways of mathematically modellingthe energy-conserving physical systems.Before going into these representations we �rst note that a generalized Dirac structure D on ann-dimensional manifold X de�nes the smooth distributionsG0 = fX 2 TX j (X; 0) 2 DgG1 = fX 2 TX j 9� 2 T �X s.t. (X;�) 2 Dg (3.1)and the smooth co-distributionsP0 = f� 2 T �X j (0; �) 2 DgP1 = f� 2 T �X j 9X 2 TX s.t. (X;�) 2 Dg (3.2)De�ne for any smooth distribution G the smooth co-distribution annG asannG = f� 2 T �X j h� jXi = 0 for all X 2 Gg (3.3)and for any smooth co-distribution P the smooth distribution ker P asker P = fX 2 TX j h� jXi = 0 for all � 2 Pg (3.4)The smooth (co-)distributions G0, G1 and P0, P1 are related as follows.7

Proposition 3.1 Let D be a generalized Dirac structure on X and de�ne G0, G1, P0, P1 as in(3.1), (3.2). Then1. G0 = ker P1, P0 = annG12. P1 � annG0, G1 � ker P0, with equality if G1, respectively P1, is constant-dimensional.Proof:1. Z 2 G0 if and only if (Z; 0) 2 D, if and only ifh0 jXi + h� jZi = 0; for all (X;�) 2 Dor equivalently h� jZi = 0 for all � 2 P1. Thus G0 = ker P1. Similarly � 2 P0 if and only if(0; �) 2 D, if and only if h� jXi = 0 for all X 2 G1, which implies P0 = annG1.2. Follows from property 1 and the inequalities P � ann ker P , G � ker annG, for any smooth(co-)distribution P and G, with equality if P and G are constant-dimensional [NS]. �Remark 3.1 The distribution G1 and the co-distribution P1 have the following interpretation.Consider the implicit generalized Hamiltonian system (2.11) corresponding to a generalized Diracstructure D and a Hamiltonian H. Then the distribution G1 describes the set of admissible ows_x. In particular, if G1 is constant-dimensional and involutive then there are (n � dimG1) inde-pendent conserved quantities for (2.11). Dually the co-distribution P1 describes the set of algebraicconstraints of (2.11), i.e. @H@x (x) 2 P1(x) (3.5)De�nition 3.1 A point x 2 X is a regular point for the Dirac structure D on X if the dimensionof G1 and P1 (and hence, see Proposition 3.1, of G0, P0) is constant in a neighborhood of x.At every regular point x 2 X we haveD?(x) = f(v; v�) 2 TxX � T �xX j hv� j vi+ hv� jvi = 0 for all (v; v�) 2 D(x)g (3.6)and, since D?(x) = D(x), we may regard D(x) � TxX � T �xX as a constant Dirac structure onTxX (see De�nition 2.2). Invoking Proposition 2.1 we deduce that dimD(x) = n for every regularpoint x 2 X . Since the set of regular points is open and dense in X , and D is a vector subbundle,it thus follows that dimD(x) = n; for all x 2 X (3.7)and therefore we may regard D(x) � TxX � T �xX as a constant Dirac structure on TxX for everyx 2 X . In particular it follows, since D is a smooth vector subbundle, that locally about everypoint in X we may �nd n�n matrices E(x) and F (x), depending smoothly on x, such that locallyD(x) = f(v; v�) 2 TxX � T �xX jF (x)v = E(x)v�grank[F (x) : �E(x)] = n (3.8)Furthermore, because D = D? necessarily (see [SM2])E(x)F T (x) + F (x)ET (x) = 0 (3.9)We will refer to this local representation (3.8), (3.9) of a Dirac structure as Representation I.Given a Hamiltonian H : X ! R the corresponding implicit generalized Hamiltonian system inRepresentation I is locally given as F (x) _x = E(x)@H@x (x) (3.10)8

Example 3.1 ([SM2], see also [MSB2]) An LC-circuit is composed of a set of (multiport) in-ductors and capacitors interconnected through their ports by the network graph. An n-port inductoris de�ned by ux linkage variables � 2 Rn (the energy variables) and an energy function HL(�).The port variables are the voltages vL 2 Rn and the currents iL 2 Rn de�ned asvL = _�; iL = @HL@� (3.11)Similary, an n-port capacitor is de�ned by charge variables q 2 Rn and energy function HC(q),with port variables the currents iC 2 Rn and voltages vC 2 Rn de�ned asiC = _q; vC = @HC@q (3.12)By Kirchho�'s laws we obtain nL + nC independent equationsFCiC +ECiL = 0; FLvL + ELvC = 0 (3.13)for certain matrices FC , FL, EC and EL satisfying (Tellegen's theorem)ECFTL + FCETL = 0 (3.14)Using (3.11), (3.12), and de�ning the total energy H(q; �) = HL(�)+HC(q), we may rewrite (3.13)as the implicit generalized Hamiltonian system� FC 00 FL �| {z }F � _q_� � = � 0 �EC�EL 0 �| {z }E " @H@q@H@� # (3.15)where EF T + FET = 0 by (3.14).Two other useful types of representations of generalized Dirac structures, which admit a globaland coordinate-free de�nition, can be given provided an extra regularity condition is satis�ed. Wewill denote them as Representation II, respectively Representation III.Theorem 3.1 (Representation II) Let X be an n-dimensional manifold. Let G be a constant-dimensional distribution on X , and J(x) : T �xX ! TxX , x 2 X , a skew-symmetric vector bundlemap. Then D = f(X;�) 2 TX � T �X jX(x) � J(x)�(x) 2 G(x); x 2 X ; � 2 annGg (3.16)de�nes a generalized Dirac structure. Conversely, let D be any generalized Dirac structure havingthe property that the co-distribution P1 (see (3.2)) is constant-dimensional. Then there exists askew-symmetric vector bundle map J(x) : P1(x) ! (P1(x))�, x 2 X , which locally can be extendedto a skew-symmetric vector bundle map J(x) : T �xX ! TxX , x 2 X , such that D is given by (3.16)with G := ker P1.Proof (see also [C1] for the constant case): Let D be given by (3.16). We have to show thatD? = D.1. Take (X;�) = (J�+ Z;�) 2 D, with Z 2 G. Then for all (X; �) = (J�+ Z ; �) 2 D, Z 2 Gh� j Xi + h� jXi = h� j J�i + h� j J�i+ h� j Zi+ h� jZi = 0because J(x) is skew-symmetric, and �; � 2 annG.2. Take (X;�) 2 D?, that is for all (X; �) = (J�+ Z; �) 2 D, Z 2 G, � 2 annG0 = h� j Xi+ h� jXi = h� jJ�i+ h� j Zi + h� jXiFirst let Z = 0. Then 0 = h� jJ�i+ h� jXi = h� jX � J�ifor all � 2 annG, implying that X � J� 2 ker annG = G, since G is constant-dimensional.Now let � = 0. Then 0 = h� j Zifor all Z 2 G, implying that � 2 annG. 9

Conversely, let D be a generalized Dirac structure on X , with P1 constant-dimensional. Then wede�ne for every x 2 X a linear mapJ(x) : P1(x) � T �xX ! (P1(x))� � TxXas follows. Let v� 2 P1(x), that is, there exists v 2 TxX such that (v; v�) 2 D(x). Then de�neJ(x)v� = v 2 (P1(x))� (3.17)To see that J(x) is well-de�ned, let also (v; v�) 2 D(x). Then (v � v; 0) 2 D(x), which meansv � v 2 G0(x) = ker P1(x), and thus v and v de�ne the same linear function on P1(x). Skew-symmetry of the map J(x) : P1(x)! (P1(x))� follows fromhv� jvi + hv� j vi = 0for all (v; v�); (v; v�) 2 D(x). Finally we may locally extend J(x) to a skew-symmetric map fromTxX to T �xX . Now, let (v; v�) 2 D(x). Then by (3.17) v = J(x)v� modulo G(x) := ker P1(x),while v� 2 P1(x), and thus D is indeed given by (3.16). �Remark 3.2 Note (see (3.17)) that the kernel of J(x) : P1(x)! (P1(x))� is given by P0(x).Given a Hamiltonian H : X ! R the equations of the implicit generalized Hamiltonian systemcorresponding to Representation II now take the form_x = J(x)@H@x (x) + g(x)�0 = gT (x)@H@x (x) (3.18)where g(x) is any full rank matrix such that Im g(x) = G(x). The variables � can be seen asLagrange multipliers, required to keep the constraint equations gT (x)@H@x (x) = 0 to be satis�edfor all time. Note that (3.18) can be also interpreted as a port-controlled generalized Hamiltoniansystem (see Section 2) with the e�orts (or outputs) e set equal to zero.\Dualizing" Representation II we obtainTheorem 3.2 (Representation III) Let X be an n-dimensional manifold. Let P be a constant-dimensional co-distribution on X , and !(x) : TxX ! T �xX , x 2 X , a skew-symmetric vector bundlemap. Then D = f(X;�) 2 TX � T �X j�(x)� !(x)X(x) 2 P (x); x 2 X ; X 2 ker Pg (3.19)de�nes a generalized Dirac structure. Conversely, let D be any generalized Dirac structure havingthe property that the distribution G1 (see (3.1)) is constant-dimensional. Then there exists a skew-symmetric vector bundle map !(x) : G1(x) ! (G1(x))�, x 2 X , which locally can be extended to askew-symmetric vector bundle map !(x) : TxX ! T �xX , x 2 X , such that D is given by (3.19) withP := annG1.Proof: Completely dual to the proof of Theorem 3.1 �Remark 3.3 (see Remark 3.2) The kernel of !(x) : G1(x)! (G1(x))� is given by G0(x).The equations of an implicit generalized Hamiltonian system corresponding to RepresentationIII and a Hamiltonian H take the form@H@x (x) = !(x) _x+ pT (x)�0 = p(x) _x (3.20)where p(x) is any full rank matrix such that Im p(x) = P (x). A main feature of (3.20) in comparisonwith (3.18) is that in (3.20) the ow constraints pT (x) _x = 0 are made explicit, while in (3.18) thealgebraic constraints gT (x)@H@x (x) = 0 are distinguished.10

Example 3.2 Let Q be an n-dimensional con�guration manifold of a mechanical system. Classical(kinematic) constraints are given in local coordinates q = (q1; : : : ; qn) for Q asAT (q) _q = 0 (3.21)with A(q) an n�k matrix, k � n, with entries depending smoothly on q. We will assume that A(q)has rank equal to k everywhere. The constrained Hamiltonian equations on T �Q are classicallygiven as (see e.g. [SM1])� _q_p � = � 0 In�In 0 �| {z }J " @H@q (q; p)@H@p (q; p) #+ � 0A(q) ��0 = � 0 AT (q) � " @H@q (q; p)@H@p (q; p) # (3.22)Here the constraint forces A(q)�, with � 2 Rk, are uniquely determined by the requirement thatthe constraints (3.21) have to be satis�ed for all time. It straightforward to see that an equivalentdescription of the equations (3.22) is given as follows" @H@q (q; p)@H@p (q; p) # = � 0 �InIn 0 �| {z }! � _q_p �+ � A(q)0 ��0 = � AT (q) 0 � � _q_p � (3.23)Let G and P be the distribution, respectively the co-distribution, on T �Q spanned by the columnsof the matrix � 0A(q)�, respectively the rows of the matrix �AT (q) 0�. Then, since both J and !are skew-symmetric, it follows from Theorem 3.1 and 3.2 that the pairs (J;G) and (!; P ) de�neRepresentation II, respectively Representation III, of the same generalized Dirac structure. We willrefer to this generalized Dirac structure as DA.As the last part of this section we will now brie y show how we can directly go from Repre-sentation I to a local version of Representation II or III, and vice versa. This is particularly usefulin analysis, where some aspects may be more easily studied in one representation, while others aremore easy to address in a di�erent representation. The transformation from Representation II or IIIto I is direct, and consists in eliminating the Lagrange multipliers �. Indeed, consider the implicitgeneralized Hamiltonian system (3.18) corresponding to Representation II. Since rank g(x) = kfor all x 2 X , we can locally �nd an (n � k) � n matrix s(x) of constant rank n � k such thats(x)g(x) = 0. Premultiplying the �rst n equations of (3.18) by s(x) then transforms (3.18) into thefollowing n equations �s(x)0 � _x = �s(x)J(x)gT (x) � @H@x (x) (3.24)which is easily seen to be of the form (3.10) with F (x) = �s(x)0 � and E(x) = �s(x)J(x)gT (x) � satisfying(3.8), (3.9). The transformation from Representation III to I is completely similar.Example 3.3 Consider again the mechanical system with kinematic constraints in Example 3.2.Since rankA(q) = k for all q 2 Q, we can locally �nd an (n � k) � n matrix S(q) of constant rankn� k such that S(q)A(q) = 0. Premultiplying the �rst 2n equations of (3.22) by the (2n� k)� 2nmatrix �In 00 S(q)� (3.25)of constant rank 2n� k, then transforms (3.22) into the following 2n equations24In 00 S(q)0 0 35� _q_p � = 24 0 In�S(q) 00 AT (q)35"@H@q (q; p)@H@p (q; p)# (3.26)11

The transformation from Representation I to II or III is more substantial. Consider Represen-tation I as given by (3.8), (3.9). Sinceker [F (x) : �E(x)] = Im � ET (x)�F T (x)� (3.27)we deduce that locally G1(x) = ImET (x); P1(x) = ImF T (x) (3.28)(while G0(x) = ker F (x), P0(x) = ker E(x) if F (x), respectively E(x) has constant rank). Inorder to obtain Representation II we need to assume that P1 has constant dimension (see Theorem(3.1)), or equivalently by (3.28), F (x) has constant rank. Then we may always locally transformthe equations F (x)v = E(x)v� into the form�F1(x)0 � v = �E1(x)E2(x)� v� (3.29)where F1(x) has full row rank for every x in this neighborhood. Since0 = E(x)F T (x) + F (x)ET (x) = �E1(x)F T1 (x) + F1(x)ET1 (x) F1(x)ET2 (x)E2(x)F T1 (x) 0 � (3.30)it follows that E1(x)F T1 (x) + F1(x)ET1 (x) = 0 (3.31)and ET2 (x)F1(x) = 0, or actually since rank [F (x) : �E(x)] = nker F1(x) = ImET2 (x) (3.32)By injectivity of F T1 (x) it follows that there exists an n � n matrix J(x) satisfying J(x)F T1 (x) =�ET1 (x), which is by (3.31) skew-symmetric on ImF T1 (x), and extendable to a skew-symmetricmatrix on Rn. Thus the equations (3.29) can be written asv � J(x)v� 2 kerF1(x) = ImET2 (x)0 = E2(x)v� (3.33)or equivalently, de�ning the constant rank matrix g(x) := ET2 (x)v = J(x)v� + g(x)�0 = gT (x)v� (3.34)which is Representation II. Representation III can be obtained similarly by manipulating insteadof F (x) the constant rank matrix E(x).4 Closedness of generalized Dirac structuresThe Dirac structures D of De�nition 2.1 are called generalized because they do not necessarilysatisfy the following closedness (or integrability) condition.De�nition 4.1 ([D2]) A generalized Dirac structure D on X is called closed (or simply a Diracstructure) if for arbitrary (X1; �1), (X2; �2) and (X3; �3) 2 D there holdshLX1�2 jX3i+ hLX2�3 jX1i+ hLX3�1 jX2i = 0 (4.1)The following theorem gives a very useful characterization of closedness of a generalized Diracstructure.Theorem 4.1 (cf. [D2], Theorem 2.1, see also [C1]) D is closed if and only if([X1; X2]; iX1d�2 � iX2d�1 + dh�2 jX1i) 2 D; 8(X1; �1); (X2; �2) 2 D (4.2)12

Proof: First note that the following identities (see e.g. [AMR])LX� = diX�+ iXd� (4.3)i[X;Y ]� = LX iY �� iY LX� (4.4)are satis�ed for all vector �elds X;Y and k-forms � on X . (The formula (4.3) is also known asCartan's magic formula.) Hence,hLX� jY i = hdh� jXi jY i+ d�(X;Y ) (4.5)h� j [X;Y ]i = �hdh� jXi jY i + hLY � jXi (4.6)for all vector �elds X;Y and one-forms � on X .Take now arbitrary (X1; �1); (X2; �2); (X3; �3) 2 D. ThenhiX1d�2 � iX2d�1 + dh�2 jX1i jX3i+ h�3 j [X1; X2]i= hiX1d�2 jX3i � hiX2d�1 jX3i+ hdh�2 jX1i jX3i+ h�3 j [X1; X2]i= d�2(X1; X3) + hdh�2 jX1i jX3i � d�1(X2; X3) + h�3 j [X1; X2]i= hLX1�2 jX3i+ d�1(X3; X2) + hLX2�3 jX1i � hdh�3 jX1i jX2i= hLX1�2 jX3i+ hLX2�3 jX1i+ hLX3�1 jX2i (4.7)since d�1(X2; X3) = �d�1(X3; X2), and dh�3 jX1i + dh�1 jX3i = 0 because (X1; �1); (X3; �3) 2D. Thus, D is closedmhLX1�2 jX3i+ hLX2�3 jX1i+ hLX3�1 jX2i = 0for all (X1; �1); (X2; �2); (X3; �3) 2 DmhiX1d�2 � iX2d�1 + dh�2 jX1i jX3i + h�3 j [X1; X2]i = 0for all (X1; �1); (X2; �2); (X3; �3) 2 Dm([X1; X2]; iX1d�2 � iX2d�1 + dh�2 jX1i) 2 Dfor all (X1; �1); (X2; �2) 2 Dwhere the last equivalence follows from the fact that D = D?. �Remark 4.1 Courant [C1] uses property (4.2) as the de�nition of closedness (or integrability) ofa generalized Dirac structure.Closedness only needs to be checked on a set of pairs (Xi; �i) which span the generalized Diracstructure D, as follows from the following lemma.Lemma 4.1 Consider a generalized Dirac structure D on a manifold X . Let(X1; �1); : : : ; (Xn; �n) 2 Dand suppose that �[Xi; Xj ]; iXid�j � iXjd�i + dh�j jXii� 2 D; i; j = 1; : : : ; n (4.8)Then also ([X;Y ]; iXd� � iY d�+ dh� jXi) 2 D, where(X;�) =Pni=1�i(Xi; �i); (Y; �) =Pni=1�i(Xi; �i) (4.9)for arbitrary �i; �i 2 C1(X ), i = 1; : : : ; n.Proof: Let = iXd� � iY d�+ dh� jXi. A straightforward calculation then gives[X;Y ] =Pni;j=1[�iXi(�j)Xj + �i�j[Xi; Xj]� �jXj(�i)Xi] (4.10) =Pni;j=1[�iXi(�j)�j + �i�j(iXid�j � iXjd�i + dh�j jXii) � �jXj(�i)�i] (4.11)13

Thus, from (4.8) it follows that ([X;Y ]; ) 2 D. �A smooth function H 2 C1(X ) is said to be admissible (see [C1]) if there exists a (smooth)vector �eld X such that (X;dH) 2 D. From the de�nition of the co-distribution P1 in equation(3.2) we see that the space of all admissible functions is given byAD = fH 2 C1(X ) jdH 2 P1g (4.12)There is a well-de�ned generalized Poisson bracket on AD given by the formulafH1;H2gD = hdH1 jX2i = �hdH2 jX1i (4.13)where (X1;dH1); (X2;dH2) 2 D. To show that f ; gD as de�ned in (4.13) is a generalized Poissonbracket is straightforward. Bilinearity of f ; gD follows from bilinearity of h j i. Skew-symmetryis a consequence of (4.13). Finally, take arbitrary (X1;dH1); (X2;dH2); (X3;dH3) 2 D. ThenfH1;H2H3gD = �hd(H2H3) jX1i = �hH3dH2 +H2dH3 jX1i= H3fH1;H2gD +H2fH1;H3gD (4.14)so f ; gD also satis�es the Leibniz identity. For a Dirac structure given by Representation II (seeTheorem 3.1), f ; gD is given as followsfH1;H2gD(x) = �@H1@x (x)�T J(x) @H2@x (x); H1;H2 2 AD (4.15)We will now characterize closedness of a generalized Dirac structure D in terms of the bracketf ; gD and the admissible functions AD. The following necessary conditions for closedness followfrom Theorem 4.1.Corollary 4.1 (cf. [C1, D2]) If D is closed then1. G0 and G1 are involutive distributions2. fH1;H2gD 2 AD3. fH1; fH2;H3gDgD + fH2; fH3;H1gDgD + fH3; fH1;H2gDgD = 0for all H1;H2;H3 2 AD.Proof:1. Let X1; X2 2 G0, i.e. (X1; 0); (X2; 0) 2 D. Then by Theorem 4.1 ([X1; X2]; 0) 2 D, whichmeans that [X1; X2] 2 G0. Involutivity of G1 also follows directly from Theorem 4.1.2. Take H1;H2 2 AD so that (X1;dH1); (X2;dH2) 2 D. Then ([X1; X2];dhdH2 jX1i) 2 Dwhich means that dhdH2 jX1i = dfH2;H1gD 2 P1 ) fH1;H2gD 2 AD (4.16)3. Take H1;H2;H3 2 AD so that (X1;dH1); (X2;dH2); (X3;dH3) 2 D. Then0 = hLX1dH2 jX3i + hLX2dH3 jX1i+ hLX3dH1 jX2i= hdhdH2 jX1i jX3i+ hdhdH3 jX2i jX1i+ hdhdH1 jX3i jX2i= hdfH2;H1gD jX3i + hdfH3;H2gD jX1i + hdfH1;H3gD jX2i= ffH2;H1gD;H3gD + ffH3;H2gD;H1gD + ffH1;H3gD;H2gD= fH1; fH2;H3gDgD + fH2; fH3;H1gDgD + fH3; fH1;H2gDgD (4.17)�If in addition the co-distribution P1 (see (3.2)) is constant-dimensional, the following theorem givesnecessary and su�cient conditions for closedness in terms of f ; gD and AD.14

Theorem 4.2 Consider a generalized Dirac structure D on a manifold X . Let P1 denote the co-distribution on X de�ned by (3.2). Assume that P1 is constant-dimensional. Then D is closed ifand only if the following three conditions are satis�ed1. G0 = ker P1 is involutive2. fH1;H2gD 2 AD3. fH1; fH2;H3gDgD + fH2; fH3;H1gDgD + fH3; fH1;H2gDgD = 0for all H1;H2;H3 2 AD.Proof: The necessity of these three conditions follows from Corollary 4.1 so we only have to showthe su�ciency part here. First note that by using Proposition 3.1 we have that P1 = annG0. SinceG0 = ker P1 is involutive and P1 is constant-dimensional, by Frobenius' theorem in a neighborhoodof any point x0 2 X there exists local coordinates x = (x1; : : : ; xn) such thatP1 = annG0 = span fdx1; : : : ;dxn�mg; (4.18)where m = dim ker P1 (= dimG0). In the sequel, every computation is done in such a neighborhoodusing local coordinates.Take now arbitrary (X1; �1); (X2; �2) 2 D. Then, since �1; �2 2 P1, we have that�1 = Pn�mi=1 �idxi (4.19)�2 = Pn�mi=1 �idxi (4.20)where �i; �i are smooth functions. Now, let the vector �elds Y1; : : : ; Yn�m be such that(Yi;dxi) 2 D; 1 � i � n�m (4.21)Since also (X1; �1) 2 D it follows thathdxk jX1i+ hPn�mi=1 �idxi jYki = 0 (4.22)so hdxk jX1i = �Pn�mi=1 �ifxi; xkgD (4.23)for 1 � k � n�m. De�ne the vector �elds Z1; Z2 asZ1 = X1 �Pn�mi=1 �iYi (4.24)Z2 = X2 �Pn�mi=1 �iYi (4.25)Then hdxk jZ1i = hdxk jX1 �Pn�mi=1 �iYii= �Pn�mi=1 �ifxi; xkgD �Pn�mi=1 �ifxk; xigD= 0 (4.26)for all 1 � k � n�m since fxi; xkgD = �fxk; xigD. This means that Z1 2 kerP1 = G0 andX1 = Pn�mi=1 �iYi + Z1 (4.27)X2 = Pn�mi=1 �iYi + Z2 (4.28)where Z1; Z2 2 G0.Now we want to calculate the term�12 = iX1d�2 � iX2d�1 + dh�2 jX1i (4.29)We have d�2 = d(Pn�mi=1 �idxi) =Pn�mi=1 d�i ^ dxi, soiX1d�2 = iX1 (Pn�mi=1 d�i ^ dxi)= Pn�mi=1 [iX1d�i ^ dxi � d�i ^ iX1dxi]= Pn�mi=1 [hd�i jPn�mj=1 �jYj + Z1idxi � hdxi jPn�mj=1 �jYj + Z1id�i]= Pn�mi;j=1[�jYj(�i)dxi � �jfxi; xjgDd�i] +Pn�mi=1 Z1(�i)dxi (4.30)15

where we used the fact that hdxi jZ1i = 0 since dxi 2 annG0. Similarly we obtainiX2d�1 =Pn�mi;j=1[�jYj(�i)dxi � �jfxi; xjgDd�i] +Pn�mi=1 Z2(�i)dxi (4.31)Moreover, dh�2 jX1i = dhPn�mi=1 �idxi jPn�mj=1 �jYj + Z1i= d[Pn�mi;j=1�i�jhdxi jYji]= d[Pn�mi;j=1�i�jfxi; xjgD]= Pn�mi;j=1[�i�jdfxi; xjgD + fxi; xjgD(�jd�i + �id�j)]= Pn�mi;j=1[�i�jdfxi; xjgD + fxi; xjgD(�jd�i � �jd�i)] (4.32)where the last equation follows from skew-symmetry of f ; gD. Inserting (4.30), (4.31) and (4.32)in (4.29) gives �12 = iX1d�2 � iX2d�1 + dh�2 jX1i= Pn�mi;j=1[(�jYj(�i) � �jYj(�i))dxi + �i�jdfxi; xjgD]+Pn�mi=1 (Z1(�i) � Z2(�i))dxi (4.33)From (4.33) we immediately see that �12 2 P1 since dfxi; xjgD 2 P1 when 1 � i; j � n�m.Now we have to take a closer look at the term [X1; X2]. A direct calculation yields[X1; X2] = [Pn�mi=1 �iYi + Z1;Pn�mj=1 �jYj + Z2]=Pn�mi;j=1f(�jYj(�i)� �jYj(�i))Yi + �i�j [Yj; Yi]g+Pn�mi=1 (Z1(�i)� Z2(�i))Yi + Z12 (4.34)where the vector �eld Z12 is given byZ12 =Pn�mi=1 (�i[Yi; Z2]� �i[Yi; Z1]) + [Z1; Z2] (4.35)Take now arbitrary Z 2 G0 and consider dfxi; xjgD 2 annG0 for 1 � i; j � n�m. Then0 = hdfxi; xjgD jZi= �Z(fxj ; xigD)= �Z(Yi(xj))= �Yi(Z(xj)) + [Yi; Z](xj)= hdxj j [Yi; Z]i (4.36)for all 1 � i; j � n�m, which means that [Yi; Z] 2 ker P1 = G0 for all 1 � i � n�m. Since G0 isinvolutive we immediately see from (4.35) that also Z12 2 G0.Now we want to show that ([Yj ; Yi];dfxi; xjgD) 2 D. We know that fxi; xjgD 2 AD, whichmeans that there exist vector �elds Yij such that (Yij ;dfxi; xjgD) 2 D, i; j = 1; : : : ; n�m. Thenhdxk j [Yj; Yi]� Yiji = [Yj; Yi](xk)� Yij(xk)= Yj(hdxk jYii) � Yi(hdxk jYji) � hdxk jYiji= hdfxk; xigD jYji � hdfxk; xjgD jYii � fxk; fxi; xjgDgD= ffxk; xigD; xjgD � ffxk; xjgD; xigD � fxk; fxi; xjgDgD= fxj; fxi; xkgDgD + fxi; fxk; xjgDgD + fxk; fxj; xigDgD= 0 (4.37)when 1 � i; j; k � n�m, which means that [Yj ; Yi]� Yij 2 ker P1 = G0. Thus,([Yj; Yi];dfxi; xjg) 2 D; i; j = 1; : : : ; n�m (4.38)and by inspection of (4.33) and (4.34) we see that ([X1; X2]; �12) 2 D, and closedness of D followsfrom Theorem 4.1. �In the following we will explicitly characterize closedness in the three di�erent representationsof a Dirac structure. 16

Theorem 4.3 (Representation I) Consider a generalized Dirac structure D on a manifold Xgiven locally in Representation I (see (3.8), (3.9)). De�ne (Xi; �i) 2 D in local coordinates byXi = ETi (x) (4.39)�i = �F Ti (x) (4.40)where ETi (x) and F Ti (x) denote the i-th column of the matrices ET (x) and F T (x) respectively.Then D is closed if and only if�[Xi; Xj ]; iXid�j � iXjd�i + dh�j jXii� 2 D(x); 8x 2 X ; i; j = 1; : : : ; n (4.41)Proof: Follows from (3.27), Theorem 4.1, and Lemma 4.1. �Theorem 4.4 (Representation II) Let X be an n-dimensional manifold. Let G be a constant-dimensional distribution on X , and J(x) : T �xX ! TxX , x 2 X , a skew-symmetric vector bundlemap. Moreover, let f ; g denote the generalized Poisson bracket corresponding to J . Then thegeneralized Dirac structure given by (see Theorem 3.1)D = f(X;�) 2 TX � T �X jX(x) � J(x)�(x) 2 G(x); x 2 X ; � 2 annGg (4.42)is closed if and only if1. G is involutive2. fH1;H2g 2 AD3. fH1; fH2;H3gg+ fH2; fH3;H1gg+ fH3; fH1;H2gg = 0for all H1;H2;H3 2 AD = fH 2 C1(X ) jdH 2 annGg.Proof: The result follows from Theorem 4.2 using the facts that G0 = G and that fH1;H2gD =fH1;H2g for all H1;H2 2 AD. �Theorem 4.5 (Representation III) Let X be an n-dimensional manifold. Let P be a constant-dimensional co-distribution on X , and !(x) : TxX ! T �xX , x 2 X , a skew-symmetric vector bundlemap. Then the generalized Dirac structure given by (see Theorem 3.2)D = f(X;�) 2 TX � T �X j�(x)� !(x)X(x) 2 P (x); x 2 X ; X 2 ker Pg (4.43)is closed if and only if1. kerP is involutive2. d!(X1; X2; X3) = 0 for all X1; X2; X3 2 kerPProof: Let (X1; �1); (X2; �2) 2 D, i.e.�i = iXi! + pi; pi 2 P; Xi 2 kerP; i = 1; 2 (4.44)De�ne as in (4.29) the 1-form �12 = iX1d�2 � iX2d�1 + dh�2 jX1i. Now, using Cartan's magicformula, we get diX1! = LX1! � iX1d! (4.45)diX1iX2! = LX1iX2! � iX1LX2! + iX1iX2d! (4.46)for all vector �elds X1; X2 on X . Hence�12 = iX1d�2 � iX2d�1 + dh�2 jX1i= iX1d(iX2! + p2)� iX2d(iX1! + p1) + diX1(iX2! + p2)= iX1diX2! + iX1dp2 � iX2diX1! � iX2dp1 + diX1iX2! + diX1p2= �iX2LX1! + LX1iX2! + iX1dp2 � iX2dp1 + iX2iX1d!= i[X1 ;X2]! + iX1dp2 � iX2dp1 + iX2 iX1d! (4.47)17

since i[X1 ;X2]! = LX1 iX2!� iX2LX1!. So, using Theorem 4.1 and the de�nition of D, we have thatD is closedm([X1; X2]; i[X1 ;X2 ]! + iX1dp2 � iX2dp1 + iX2 iX1d!) 2 D 8p1; p2 2 P; 8X1; X2 2 kerPm[X1; X2] 2 kerPiX1dp2 � iX2dp1 + iX2iX1d! 2 P � 8p1; p2 2 P; 8X1; X2 2 kerPNow, if P is a constant-dimensional co-distribution and kerP is involutive, it follows that for everyp 2 P there exists �p 2 P and a one-form � such that dp = � ^ �p. Thus, iXdp = �(X)�p 2 P forall X 2 kerP . Moreover, iX2 iX1d!(X3) = d!(X1; X2; X3) which means that iX2iX1d! 2 P if andonly if d!(X1; X2; X3) = 0 for all X3 2 ker P since P is constant-dimensional. �Remark 4.2 In [C1] it is shown that closedness of D implies condition 2 in Theorem 4.5.We will now apply the above theory to mechanical systems with kinematic constraints (seeExample 3.2).Proposition 4.1 Consider the mechanical system with kinematic constraints AT (q) _q = 0 as givenin Example 3.2. Let f ; g denote the Poisson bracket de�ned (locally) by the structure matrix J .Then the following statements are equivalent1. DA is closed2. The constraints (3.21) are holonomic3. dfH1;H2g 2 annG for all H1;H2 such that dH1;dH2 2 annGProof: 1 , 2: From Theorem 4.5 it follows that DA is closed if and only if ker P is involutivewhich is equivalent to the constraints (3.21) being holonomic. 1 , 3: Follows from Theorem 4.4since G is involutive and f ; g satis�es the Jacobi identity in this case. �The next proposition gives an interesting interpretation of closedness of generalized Dirac struc-tures that come up in connection with Lie-Poisson structures.Proposition 4.2 Let G be any n-dimensional Lie group (e.g. SE(3)), with Lie algebra g, and thedual Lie algebra g� with the Lie-Poisson bracket f ; g. Consider a constant distribution on g�, thatis a linear subspace V � g�. De�ne the Dirac structure D on g� asD = f(X;�) 2 Tg� � T �g� jX(x)� J(x)�(x) 2 V; �(x) 2 V?; x 2 g�g (4.48)where J(x) is the structure matrix of the Lie-Poisson bracket f ; g. Then D is closed if and onlyif V? � g is a subalgebra.Proof: The proof follows more or less directly from results in [MR], page 287. �Example 4.1 (X = se�(3) w R6) The motion of a rigid body with respect to a body-�xed rotationreference frame in the center of mass is given by (in the absence of gravity)M _! + ! �M! = � (4.49)m _v + ! �mv = F (4.50)where v; ! 2 R3 are the linear, respectively, the angular velocities, M is the inertia tensor and�; F 2 R3 are the torques, respectively, the forces. By de�ning �; p 2 R3 as� = M!; � = [�x;�y;�z]T (4.51)p = mv; p = [px; py; pz]T (4.52)and the Hamiltonian H(�; p) as H(�; p) = 12�TM�1�+ 12mpTp (4.53)18

it follows that (4.49) and (4.50) can be written as" _�_p # = " S(�) S(p)S(p) 0 #| {z }J(�;p) " @H@�@H@p #+ " �F # (4.54)Here � = [�x;�y;�z]T and p = [px; py; pz]T are the body angular and linear momentum respec-tively. S( � ) is de�ned by S(a)b = a � b for a; b 2 R3. J(�; p) is the structure matrix of theLie-Poisson bracket on X = se�(3) w R6.Assume that the following constraints are imposed on the systempy = pz = 0 (4.55)Let ex = [1 0 0]T , ey = [0 1 0]T , ez = [0 0 1]T . ThenV? = ker � 0 0ey ez� (4.56)which is not a subalgebra of se(3) w R6 (see e.g. [MR]). Hence, the corresponding generalized Diracstructure is not closed in this case. However, if the additional constraint px = 0 is imposed on thesystem (�xed center of mass), it is easy to see that closedness of the corresponding generalized Diracstructure follows.Similarly to the case when the Jacobi-identity is satis�ed for a generalized Poisson structure, onecan show that if the closedness condition (4.1) is satis�ed for a generalized Dirac structure thenthere exist local canonical coordinates around any regular point in which the geometric picturesimpli�es considerably (see Proposition 4.1.2 in [C1]). In our context (i.e. for generalized Diracstructures arising from physical systems) constant-dimensionality of the co-distribution P1 is oftena reasonable assumption. Thus, in the next proposition we will draw attention to the existenceand construction of canonical coordinates for Dirac structures that may be given in RepresentationII (cf. Theorem 3.1). In essence, the proof of this proposition comes down to using Frobenius'theorem and a generalized version of Darboux' theorem and proceeds along the same general lineas the proof of Proposition 4.1.2 in [C1]. However, we show directly how local canonical coordinatesmay be found for a Dirac structure in Representation II. In addition, we show more explicitly wherethe three necessary conditions in Corollary 4.1 come into play which is interesting in itself.Proposition 4.3 Let D be a generalized Dirac structure on an n-dimensional manifold X . Assumethat the co-distribution P1 (see (3.2)) is constant-dimensional so that D can always be given inRepresentation II as followsD = f(X;�) 2 TX � T �X jX(x) � J(x)�(x) 2 G0(x); x 2 X ; � 2 annG0g (4.57)where J(x) : T �xX ! TxX , x 2 X , is a skew-symmetric vector bundle map. Then, if D is closed,there exist around every regular point x0 2 X local canonical coordinates(q; p; r; s) = (q1; : : : ; qk; p1; : : : ; pk; r1; : : : ; rl; s1; : : : ; sm); 2k + l +m = nfor X in which J(x) and G0 take the simple formJ(x) = 2664 0 Ik 0 ��Ik 0 0 �0 0 0 �� � � �3775 ; G0 = span � @@s1 ; : : : ; @@sm� (4.58)where � denotes unspeci�ed elements, m = n� dimP1 and l = n � dimG1(x0).Conversely, if D is given by (4.57), (4.58) in a neighborhood of x0 2 X , then D is closed in thisneighborhood. 19

Proof: If D is closed, it follows from condition 1 in Corollary 4.1 that G0 is involutive. SinceP1 = annG0 is constant-dimensional, also G0 is constant-dimensional with dimension equal tom. Thus, by Frobenius' theorem in a neighborhood Nx0 of any point x0 2 X there exist localcoordinates (y; s) = (y1; : : : ; yn�m; s1; : : : ; sm), such thatG0 = span � @@s1 ; : : : ; @@sm� (4.59)and P1 = annG0 = span fdy1; : : : ;dyn�mg (4.60)Now, f ; gD is given in terms of J(x) as followsfF;GgD(x) = �@F@x (x)�T J(x) @G@x (x) (4.61)for all F;G 2 C1(X ) such that dF;dG 2 annG0. Moreover, since D is closed it follows fromcondition 2 in Corollary 4.1 thatdfyi; yjgD 2 annG0; i; j = 1; : : : ; n�m (4.62)which means that @fyi; yjgD@sk = 0; k = 1; : : : ;m; i; j = 1; : : : ; n�m (4.63)Hence, J(x) takes the following form in the local coordinates (y; s)J(y; s) = � �J(y) �� �� (4.64)where �J(y) = [fyi; yjgD] is the (n�m)� (n�m) upper-left submatrix of J(y; s). In addition, thedistribution G1 is given locally in the coordinates (y; s) asG1(y; s) = Im � �J(y) 00 Im� (4.65)If x0 2 X is a regular point then G1 is by de�nition constant-dimensional in a neighborhood of x0which implies that �J(y) has constant rank 2k = n � (l +m) in a neighborhood Nx0 � Nx0 of x0.De�ne (without loss of generality) the submanifold Y � X asY = f(y; s) 2 Nx0 j s = s(x0)g (4.66)y = (y1; : : : ; yn�m) are local coordinates for Y around y0 = y(x0). Since D is closed it follows fromcondition 3 in Corollary 4.1 that f ; gD de�nes a Poisson structure on Y with structure matrix�J(y). Now, using the fact that �J(y) has constant rank 2k � n � m for all y 2 Y it follows fromTheorem 6.22 in [O] (called the generalized Darboux' theorem; see also [W]), that around y0 2 Ythere exist local (q; p; r) = (q1; : : : ; qk; p1; : : : ; pk; r1; : : : ; rl) in which �J(y) takes the form�J(p; q; r) = 24 0 Ik 0�Ik 0 00 0 035 (4.67)Now (q; p; r; s) are local coordinates for X around x0 2 X in which J(x) and G0 take the simpleform (4.58).Conversely, it is easy to check that a generalized Dirac structure given by (4.57), (4.58) in aneighborhood of x0 2 X , satis�es the su�cient conditions for closedness as given in Theorem 4.2in this neighborhood. �The equations of an implicit generalized Hamiltonian system corresponding to the local repre-sentation (4.57), (4.58) and a Hamiltonian H take the form_q = @H@p (q; p; r; s)_p = �@H@q (q; p; r; s)_r = 00 = @H@s (q; p; r; s) (4.68)20

Comparing (4.68) with (2.7) we see that while (2.7) makes explicit the conserved quantities, (4.68)also makes explicit the algebraic constraints0 = @H@s1 (q; p; r; s)...0 = @H@sm (q; p; r; s) (4.69)If H is non-degenerate in the energy-variables s1; � � � ; sm, that isrank � @2H@si@sj � = m (4.70)then by the Implicit function theorem one may locally express the variables s1; � � � ; sm as functionsof q; p; r; i.e. si = si(q; p; r); i = 1; � � � ;m. De�ning the constrained HamiltonianHc(q; p; r) := H(q; p; r; s(q; p; r)) (4.71)it follows that (4.68) reduces to the same format as (2.7)_q = @Hc@p (q; p; r)_p = �@Hc@q (q; p; r)_r = 0 (4.72)which is an explicit Hamiltonian dynamics on the constrained state space Xc = f(q; p; r; s) j@H@si (q; p; r; s) = 0; i = 1; � � � ;mg. Also note that while under the assumption (4.70) the variabless1; � � � ; sm together with the HamiltonianH de�ne a (constraint) submanifold Xc of X , dually thelevel sets of the variables r1; � � � ; r` de�ne a foliation of X . Both the constraint submanifoldXc andthe foliation are invariant for the Hamiltonian dynamics. However, as shown in this section, thereare cases of interest where the generalized Dirac structure does not satisfy the closedness condi-tion (e.g. mechanical systems with nonholonomic constraints). Furthermore, also if the closednesscondition is satis�ed the actual construction of the canonical coordinates qi; pi; ri; si, may be veryinvolved, and preferably should be avoided.We remark that the representation (4.68) of an implicit Hamiltonian system with regard to aclosed Dirac structure is quite amenable for stability analysis, at least when the non-degeneracycondition (4.70) is satis�ed. Indeed, let (q0; p0; r0; s0) be an equilibrium of (4.68), that is@H@q (q0; p0; r0; s0) = 0; @H@p (q0; p0; r0; s0) = 0; @H@s (q0; p0; r0; s0) = 0 (4.73)and let us also assume that @H@r (q0; p0; r0; s0) = 0 (see later on). Under the non-degeneracy condition(4.70) the Implicit function theorem allows to express the variables s locally around q0, p0, r0, s0as functions of q, p, r leading as above to the explicit Hamiltonian dynamics (4.72). Note that ingeneral the Implicit function theorem only provides an existence result, and that �nding the actualexpression of s as function of q, p, r is in general not possible or preferably should be avoided.Now, if the Hessian matrix of Hc at (q0; p0; r0) is positive (or negative) de�nite it follows that(q0; p0; r0) is a stable equilibrium of (4.72) (see e.g. [MR]). On the other hand, this Hessian matrixof Hc can be easily expressed in the original Hamiltonian H as2664 @2H@q2 @2H@q@p @2H@q@r@2H@p@q @2H@p2 @2H@p@r@2H@r@q @2H@r@p @2H@r2 3775� 2664 @2H@q@s@2H@p@s@2H@r@s 3775�@2H@s2 ��1 h @2H@s@q @2H@s@p @2H@s@ri (4.74)evaluated at (q0; p0; r0; s0). Thus this way of checking stability can be performed without the actualcomputation of Hc. Furthermore, note that for checking de�niteness of (4.74) only the variables sneed to be explicitly computed; we may use other coordinates instead of q, p, r.Since the variables r1; : : : ; rl are invariants (or Casimirs) we may also replace in the sta-bility analysis the constrained Hamiltonian Hc by Hc(q; p; r) + �(r), with � any function ofr = (r1; : : : ; rl). Hence we may also replace H(q; p; r; s) by�H�(q; p; r; s) := H(q; p; r; s)+ �(r) (4.75)21

and substitute �H� into (4.74) in order to check de�niteness. (The addition of a function �(r) toHc when checking the de�niteness of the Hessian is known as the the Energy-Casimir method; seee.g. [MR].)5 Implicit port-controlled generalized Hamiltonian systemsAs already alluded to in Section 2, if we interconnect port-controlled Hamiltonian systems (2.1) insuch a way that some of the external variables remain free port variables, then we will end up withan implicit generalized Hamiltonian system with external (or port) variables. In order to make thisprecise we give the following de�nition (see [SM2]).De�nition 5.1 Let X be an n-dimensional manifold of energy variables, and let H : X ! R be aHamiltonian. Furthermore, let F be the linear space Rm of external ows f , with dual the spaceF� of external e�orts e. Consider a Dirac structure on the product space X � F , only dependingon x. The implicit port-controlled generalized Hamiltonian system corresponding to X , H, D andF is de�ned by the speci�cation � _x; f; @H@x (x);�e� 2 D(x) (5.1)Remark 5.1 The minus sign in front of the e�ort e comes from the natural identi�cation (�; e) 2T �X � F� ! (�;�e) 2 (TX � F)�. Physically this means that the ingoing power is countedpositively.Since by de�nition of a Dirac structure (cf. (2.9)) h� jXi � he j fi = 0 for all (X; f; �;�e) 2 D, itfollows that an implicit port-controlled Hamiltonian system satis�es the energy balancedHdt = eT f (5.2)De�nition 5.1 generalizes the notion of an (explicit) port-controlled generalized Hamiltonian system(2.1) by noting that in this case the Dirac structure D on X � F is given by the speci�cation(X; f; �;�e) 2 D i� X(x) = J(x)�(x) + g(x)fe = gT (x)�(x); x 2 X (5.3)Indeed, let (X; f; �;�e) 2 D?, that ish� jXi + h� j Xi � he j fi � he j f i = 0 (5.4)for all (X; f ; �;�e) satisfying (5.3). By �rst taking f = 0 we obtain�T (x)X(x) + �T (x)J(x)�(x) � �T (x)g(x)f = 0 (5.5)for all �, and thus X(x) = J(x)�(x) + g(x)f , and substitution in (5.4) yields�T (x)g(x)f + �T (x)g(x)f � �T (x)g(x)f � eT f = 0 (5.6)for all f , implying that e = gT (x)�(x), and thus that (X; f; �;�e) 2 D.Now let us consider, as in Section 2, k port-controlled generalized Hamiltonian systems, see(2.15), with Ej = F�j , j = 1; : : : ; k. A power-conserving partial interconnection is obtained bywriting a direct sum decomposition F1 � : : :� Fk = F i �Fp (5.7)with the subspace F i denoting the ows to be interconnected, and Fp the remaining ows at theexternal ports of the partially interconnected system. By de�ning E i := (Fp)? and Ep := (F i)?we obtain the dual direct sum decompositionE1 � : : :� Ek = E i � Ep (5.8)22

Proposition 5.1 Consider as in (2.15) k port-controlled generalized Hamiltonian systems, withdirect sum decomposition (5.7), (5.8). Consider a power-conserving partial interconnection givenby a subspace (possibly parametrized by x1; : : : ; xk)I(x1; : : : ; xk) � F i � E i (5.9)with dim I(x1; : : : ; xk) = dim F i, having the property(f i; ei) 2 I(x1; : : : ; xk) ) hei jf ii = 0 (5.10)Then the resulting partially interconnected system is an implicit port-controlled generalized Hamilto-nian system with state space X := X1�: : :Xk, Hamiltonian H(x1; : : : ; xk) := H1(x1)+: : :+Hk(xk)and generalized Dirac structure on X � Fp given as(X; fp; �;�ep) = (X1; : : : ; Xk; fp; �1; : : : ; �k;�ep) 2 D ()Xj(xj) = Jj(xj)�j(xj) + gj(xj)fjej = gTj (xj)�j(xj); xj 2 Xj; j = 1; : : : ; k(f1; : : : ; fk; e1; : : : ; ek) = (f i; fp; ei; ep) such that (f i; ei) 2 I(x1; : : : ; xk) (5.11)Proof: The proof is very similar to the proof of Proposition 2.2. Let (X; fp; �;�ep) be in D?,that is h� jXi + h� j Xi � hep j fpi � hep j fpi = 0 (5.12)for all (X; fp; �;�ep) satisfying (5.11). Letting �rst fj = ej = 0 for all j = 1; : : : ; k we obtain(2.21), and by substitution in (5.12) we obtain, similarly as in (2.22)0 = kXj=1�eTj fj + eTj fj�� hep j fpi � hep j fpi = hei j f ii + hei j f ii (5.13)for all f i; ei. By de�nition of I(x1; : : : ; xk) in (5.10) this implies, as in Proposition 2.2 (adding ifnecessarily ow vectors in the kernel of gj(xi)) that (f i; ei) 2 I(x1; : : : ; xk), and thus (X; fp; �;�ep)2 D. Since it is readily seen that D � D? it follows that D de�nes a Dirac structure. �Remark 5.2 An interesting open problem is the variational interpretation of Proposition 5.1 (andProposition 2.2). Indeed, if all the Hamiltonian sub-systems admit a variational characterization (asEuler-Lagrange equations) one could conjecture that also the (partially) interconnected Hamiltoniansystem admits "some kind of" variational characterization. It is to be expected, however, that theclosedness conditions as treated in this and the previous section will play an important role in sucha characterization, since already for classical mechanical systems with kinematic constraints it isknown (see e.g. [AKN, BC]) that they cannot be formulated as standard Euler-Lagrange equationsin case the constraints are nonholonomic. Also, the formulation (4.68) of an implicit Hamiltoniansystem satisfying the closedness condition suggests a connection with variational principles via the�rst-order condition of Pontryagin's Maximum principle. In the case of electrical circuits, where theinterconnections are de�ned by Kirchho�'s laws and the closedness conditions are trivially satis�ed(see Example 3.1), some important work concerning a variational formulation of Kirchho�'s lawsand the resulting variational characterization of the overall circuit has been done, see e.g. [JE, M1],and it seems of interest to extend these ideas to the general situation considered in Proposition 5.1.In the rest of this section we will not elaborate on general implicit port-controlled Hamiltoniansystems and their di�erent representations, but instead concentrate on a special subclass whicharises naturally in the control of mechanical systems. Consider the following port-controlled gen-eralized Hamiltonian system with constraints given by_x = J(x)@H@x (x) + g(x)f + b(x)�e = gT (x)@H@x (x)0 = bT (x)@H@x (x) (5.14)23

where x 2 X , f 2 F := Rm and g(x) = [g1(x) : : : gm(x)] is the n �m matrix of input vector �eldsgj. b(x) = [b1(x) : : : bk(x)] is the n� k matrix of constraint vector �elds. Throughout this sectionwe will assume that b(x) has rank equal to k everywhere. It is easily seen that e.g. an actuatedmechanical system with kinematic constraints will �t into the description (5.14). By rewriting(5.14) as follows � _xf � = � J(x) 00 0 �| {z }~J(x) � @H@x (x)�e �+ � g(x) b(x)Im 0 � ~�0 = � gT (x) ImbT (x) 0 �� @H@x (x)�e � (5.15)where ~� 2 Rm+k, it follows from Theorem 3.1 that (5.15) de�nes Representation II of a generalizedDirac structure D on X � F . Thus (5.14) is an implicit port-controlled generalized Hamiltoniansystem.We will now study D further as given in representation (5.15). In what follows we will usef ; g and f ; gX�F to denote the generalized Poisson brackets on X , respectively X � F , withstructure matrices J(x) and ~J(x) (see (5.15)) respectively. In addition we will let B denote theconstant-dimensional distribution on X given byB(x) = Imb(x); x 2 X (5.16)From (5.15) we immediately see that the distribution G0 on X � F de�ned by D (see (3.1)) isgiven by G0(x; y) = Im � g(x) b(x)Im 0 � ; (x; y) 2 X � F (5.17)Note that G0 is constant-dimensional with dimension equal to m + k since rank b(x) = k for allx 2 X . The following lemma, for which a proof is straightforward, gives necessary and su�cientconditions for G0 being involutive,Lemma 5.1 G0 is involutive if and only if [X;Y ] 2 B for all X;Y 2 fg1; : : : ; gm; b1; : : : ; bkgThe next lemma gives three necessary conditions for the closedness of D.Lemma 5.2 If the generalized Dirac structure D on X � F is closed then1. fH1; fH2;H3gg+ fH2; fH3;H1gg+ fH3; fH1;H2gg = 02. LgjfH1;H2g = fLgjH1;H2g+ fH1; LgjH2g, j = 1; : : : ;m3. dfH1;H2g 2 annBfor all H1;H2;H3 2 C1(X ) such that dH1;dH2;dH3 2 annB.Proof: Assume that D is closed, i.e. satis�es (4.1). Using Cartan's magic formula, the closednesscondition (4.1) can be written ashdh�2 jX1i jX3i + hdh�3 jX2i jX1i + hdh�1 jX3i jX2i+ d�2(X1; X3) + d�3(X2; X1) + d�1(X3; X2) = 0 (5.18)Consider H1, H2 and H3 2 C1(X ) where dHi 2 annB, i = 1; 2; 3. Let j 2 f1; : : : ;mg and de�neX1(x; y) = � XH1 (x)0 � ; �1(x; y) = " @H1@x (x)�LTg H1(x) # (5.19)X2(x; y) = � XH2 (x)0 � ; �2(x; y) = " @H2@x (x)�LTg H2(x) # (5.20)X3(x; y) = � XH3 (x) + �gj(x)�Yj � ; �3(x; y) = " @H3@x (x)�LTg H3(x) # (5.21)24

for (x; y) 2 X �F where Yj = @@yj , � 2 R andXHi (x) = J(x)@Hi@x (x); LTg Hi(x) = gT (x)@Hi@x (x) (5.22)Thus, (Xi; �i) 2 D, i = 1; 2; 3. Now, it is easy to see that h�i jXji = fHi;Hjg, i; j = 1; 2; 3, whichimplies that hdh�2 jX1i jX3i = ffH2;H1g;H3g+ �LgjfH2;H1g (5.23)hdh�3 jX2i jX1i = ffH3;H2g;H1g (5.24)hdh�1 jX3i jX2i = ffH1;H3g;H2g (5.25)Moreover, we have that�i = @Hi@x1 dx1 + : : :+ @Hi@xndxn � Lg1Hidy1 � : : :� LgmHidym; i = 1; : : : ; 3 (5.26)which means that d�i = � mXl=1 nXk=1 @LglHi@xk dxk ^ dyl; i = 1; : : : ; 3 (5.27)Hence, d�2(X1; X3) = �� h @LgjH2@x iT XH1 = ��fLgjH2;H1g (5.28)and similarlyd�1(X3; X2) = �d�1(X2; X3) = �fLgjH1;H2g = ��fH2; LgjH1g (5.29)In addition, it follows that d�3(X2; X1) = 0. Therefore, from the integrability condition (5.18) wehave thatffH2;H1g;H3g+ ffH3;H2g;H1g+ ffH1;H3g;H2g+ � �LgjfH2;H1g � fLgjH2;H1g � fH2; LgjH1g� = 0 (5.30)for all � 2 R, implying for � = 0 condition 1, and for � = 1 condition 2.Now, a direct calculation yields�12 = iX2d�1 � iX1d�2 + dh�1 jX2i =� mXl=1 (fLglH1;H2g++fH1; LglH2g)dyl + nXk=1 @fH1;H2g@xk dxk (5.31)from which condition 3 (and condition 2) follows directly since �12 2 annG0 (see Theorem 4.1). �Before being able to give the su�cient and necessary conditions for D being closed, we also needthe following result.Lemma 5.3 If for arbitrary H1;H2 2 C1(X ) such that dH1;dH2 2 annB there holds1. LgjfH1;H2g = fLgjH1;H2g+ fH1; LgjH2g, j = 1; : : : ;m2. dfH1;H2g 2 annBthen f ~H1; ~H2gX�F 2 AD for all ~H1; ~H2 2 AD.Proof: Take arbitrary ~H2; ~H2 2 AD. From (5.17) we see that this is equivalent to0 = bT (x)@ ~Hi@x (x; y) (5.32)@ ~Hi@y (x; y) = �gT (x)@ ~Hi@x (x; y) (5.33)25

for i = 1; 2. Let j 2 f1; : : : ;mg and de�negj(x; y) = � gj(x)0 � (5.34)Then (5.33) can be written as @ ~Hi@yk = �Lgk ~Hi; k = 1; : : : ;m (5.35)for i = 1; 2. Now, f ~H1; ~H2gX�F = nXk;l=1 Jkl(x)@ ~H1@xk @ ~H2@xl (5.36)so @f ~H1; ~H2gX�F@yj = � fLgj ~H1; ~H2gX�F � f ~H1; Lgj ~H2gX�F (5.37)Since fH1;H2g = nXk;l=1 Jkl(x)@H1@xk @H2@xl ; H1;H2 2 C1(X ) (5.38)it follows from condition 1 thatLgjf ~H1; ~H2gX�F = fLgj ~H1; ~H2gX�F + f ~H1; Lgj ~H2gX�F (5.39)which inserted in (5.37) yields @f ~H1; ~H2gX�F@yj = �Lgjf ~H1; ~H2gX�F (5.40)Moreover, from condition 2 it follows thatbT (x)@f ~H1; ~H2gX�F@x = 0 (5.41)Thus, from (5.40) and (5.41) we see that f ~H1; ~H2gX�F 2 AD. �We are now ready to present the necessary and su�cient conditions for (5.15) de�ning a Diracstructure on X �F .Theorem 5.1 The generalized Dirac structure D on X � F as de�ned by (5.15) is closed if andonly if1. [X;Y ] 2 B for all vector �elds X;Y 2 fg1; : : : ; gm; b1; : : : ; bkg2. LgjfH1;H2g = fLgjH1;H2g+ fH1; LgjH2g, j = 1; : : : ;m3. dfH1;H2g 2 annB4. fH1; fH2;H3gg+ fH2; fH3;H1gg+ fH3; fH1;H2gg = 0for all H1;H2;H3 2 C1(X ) such that dH1;dH2;dH3 2 annB.Proof: The necessary and su�cient conditions for closedness of D follows immediately by com-bining the results in Lemma 5.1, Lemma 5.2, Lemma 5.3 and then using Theorem 4.4. �Corollary 5.1 (B = 0) Let b(x) = 0 (no constraints) in (5.14). Then the generalized Dirac struc-ture D on X �F as de�ned by (5.15) (with b(x) = 0) is closed if and only if1. [gi; gj] = 0, i; j = 1; : : : ;m2. LgjfH1;H2g = fLgjH1;H2g+ fH1; LgjH2g for all H1;H2 2 C1(X ), j = 1; : : : ;m26

3. f ; g satis�es the Jacobi identityHence, the closedness condition (4.1) for the generalized Dirac structure on X �F arising fromthe constrained port-controlled Hamiltonian system (5.14) translates (among other things) intostrong conditions on the input vector �elds gj.The conditions 2-4 in Theorem 5.1 may be succinctly expressed by requiring that the generalizedPoisson bracket f ; g of F;G 2 C1(X ) where dF , dG 2 annB is preserved by the dynamics of(5.14) for every choice of internal energy H such that dH 2 annB and for every f 2 F . Indeed,requiring that ddtfF;Gg = f ddtF;Gg+ fF; ddtGg (5.42)for all F;G 2 C1(X ) such that dF;dG 2 annB where ddt denotes the time-derivative along (5.14),is equivalent toffF;Gg;Hg+ LgfF;Ggf + LbfF;Gg� =ffF;Hg; Gg+ f(LgF )f;Gg+ fF; fG;Hgg+ fF; (LgG)fg (5.43)for all H 2 C1(X ) such that dH 2 annB and f 2 F . Letting �rst H = 0 and f = 0, we obtainLbfF;Gg� = 0; 8� 2 Rk (5.44)which means that dfF;Gg 2 annB. Moreover, letting f = 0 leads toffF;Gg;Hg= ffF;Hg; Gg+ fF; fG;Hg (5.45)which is nothing else than the Jacobi-identity. Thus, (5.43) amounts toLgfF;Ggf = f(LgF )f;Gg+ fF; (LgG)fg; 8f 2 F (5.46)which is equivalent to LgjfF;Gg = fLgjF;Gg+ fF;LgjGg; j = 1; : : : ;m (5.47)The next example should give an idea of what the conditions in Theorem 5.1 imply for the (local)mathematical structure of system (5.14).Example 5.1 Consider the port-controlled generalized Hamiltonian system with constraints givenin (5.14). Assume that the conditions 1-4 in Theorem 5.1 are all satis�ed. By condition 1 itfollows that the constant-dimensional distribution B is involutive. Hence, by Frobenius' theorem ina neighborhood of any point x0 2 X there exist local coordinates (y; s) = (y1; : : : ; yn�k; s1; : : : ; sk),such that annB = span fdy1; : : : ;dyn�kg (5.48)and B = span � @@s1 ; : : : ; @@sk� (5.49)Condition 3 implies that @fyi; yjg@sl = 0; l = 1; : : : ; k; i; j = 1; : : : ; n� k (5.50)Hence, J(x) takes the following form in the local coordinates (y; s)J(y; s) = �Jyy(y) �� �� (5.51)where Jyy(y) = [fyi; yjg] is the (n�k)� (n�k) upper-left submatrix of J(y; s). From condition 1 italso follows that [bi; gj] 2 B which implies that in the coordinates (y; s) the matrix of input vector�elds takes the form g(x; y) = � gy(y)gs(y; s) � (5.52)27

Furthermore, since [gi; gj](y; s) = � [gyi; gyj ](y)� � (5.53)while [gi; gj] 2 B it follows that [gyi; gyj ] = 0, i; j = 1; : : : ;m. Assume additionally that thedistribution B + G is constant-dimensional with dimension equal to m + k. Then the submatrixgy(y) of g(x; y) has constant rank equal to m � n� k. Thus (see e.g. Theorem 2.36 in [NS]), thereexists a local transformation (y1; : : : ; yn�k)! (~y1; : : : ; ~yn�k) such thatgyj = @@~yj ; j = 1; : : :m (5.54)In these coordinates condition 2 amounts to@f~yi; ~yjg@~yl = �@~yi@~yl ; ~yj� +�~yi; @~yj@~yl � = 0; l = 1; : : : ;m; i; j = 1; : : :n � k (5.55)which means that f~yi; ~yjg is independent of the �rst m local coordinates ~y1; : : : ; ~ym. Let nowz = (~ym+1 ; : : : ; ~yn�k) and w = (~y1; : : : ; ~ym). Then from the discussion above we can conclude that(z; w; s) are local coordinates for X around x0 in which (5.14) takes the form� _z_w � = � Jzz(z) Jzw(z)�JTzw(z) Jww(z)� " @H@z (z; w; s)@H@w (z; w; s) #+ � 0Im � f_s = Jsz(z; w; s)@H@z (z; w; s) + Jsw(z; w; s)@H@w (z; w; s) + gs(z; w; s)f + bs(z; w; s)�e = @H@w (z; w; s)0 = @H@s (z; w; s) (5.56)where the last equation follows from the fact that the k�k matrix bs(z; w; s) has full rank. Note thatthe equation for _s can be left out from (5.56) because it is only needed to determine the Lagrangemultipliers � 2 Rk. Finally from condition 4 it follows that the matrix� Jzz(z) Jzw(z)�JTzw(z) Jww(z)� (5.57)satis�es the Jacobi-identity (in the (z,w)-coordinates).Finally, in the next example we will relate the results in this paper (in particular this section)to \passivity-based control" of actuated mechanical systems with kinematic constraints.Example 5.2 Consider a mechanical system with kinematic constraints AT (q) _q = 0 as in Exam-ple 3.2. Additionally, let the system be actuated by generalized external forces u = (u1; : : : ; um)corresponding to generalized con�guration coordinates C1(q); : : : ; Cm(q). The dynamical equationsof motion are given as� _q_p � = � 0 In�In 0 � " @H@q (q; p)@H@p (q; p) #+Pmi=1 � 0@Ci@q (q) �ui + � 0A(q) ��yi = h@Ci@q (q)iT @H@p (q; p) �= dCidt (q)� ; i = 1; : : : ;m0 = AT (q)@H@p (q; p) (= AT (q) _q) (5.58)This is a port-controlled generalized Hamiltonian system with constraints as in (5.14) with external ows the vector (u1; : : : ; um) of external forces, and external e�orts the vector (y1; : : : ; ym) ofcorresponding generalized velocities. It can be veri�ed, as in Proposition 4.1, that the underlyinggeneralized Dirac structure satis�es the conditions of Theorem 5.1 (that is, is closed) if and only ifthe kinematic constraints AT (q) _q are holonomic.28

Now, consider an additional port-controlled Hamiltonian system (the \controller")_� = ucyc = @P@� (�); �; uc; yc 2 Rm (5.59)with Hamiltonian P . (Note that this is of the type (2.1) with J = 0, g = identity matrix, x = �,f = uc, and e = yc.) Feedback interconnection as in Example 2.1 leads to the implicit generalizedHamiltonian system24 _q_p_� 35 = 264 0 In 0�In 0 �@C@q (q)0 @TC@q (q) 0 3752664 @H@q (q; p)@H@p (q; p)@P@� (�) 3775+ 24 0A(q)0 35�0 = AT (q)@H@p (q; p) (5.60)with @C@q (q) denoting the matrix with i-th column @Ci@q (q). The co-distribution P0 of the underlyinggeneralized Dirac structure can be readily seen to be given asP0 = span fdCi � d�i j i = 1; : : : ;mg (5.61)expressing the fact (see also Remark 3.1) that the functionsCi(q) � �i; i = 1; : : : ;m (5.62)are independent conserved quantities for the closed-loop dynamics (5.60). It follows that along(5.60) �i(t) = Ci(q(t)) + ci; for all t; i = 1; : : : ;m (5.63)with the constants ci solely depending on the initial conditions of the \controller" (5.59).Substituting (5.63) into (5.60), and noting that@C@q (q)@P@� (C1(q) + c1; : : : ; Cm(q) + cm) = @P@q (C1(q) + c1; : : : ; Cm(q) + cm) (5.64)it follows that the dynamics of the (q; p)-part of (5.60) (the original mechanical system) is given as� _q_p � = � 0 In�In 0 � " @Hnew@q (q; p)@Hnew@p (q; p) #+ � 0A(q) ��0 = AT (q)@Hnew@p (q; p) (5.65)where Hnew is the \new" Hamiltonian de�ned byHnew(q; p) = H(q; p) + P (C1(q) + c1; : : : ; Cm(q) + cm) (5.66)Thus by appropriately choosing the Hamiltonian P (�) of the \controller sub-system" (5.59), wemay shape the Hamiltonian H(q; p) of the constrained mechanical system (5.58) by addition ofthe potential energy P (C1(q) + c1; : : : ; Cm(q) + cm), with c1; : : : ; cm only depending on the initialcondition of (5.59) (that is, by properly initialization we may set c1 = : : : = cm = 0). This ideaof shaping the internal energy is one of the main ideas of \passivity-based control". We have thusdemonstrated that this can be accomplished by power-conserving (in-fact, feedback) interconnectionof (5.58) with a controller sub-system (5.59).In particular, if H and C1, : : : , Cm are such that P can be chosen in such a manner that Hnewas de�ned by (5.66) has a strict minimum at some desired equilibrium point (q0; p0), then (q0; p0)will be a (Lyapunov) stable equilibrium of (5.65) (and, because of (5.63), also the �-dynamics willbe stable). To be more precise we only need the function Hnew restricted to the constraint manifoldf(q; p) jAT (q)@Hnew@p (q; p) = 0g to have a strict minimum at (q0; p0).29

It can be veri�ed that the underlying generalized Dirac structure of (5.60) is closed if and onlyif the kinematic constraints AT (q) _q = 0 are holonomic. If this happens to be the case then checkingthat Hnew restricted to the constraint manifold has a strict minimum may be performed as indicatedat the end of Section 4.Within the same philosophy one may pursue asymptotic stability by adding, apart from theenergy-shaping Hamiltonian controller (2.24), energy-dissipating elements to the system. In partic-ular, one may replace the feedback interconnection uc = y, u = �yc as above by the power-conservingpartial interconnection (with free external ow v and external e�ort y)uc = yu = �yc + v (5.67)and then terminate this port by an energy-dissipating elementv = �@R@y (y) (5.68)for some (Rayleigh) dissipation function R. For the asymptotic stability analysis of the resultingclosed-loop system one again need to distinguish between holonomic and nonholonomic kinematicconstraints AT (q) _q = 0. (In fact, in the nonholonomic case there is a fundamental obstruction toasymptotic stabilization, since Brockett's necessary conditions is not satis�ed; see e.g. [MS3] forthe references.)6 ConclusionsIt has been shown that a power-conserving interconnection of port-controlled generalized Hamilto-nian systems leads to an implicit generalized Hamiltonian system, and a power-conserving partialinterconnection to an implicit port-controlled Hamiltonian system. The crucial concept is the no-tion of a (generalized) Dirac structure, de�ned on the space of energy-variables or on the productof the space of energy-variables and the space of ow-variables in the port-controlled case. Threenatural representations of generalized Dirac structures have been treated. Necessary and su�cientconditions for closedness of a Dirac structure in all three representations have been obtained. Thishas been illustrated on mechanical systems with kinematic constraints and constrained systemson dual Lie algebras. Canonical coordinates for (closed) Dirac structures have been discussed, aswell as their use for stability analysis of implicit Hamiltonian systems. Finally the theory has beenapplied to implicit port-controlled generalized Hamiltonian systems, such as actuated mechani-cal systems with kinematic constraints, and it has been shown in particular that the closednesscondition for the Dirac structure leads to strong conditions on the input vector �elds.AcknowledgementsWe would like to thank Bernhard Maschke (Conservatoire National des Arts et M�etiers, Paris) forstimulating discussions. We also thank Peter Crouch (Arizona State Univ.) and Tony Bloch (Univ.of Michigan) for pointing out to the second author the relevance of Dirac structures.References[AKN] V.I. Arnold, V.V. Kozlov, and A.I.Neishtadt, Mathematical aspects of classical and ce-lestial mechanics, Springer-Verlag, 1997.[AMR] R. Abraham, J.E. Marsden, and T. Ratiu,Manifolds, Tensor Analysis, and Applications,Springer-Verlag, 1988.[BC] A.M. Bloch and P.E. Crouch, Nonholonomic control systems on Riemannian manifolds,SIAM J.Control and Optimization, 33 (1995), pp. 126-148.[BKMM] A.M. Bloch, P.S. Krishnaprasad, J.E. Marsden, and R.M. Murray, Nonholonomic me-chanical systems with symmetry, Archive for Rational Mechanics and Analysis, 136(1996), pp. 21-99. 30

[C1] T.J. Courant, Dirac manifolds, Trans. American Math. Soc., 319 (1990), pp. 631-661.[C2] P.E. Crouch, Handwritten notes (1994).[CW] T.J. Courant and A.Weinstein, Beyond Poisson structures, Seminaire sud-rhodanien degeometrie 8.Travaux en Cours 27 (1988), Hermann, Paris.[D1] I. Dorfman, Dirac structures of integrable evolution equations, Physics Letters, A 125(1987), pp. 240-246.[D2] I. Dorfman, Dirac Structures and Integrability of Nonlinear Evolution Equations, Chich-ester: John Wiley, 1993.[D3] P.Dirac, Generalized Hamiltonian dynamics, Canadian Journal of Mathematics, 2 (1950),pp. 129-148.[JE] D.L. Jones and F.J. Evans, Variational analysis of electrical networks, Journal of theFranklin Institute, 295 (1973), pp. 9-23.[M1] A.G.J. MacFarlane, An integral formulation of a canonical equation set for non-linearelectrical networks, Int.J.Control, 11 (1970), pp. 449-470.[M2] B.M. Maschke, Elements on the Modelling of Multibody systems, in Modelling and Controlof Mechanisms and Robot system, C. Melchiorri and A. Tornambe, eds., World Scienti�cPublishing Ltd., 1996.[MBS] B.M. Maschke, C. Bidard, and A.J. van der Schaft, Screw-vector bond graphs for thekinestatic and dynamic modeling of multibody systems, Proc. ASME Int. Mech. Engg.Congress a. Expo., 55-2 (1994), Chicago, USA, pp. 637-644.[MR] J.E. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry, Springer-Verlag,1994.[MS1] B.M. Maschke and A.J. van der Schaft, Port-controlled Hamiltonian systems: Modellingorigins and system theoretic properties, Proc. IFAC Symp. NOLCOS, Bordeaux, France,1992, M. Fliess, ed., pp. 282-288.[MS2] B.M. Maschke and A.J. van der Schaft, System-theoretic properties of port-controlledHamiltonian systems, in Systems and Networks: Mathematical Theory and Applications,vol. II., Berlin: Akademie Verlag, 1994, pp. 349-352.[MS3] B.M. Maschke and A.J. van der Schaft, A Hamiltonian approach to stabilization of non-holonomic mechanical systems, Proc. 33rd IEEE CDC, Orlando, (1994), pp. 2950-2954.[MSB1] B.M. Maschke, A.J. van der Schaft, and P.C. Breedveld, An intrinsic Hamiltonian formu-lation of network dynamics: Non-standard Poisson structures and gyrators, J. FranklinInst., 329 (1992), pp. 923-966.[MSB2] B.M. Maschke, A.J. van der Schaft, and P.C. Breedveld, An intrinsic Hamiltonian for-mulation of the dynamics of LC-circuits, IEEE Trans. Circ. a. Syst., 42 (1995), pp. 73-82.[NS] H. Nijmeijer and A.J. van der Schaft, Nonlinear Dynamical Control Systems, Springer-Verlag, 1990.[O] P.J. Olver, Applications of Lie Groups to Di�erential Equations, Second Edition,Springer-Verlag, 1993.[OS] R. Ortega and M.W. Spong, Adaptive motion control of rigid robots: A tutorial, Auto-matica, 25 (1989), pp. 877-888.[S] J.J. Slotine, Putting Physics in Control: the Example of Robotics, IEEE Contr. Syst.Mag., 8, 1988, pp. 12-18.[SM1] A.J. van der Schaft and B.M. Maschke, On the Hamiltonian formulation of nonholonomicmechanical systems, Rep. Math. Phys., 34 (1994), 225-233.31

[SM2] A.J. van der Schaft and B.M. Maschke, The Hamiltonian formulation of energy conservingphysical systems with external ports, Archiv f�ur Elektronik und �Ubertragungstechnik, 49(1995), 362-371.[SM3] A.J. van der Schaft and B.M. Maschke, Mathematical modelling of constrained Hamilto-nian systems, Preprints 3rd NOLCOS, Tahoe City, CA, 1995.[TA] M. Takegaki and S. Arimoto,A new feedback method for dynamic control of manipulators,Trans. ASME, J. Dyn. Syst. Meas. Contr., 103 (1981), pp. 119-125.[W] A.Weinstein, The local structure of Poisson manifolds, J. Di�. Geom., 18 (1983), 523-557.

32