Tensor products of Dirac structures and interconnection in Lagrangian mechanical systems

Many mechanical systems are large and complex, despite being composed of simple subsystems. In order to understand such large systems it is natural to tear the system into these subsystems. Conversely we must understand how to invert this tearing. In other words, we must understand interconnection. Such an understanding has already successfully understood in the context of Hamiltonian systems on vector spaces via the port-Hamiltonian systems program. In port-Hamiltonian systems theory, interconnection is achieved through the identification of shared variables, whereupon the notion of composition of Dirac structures allows one to interconnect two systems. In this paper we seek to extend the port-Hamiltonian systems program to Lagrangian systems on manifolds and extend the notion of composition of Dirac structures appropriately. In particular, we will interconnect Lagrange-Dirac systems by modifying the respective Dirac structures of the involved subsystems. We define the interconnection of Dirac structures via an interaction Dirac structure and a tensor product of Dirac structures. We will show how the dynamics of the interconnected system is formulated as a function of the subsystems, and we will elucidate the associated variational principles. We will then illustrate how this theory extends the theory of port-Hamiltonian systems and the notion of composition of Dirac structures to manifolds with couplings which do not require the identification of shared variables. Lastly, we will close with some examples: a mass-spring mechanical systems, an electric circuit, and a nonholonomic mechanical system.


Introduction
A large class of physical and engineering problems can be described in terms of Lagrangian and Hamiltonian systems, such as controlled mechanical systems, externally forced and dissipative systems, collisions, electric circuits, stochastic systems, and field theories such as electromagnetism and elasticity. Such systems are usually categorized into Lagrangian or Hamiltonian systems with constraints, whether holonomic or nonholonomic; however, one may be faced with serious problems on ever larger scales becoming more and more heterogeneous, involving a mixture of mechanical and electrical components, with flexible and rigid parts, and elastic or magnetic couplings (see, for instance, Yoshimura [1995]; Bloch [2003] and Afshari, Bhat, Hajimiri and Marsden [2006]). Dirac structures are known as a powerful tool for providing a natural geometric framework for describing the structure of such diverse Lagrangian and Hamiltonian systems on the product of configuration spaces by introducing nontrivial constraints that model interactions between subsystems, which is called an interconnection. The notion of an interconnection was originally developed by Kron [1963] in his book of "Diakoptics" for modeling complicated physical systems. The word "diakoptics" denotes a procedure of tearing a dynamical system into disconnected subsystems as well as interconnecting them to reconstruct the original system. The original system may be regarded as an interconnected system of subsystems, with constraints induced from the interconnection. Interconnection has been studied through the lens of power conservation motivated by the bond graph theory established by Paynter [1961]. In electric circuit theory, the interconnection network has been widely employed through nonenergic multiports (see Brayton [1971]; Wyatt and Chua [1977]), where Kirchhoff's current law gives a constraint distribution on a configuration charge space such that Tellegen's theorem holds. In mechanics, it was shown by Yoshimura [1995] that kinematical constraints due to mechanical joints, coordinate transformations as well as force equilibrium conditions in d'Alembert principle can be represented by the interconnection modeled by nonenergic multiports using the bond graph theory in the context of Lagrangian mechanics.
It was demonstrated by van der Schaft and  that the interconnection can be represented by Dirac structures induced from Poisson structures and also that nonholonomic systems and L-C circuits can be represented by implicit Hamiltonian systems. On the Lagrangian side, it was shown by Yoshimura and Marsden [2006a] that nonholonomic mechanical systems and L-C circuits (as degenerate Lagrangian systems) can be formulated by implicit Lagrangian systems associated with the Dirac structures on the cotangent bundles induced from given constraint distributions. From the view point of variational structures, it was shown by Yoshimura and Marsden [2006b] that the standard implicit Lagrangian systems, namely, implicit Euler-Lagrange equations for unconstrained systems can be derived from the Hamilton-Pontryagin principle and also that the Lagrange-Dirac dynamical systems can be formulated in the context of Lagrange-d'Alembert-Pontryagin principle.
This bracket is the one that Courant [1990] originally developed and it does not necessarily satisfy the Jacobi identity. It was shown by Dorfman [1993] that the integrability condition of the Dirac structure D ⊂ T M ⊕ T * M given in equation (2.2) can be expressed as which is the closedness condition of the Courant bracket (see also Dalsmo and van der Schaft [1998] and Jotz and Ratiu [2008]). The following Lemma is useful and the proof may be found in Yoshimura and Marsden [2006a] and Courant [1990]. Induced Dirac Structures. One of the most important and interesting Dirac structures in mechanics is an induced Dirac structure from kinematic constraints, whether holonomic or nonholonomic. Such constraints are generally given by a distribution on a configuration manifold (as to the details, refer to Yoshimura and Marsden [2006a]).
Let Q be a configuration manifold. Let T Q and T * Q be the tangent and cotangent bundles of Q. Let ∆ Q ⊂ T Q be a regular distribution on Q and define a lifted distribution on T * Q by where π Q : T * Q → Q is the cotangent projection so that its tangent is a map T π Q : T T * Q → T Q. Let Ω be the canonical two-form on T * Q. The induced Dirac structure D on T * Q is the subbundle of T T * Q ⊕ T * T * Q, whose fiber is given for each (q, p) ∈ T * Q as This is, of course, a special instance of the construction (2.1). Notice that the induced Dirac structure can be restated by using the bundle map Ω : T T * Q → T * T * Q as follows: Remark. If there exists no constraint, namely, the case in which ∆ Q = T Q, the Dirac structure D corresponds to the canonical Dirac structure, which is given by a graph of the bundle map Ω associated to Ω.
Local Expressions. Let V be a model space for Q and U be the range in V of a chart on Q. Then, the range of the induced chart on T Q is represented by U × V , while T * Q is locally given by U × V * . In this representation, (q,q) are local coordinates of T Q and (q, p) are local coordinates of T * Q. Further, T T * Q is represented by In this local representation, (q, p,q,ṗ) are the corresponding coordinates of T T * Q, while (q, p, β, v) are the local coordinates of T * T * Q. Using π Q : T * Q → Q locally denoted by (q, p) → q and T π Q : (q, p,q,ṗ) → (q,q), it follows that ∆ T * Q = {w = (q, p,q,ṗ) | q ∈ U,q ∈ ∆(q)} and the annihilator of ∆ T * Q is locally represented as Since we have the local formula Ω (q, p) · w = (q, p, −ṗ,q), the condition α − Ω (q, p) · w ∈ ∆ • T * Q reads β +ṗ ∈ ∆ • (q) and v −q = 0. Thus, the induced Dirac structure is locally represented by

Lagrange-Dirac Dynamical Systems
In this section, let us recall the definition of implicit Lagrangian systems or Lagrange-Dirac dynamical systems by following Yoshimura and Marsden [2006a,b].
Lagrange-Dirac Dynamical Systems. Let L : T Q → R be a Lagrangian, possibly degenerate. Let D be a Dirac structure on T * Q induced from a given distribution ∆ Q ⊂ T Q.
Recall that a partial vector field X : T Q ⊕ T * Q → T T * Q is defined as a mapping such that τ T * Q • X = pr T * Q , where pr T * Q : T Q ⊕ T * Q → T * Q; (q, v, p) → (q, p) and τ T * Q : T T * Q → T * Q; (q, p,q,ṗ) → (q, p) is the tangent projection. 2 The partial vector field X : T Q ⊕ T * Q → T T * Q is a map that assigns to each point (q, v, p) ∈ T Q ⊕ T * Q, a vector in T T * Q at the point (q, p) ∈ T * Q; we write X as where we note thatq = dq/dt andṗ = dp/dt are understood to be functions of (q, v, p).
An implicit Lagrangian system or a Lagrange-Dirac dynamical system is the triple (E L , D, X) that satisfies, for each (q, v, p) ∈ T Q⊕T * Q and with P = FL(∆ Q ), namely, (q, p) = (q, ∂L/∂v), may be regarded as a linear function on T (q,p) P such that (q, p) = (q, ∂L/∂v) ∈ P = FL(∆ Q ).
Local Expressions. It follows from (2.3) that the Lagrange-Dirac dynamical system (X, dE L | T P ) ∈ D is locally given by Remark. If a partial vector field X(q, v, p) = (q, p,q,ṗ) satisfies the condition for a Lagrange-Dirac dynamical system (X, dE L | T P ) ∈ D, then the Legendre transformation (q, p) = (q, ∂L/∂v) ∈ P = FL(∆ Q ) is consistent with the equality of base points and the condition itself gives the second-order conditionq = v. In other words, the partial vector field of the Lagrange-Dirac dynamical system is uniquely given on the graph of the Legendre transformation.
For the case in which no kinematic constraint is imposed, i.e., ∆ Q = T Q, we can develop the standard implicit Lagrangian system, which is expressed in local coordinates as which we shall call implicit Euler-Lagrange equations. Note that the implicit Euler-Lagrange equations include the Euler-Lagrange equationsṗ = ∂L/∂q, the Legendre transformation p = ∂L/∂v as well as the second-order conditionq = v.
Example: Harmonic Oscillators. We illustrate the implicit Euler-Lagrange equations by the example of a linear harmonic oscillator. In this case, the configuration space is Q = R representing the position of a particle on the line. The Lagrangian is given by L = v 2 /2 − q 2 /2. We find the generalized energy to be E L (q, v, p) = pv − v 2 /2 + q 2 /2. The canonical Dirac structure D(≡ graph(Ω )) can be easily constructed by using the canonical symplectic structure Ω on T * Q. The Lagrange-Dirac system is given by (E L , D, X), which satisfies and it follows dE L (q, v, p)| T P = Ω (q, p) · (q,ṗ). We see dE L (q, v, p)| T P = vdp + qdq and Ω (q, p)(q,ṗ) = −ṗdq +qdp. So, the dynamics is equivalent to the following equations: together with the Legendre transform p = v.
Lagrange-Dirac Dynamical Systems with External Forces. Here, we consider the case of a system with external force fields. Let pr Q : Given a Lagrangian L : T Q → R (possibly degenerate), a Lagrange-Dirac dynamical system with an external force field is defined by a quadruple (E L , D, X, F ), which satisfies, in coordinates (q, v, p) ∈ T Q ⊕ T * Q, (3.2) It follows that the local expression of a Lagrange-Dirac system in equation (3.2) may be given byq and with p = ∂L ∂v .
The curve (q(t), v(t), p(t)), t 1 ≤ t ≤ t 2 in T Q ⊕ T * Q that satisfies the condition (3.2) is a solution curve of (E L , D, X, F ).
Example: The Harmonic Oscillator with Damping. As before, consider the harmonic oscillator in the setting of Q = R, L = v 2 /2 − q 2 /2, E L = pv − v 2 /2 + q 2 /2 and D(≡ graph(Ω )) and add a damping force F : T Q ⊕ T * Q → T * Q defined by F (q, v, p) = −(rv)dq, where r ∈ R + is a positive damping constant. Then,F = (q, v, p, rv, 0, 0). The formulas in equation (3.3) give us the equations: The Hamilton-Pontryagin Principle. Lagrange-Dirac dynamical systems can describe general scenarios such as non-holonomic degenerate Lagrangian systems because of a firm basis in variational structures. Given a Lagrangian L : T Q → R, unconstrained equations of motion are described by Hamilton's principle: From the viewpoint of Dirac geometry, we start rather from the Hamilton-Pontryagin principle, which is given by the stationary condition of the action integral on the space of curves whereq(t) = dq(t)/dt. The above variational principle can be restated by using the generalized energy E L : Keeping the endpoints q(t 0 ) and q(t 1 ) of q(t) fixed, we can obtain the implicit Euler-Lagrange equations.
The Lagrange-d'Alembert-Pontryagin Principle. Consider a mechanical system with kinematic constraints that are given by a constraint distribution ∆ Q on Q. Then, the motion of the mechanical system c : [t 1 , t 2 ] → Q is said to be constrained ifċ(t) ∈ ∆ Q (c(t)) for all t, t 1 ≤ t ≤ t 2 . Assume that the distribution ∆ Q is not involutive; that is, [X(q), Y (q)] / ∈ ∆ Q (q) for any two vector fields X, Y on Q with values in ∆ Q . Further, let L be a (possibly degenerate) Lagrangian on T Q and let F : T Q ⊕ T * Q → T * Q be an external force field. The Lagrange-d'Alembert-Pontryagin principle for a curve (q(t), v(t), p(t)), for a given variation δq(t) ∈ ∆ Q (q(t)) and with the constraint v(t) ∈ ∆ Q (q(t)). Keeping the endpoints of q(t) fixed, it follows for a chosen variation δq(t) ∈ ∆ Q (q(t)), for all δv(t) and δp(t), and with v(t) ∈ ∆ Q (q(t)).
Proposition 3.1. The Lagrange-d'Alembert-Pontryagin principle gives the local expression of equations of motion for nonholonomic mechanical systems with external forces such thaṫ which is satisfied for a given variation δq(t) ∈ ∆ Q (q(t)), for all δv(t) and δp(t), and with the constraint v(t) ∈ ∆ Q (q(t)). Thus, we obtain equation (3.5).
Coordinate Representation. Let Q be a finite dimensional smooth manifold whose dimension is n. Choose local coordinates q i on Q so that Q is locally represented by an open set U ⊂ R n . Then, T Q is locally given by (q i , v i ) ∈ U × R n . Similarly T * Q may be locally denoted by (q i , p i ) ∈ U × R n . The constraint set ∆ Q defines a subspace on each fiber of T Q, which we denote by ∆ Q (q) ⊂ R n at each point q ∈ U . If the dimension of ∆ Q (q) is n − m, then we can choose a basis e m+1 (q), e m+2 (q), . . . , e n (q) of ∆ Q (q). Recall that the constraint sets can be also represented by the annihilator ∆ • Q (q), which is spanned by m one-forms ω 1 , ω 2 , . . . , ω m . It follows that equation (3.5) can be represented, in coordinates, by employing the Lagrange multipliers µ a , a = 1, ..., m, as follows: where we employ the local expression ω a = ω a i dq i .

Interconnection of Dirac Structures
There exist various types of the power conserving interconnections between distinct dynamical systems such as two rigid bodies connected by ball-socket joints, mechanical systems connected by transmission gears, transducers between electric circuits and mechanical systems such as D-C motors, fluid and rigid body interactions, etc. All the examples listed above may be represented by Dirac structures. In this section, we will show how distinct Dirac structures can be interconnected via an interconnection Dirac structure to yield another Dirac structure of a fully interconnected dynamical system.
The Direct Sum of Dirac Structures. Before going into details on the case of n distinct Dirac structures, we will begin with the case of n = 2 to introduce the direct sum of Dirac structures on manifolds. Let (M ) be the set of (almost) Dirac structures over a manifold M . 3 Let M 1 and M 2 be distinct smooth manifolds. To transition from systems on M 1 and M 2 to one on M = M 1 × M 2 , we shall define a direct sum associated to distinct Dirac structures.
Definition 4.1 (Direct Sum of Dirac Structures). Let D 1 ∈ (M 1 ) and D 2 ∈ (M 2 ) be given distinct Dirac structures on M 1 and M 2 respectively. Define the Dirac sum of D 1 and D 2 by . Thus, noting that m is arbitrary, one can prove the theorem by Lemma 2.1.
The Dirac Sum of Induced Dirac Structures. Let Q 1 and Q 2 be distinct configuration spaces. Let ∆ Q1 ⊂ T Q 1 and ∆ Q2 ⊂ T Q 2 be constraint distributions, and we can define the induced Dirac structures D 1 and D 2 , where we assume that ∆ Q1 = ∆ Q2 and ∆ Q1 ∩∆ Q2 = ∅.
Let Q = Q 1 × Q 2 be an extended configuration manifold and ∆ where T π Q : T T * Q → T Q is the tangent map of the cotangent bundle projection π Q : T * Q → Q, while the annihilator of ∆ T * Q is defined by, for each (q, p) ∈ T * Q, Let Ω i be the canonical symplectic structures on T * Q i and Ω i : 2. Now, we may decompose (w, α) as w = (w 1 , w 2 ) and α = T * (q,p) pr T * Q1 (α 1 ) + T * (q,p) pr T * Q2 (α 2 ) such that (w 1 , α 1 ) ∈ D 1 (q 1 , p 1 ) and (w 2 , α 2 ) ∈ D 2 (q 2 , p 2 ). Define the distributions ∆ T * Q1 = pr T T * Q1 (D 1 ) and ∆ T * Q2 = pr T T * Q2 (D 2 ). Then, It follows that An Interconnection Dirac Structure. In order to intertwine distinct Dirac structures, we will introduce a special Dirac structure called interconnecting Dirac structure. A simple case may be given by a distribution and its annihilator.
Suppose there exists a constraint distribution ∆ c ⊂ T Q due to the interconnection, such that where ω Q is a one-form on Q associated with the interconnection. The annihilator ∆ where T π Q : T T * Q → T Q is the tangent map of the cotangent bundle projection π Q : T * Q → Q, while the annihilator of ∆ int is given by, for each (q, p) ∈ T * Q, Then, the interconnection Dirac structure associated to the distribution ∆ int on T * Q can be defined by, for each (q, p) ∈ T * Q, Example: Two Particles Moving with Contact. Consider two masses moving with contact in a vector space V = R 2 , whose velocities are given by (v 1 , v 2 ) ∈ V . Since the two particles are in contact and their velocities are common, it follows This constraint is enforced through the associated constraint forces (F 1 , F 2 ) ∈ V * at the contact point, where F i is the force associated with the i-th particle. This implies Newton's third law and it follows Remark. We can consider a more general interconnection Dirac structure than stated in the above, such as those given by gyrators, transformers and so on, which are expressed by two-forms or Poisson structures. In this paper, we will mainly focus on the simple case of the interconnection Dirac structure D int induced from a given distribution ∆ c Strategy for the Interconnection of Distinct Dirac Structures. Before going into details, let us outline our strategy for interconnecting Dirac structures. Let D 1 and D 2 be two distinct Dirac structures on distinct manifolds M 1 and M 2 . We will interconnect D 1 and D 2 by introducing an interconnection Dirac structure D int on M = M 1 × M 2 , by which constraints due to the interconnection are imposed. To do this, we first make the direct sum of Dirac structures, namely D 1 ⊕ D 2 , which is a single Dirac structure on M . Then, by introducing the bowtie product of Dirac structures on M , we define an interconnected Dirac structure as (D 1 ⊕ D 2 ) D int . The bowtie product will be shown to make ( (M ), ) a commutative category with identity element T M ⊕{0}. Under the condition that the fiber The Bowtie Product. Let P = M × M and let ι :M → P be a submanifold given Courant [1990] that one may pull back the Dirac structure D a ⊕ D b ⊂ T P ⊕ T * P on P toM as is a smooth sub-bundle by either of the following conditions: Corollary 4.4. The subbundle ι * (D a ⊕ D b ) is a Dirac structure onM .
Let ς : M →M be the map m → (m, m) (this is a diffeomorphism between M andM ).
Definition 4.5. Let D a and D b be Dirac structures over M . Then, we define a tensor product called the bowtie product of D a and D b by or alternatively given by: Remark. In this paper, we focus on how to use the bowtie product to make a Dirac structure for an interconnected Lagrange-Dirac dynamical system. While the bowtie product was initially proposed in Yoshimura, Jacobs and Marsden [2010] for this purpose using formula (4.3), it was revealed that an equivalent tensor product (denoted by ) was also developed in the context of generalized complex geometry (see Gualtieri [2007] and Alekseev, Bursztyn and Meinrenken [2009]) using formula (4.4). 5 We will make explicit the equivalence of and (i.e. (4.3) and (4.4)) in the Appendix.
Remark. For standard Lagrange-Dirac mechanics, we usually employ M = T * Q, D a is the canonical Dirac structure on M , and D b = ∆ ⊕ ∆ • for some distribution ∆ ⊂ T M is an interconnection Dirac structure. Though we will not explore this in detail, one of the particularly interesting interconnection Dirac structures is the case in which D b can be developed by Lagrangians that are linear in velocities of the form L = α, u , where u ∈ T Q and α is a one-form on Q causing magnetic terms to appear.
In order to show interconnection of Dirac structures is associative, we will use a special restricted two-form Ω ∆ induced by a Dirac structure D on M with a distribution ∆ = This two-form will be employed in the proofs for theorems 4.9 and 4.11. (4.5) This two-form was initially introduced by Courant [1990] for linear Dirac structures (see also Dufour and Wade [2004]). We can easily generalize it to the case of general manifolds since Ω ∆ is defined fiber-wise, and hence we have that D(x) is a linear Dirac structures on each fiber T x M × T * x M for all x ∈ M . Here, we present a proof for completeness.
Proof. We desire to show that equation (4.5) defines a unique bi-linear anti-symmetric map Ω ∆ . First we prove linearity.
Thus, Ω ∆ is linear in the first argument. Second, in order to prove that Ω ∆ is anti-symmetric, take any Thus, we have shown anti-symmetry from which bi-linearity follows. 5 We appreciate Henrique Bursztyn for pointing out this fact in private communication.
Definition 4.8. Given a Dirac structure D ∈ (M ), we call the two-form Ω ∆ of Lemma 4.7 the Dirac two-form.
Recall from equation (2.1) that, for each x ∈ M , D(x) may be given by which may be also stated by where Ω (x) : T x M → T * x M is the skew-symmetric bundle map that is a natural extension of the skew-symmetric map Ω ∆ (x) : where D c is a Dirac structure with ∆ c and Ω c . Then, it follows that D a D b ⊂ D c . Equality follows from noting that both D a D b (x) and D c (x) are subspaces of T x M × T * x M with the same dimension.
If Ω b = 0, then it follows that D b = ∆ b ⊕ ∆ • b and also that D c = D a D b is induced by the constraint ∆ a ∩ ∆ b and the two-form Ω a .
Then the bowtie product is associative and commutative; namely we have and Proof. First we prove commutativity. Recall that any Dirac structure may be constructed by its associated constraint distribution ∆ = pr T M (D) and the Dirac two-form Ω ∆ . Let Ω a , Ω b ,and Ω c be the Dirac two-forms corresponding to D a , D b , and D c respectively. Then we find by Theorem 4.9 that D a D b is defined by the distribution ∆ ab = ∆ a ∩ ∆ b and the Dirac two-form Ω ∆ ab = (Ω ∆a + Ω ∆ b )| ∆ ab . By commutativity of + and ∩, we find the same distribution and the two-form for If Ω ∆ (ab)c and Ω ∆ a(bc) are respectively the Dirac two-forms for (D a D b ) D c and D a (D b D c ), we find Remark. We have shown that acts on pairs of Dirac structures with clean intersections to give a new Dirac structure and also that that it is an associative and commutative product. It is easy to verify from equation ( Interconnection of Two Distinct Induced Dirac Structures. Let D 1 and D 2 be Dirac structures on distinct manifolds Q 1 and Q 2 induced from constraint distributions ∆ Q1 ⊂ T Q 1 and ∆ Q2 ⊂ T Q 2 as before. Let ∆ c be a given distribution on Q = Q 1 × Q 2 due to the interconnection of D 1 and D 2 . We have mathematical ingredients for interconnecting the induced Dirac structures D 1 and D 2 through the interconnection Dirac structure D int on Q. Recall that an interconnection Dirac structure is given by It follows from Theorem 4.9 that D = (D 1 ⊕ D 2 ) D int may be constructed from the constraint ∆ = (∆ 1 × ∆ 2 ) ∩ ∆ int and Ω = Ω 1 ⊕ Ω 2 , where Ω 1 and Ω 2 are respectively the canonical two-forms on T * Q 1 and T * Q 2 .
Proposition 4.12. Assuming ∆ has constant rank, the interconnection of two distinct induced Dirac structures D 1 and D 2 through D int = ∆ int ⊕ ∆ • int , namely, This may be viewed as a Corollary to Theorem 4.9 and Proposition 4.1.
Interconnection of n Distinct Dirac Structures. Let us consider the interconnection of n distinct Dirac structures D 1 , D 2 , . . . , D n on distinct manifolds M 1 , M 2 , . . . , M n , in which n > 2. Recall that the Dirac sum ⊕ is associative as ( By choosing the appropriate interconnection Dirac structure D int ∈ (M 1 × · · · × M n ) and assuming that rank(pr T M (⊕D i ) ∩ pr T M (D int )) is constant, we can simply develop the interconnected Dirac structure for N distinct Dirac structures as The condition that rank (pr T M (⊕D i ) ∩ pr T M (D int )) is constant is equivalent to the assumption that rank(∆) is constant, which can be shown by resorting to local coordinate expressions.
A Link Between Composition and Interconnection of Dirac Structures. The notion of composition of Dirac structures was introduced in Cervera, van der Schaft, and Banos [2007] in the context of port-Hamiltonian systems. Let V 1 , V 2 and V s be vector spaces. Let D 1 be a linear Dirac structure on V 1 ⊕ V s and D 2 be a linear Dirac stucture on V s ⊕ V 2 . The composition of D 1 and D 2 is given by and V * s denote the dual space of V 1 , V 2 and V s . It was also shown that the set D 1 ||D 2 is itself a Dirac structure on V 1 × V 2 . In this section, we shall see the link between the composition and the interconnection of Dirac structures.
The composition developed in Cervera, van der Schaft, and Banos [2007] is constructed on vector-spaces. So, we also focus on the vector space case in this paper.
Next, we recall the push-forward map associated to a Dirac structure. Let U and W be vector spaces. Let ϕ : U → W be a linear map and let D U be a linear Dirac structure on U . The push-forward of D U to W by ϕ is the set It is worth noting that the push-forward of a Dirac structure ϕ * D U is itself a Dirac structure D W ; namely, we write it as ϕ * D U = D W .
We are now ready to provide the link between the bowtie product and the composition of Dirac structures.

Interconnected Lagrange-Dirac Dynamical Systems
Given distinct mechanical systems, one can develop dynamics of each mechanical system in the context of both the Lagrange-d'Alembert-Pontryagin variational structures and the induced Dirac structures over separate configuration manifolds. In this section, we will show how dynamics of an interconnected systems can be formulated in the context of the induced Dirac structures as well as in the context of the variational structures.
Variational Structures for Interconnected Mechanical Systems. Let us introduce an interconnection of the Lagrange-d'Alembert-Pontryagin principles for distinct mechanical systems.
Definition 5.1. Let Q i be distinct configuration manifolds corresponding to distinct mechanical systems i = 1, ..., n. Let E Li : T Q i ⊕ T * Q i → R be the generalized energies associated to given Lagrangians (possibly degenerate) L i on T Q i and F i : T Q i ⊕ T * Q i → T * Q i be external force fields. Let ∆ Qi be a distribution on Q i for i = 1, ..., n.
Keeping the endpoints of q(t) fixed, it follows that one can simply derive the Lagranged'Alembert-Pontryagin equations as follows: Proposition 5.3. The Lagrange-d'Alembert-Pontryagin principles in (5.1) for curves (q i (t), v i (t), p i (t)), t 1 ≤ t ≤ t 2 in T Q i ⊕ T * Q i together with the interconnection condition in (5.2) is equivalent with the Lagrange-d'Alembert-Pontryagin principle in (5.4).
(ii) Curves (q i (t), v i (t), p i (t)), t 1 ≤ t ≤ t 2 ; i = 1, ..., n satisfy the Lagrange-d'Alembert-Pontryagin principles , arbitrary δv i (t) and δp i (t), together withq i (t) ∈ ∆ Qi (q i (t)) and fixed endpoints of q i (t), such that the force vector F (q(t), v(t), p(t)) = F 1 (q 1 (t), v 1 (t), p 1 (t)) ⊕ · · · ⊕ F n (q n (t), v n (t), p n (t)) ∈ T * q(t) Q and the velocity vectoṙ q(t) = (q 1 (t), ...,q n (t)) ∈ T q(t) Q satisfẏ Interconnected Lagrange-Dirac Dynamical Systems. Next, consider the interconnection of distinct Lagrange-Dirac dynamical systems corresponding to the above variational structures. To do this, we first consider the interconnection of distinct Dirac structures D i on T * Q i , i = 1, ..., n, each of which is induced from given distinct distributions ∆ Qi ⊂ T Q i . Let (E Li , D i , X i , F i ) be n distinct Lagrange-Dirac systems, where E Li denote the generalized energies on T Q i ⊕T * Q i associated to given Lagrangians L i on T Q i , X i : T Q i ⊕T * Q i → T T * Q i are the partial vector fields, and the external force fields F i : As before, for an interconnected system, let Q = Q 1 × · · · × Q n and ∆ c be a given distribution on Q associated to the interconnection. Let ∆ = (∆ Q1 × · · · × ∆ Qn ) ∩ ∆ c ⊂ T Q. Then, define an interconnection Dirac structure by D int = ∆ int ⊕ ∆ • int as in (4.2), where ∆ int = (T π Q ) −1 (∆ c ). Set a generalized energy E L = E L1 + · · · + E Ln : T Q ⊕ T * Q → R associated to the Lagrangian L = L 1 + · · · + L n : T Q → R. Furthermore, define a partial vector field on T Q ⊕ T * Q by X = X 1 ⊕ · · · ⊕ X n : T Q ⊕ T * Q → T T * Q. Define also an external force field by F = F 1 ⊕ · · · ⊕ F n : T Q ⊕ T * Q → T * Q, which induces the horizontal one-formF =F 1 ⊕ · · · ⊕F n on T Q ⊕ T * Q such that F , w = F, T pr Q (w) for all w ∈ T (T Q ⊕ T * Q), where pr Q : T Q ⊕ T * Q → Q is the natural projection. Recall from (4.6) that the interconnection of distinct Dirac structures D 1 , ..., D n through D int is given by D = (D 1 ⊕ · · · ⊕ D n ) D int .
(5.6) It follows from the condition (5.5) that and also from condition (5.6) it follows thatq ∈ ∆ c (q), w = 0 and F ∈ ∆ • c (q). Furthermore, noting ∂L/∂q i = ∂L i /∂q i , it follows that the curves (q i (t), v i (t), p i (t)), t 1 ≤ t ≤ t 2 satisfy the conditions , v i (t)) ∈ P i . One can easily check the converse.
where (q i , p i = (q i , ∂L i /∂v i ) andF i is the horizontal one-forms on T Q i ⊕ T * Q i of force fields F i : T Q i ⊕ T * Q i → T * Q i , such that the condition of interconnection holds F (q, v, p)(t) ∈ ∆ • c (q(t)) andq(t) ∈ ∆ c (q(t)).

Examples
In this section we provide several examples of how interconnection can be carried out with the bowtie product. We have chosen rather simple examples, however the advantage of using the bowtie product is based on the procedure of tearing and interconnecting systems, which enables us to treat more complicated systems in a hierarchical structure. In future, we will seek more complex uses of the bowtie product.
(I) Example: Mass-Spring Mechanical Systems. Let us consider an illustrative example of a mass-spring system in the context of the interconnection of Dirac structures and associated Lagrange-Dirac systems. In this example, there exists no external force. Let m i and k i be the i-th mass and spring (i = 1, 2, 3).

Figure 6.1: A Mass-Spring System
Tearing and Interconnection. Inspired by the concept of tearing and interconnection originally developed by Kron [1963], the mechanical system can be torn apart into two subsystems as in Fig 6.2, each of which can be regarded as a modular unit of the whole system. Let us call each disconnected system a "primitive system" by following Kron [1963]. Note that the procedure of tearing inevitably yields interactive boundaries 6 associated to the two disconnected subsystems, where it induces the following condition: f 2 +f 2 = 0,ẋ 2 =ẋ 2 . (6.1) Without the above condition, there exists no energetic interaction between the disconnected systems. In other words, the original mechanical system can be reconstructed by intercon-Tearing Sub-system 1 Sub-system 2 The continuity conditions in (6.1) imply the continuity of power flow; namely, the power invariance holds as P 2 +P 2 = 0, where P 2 = f 2 , v 2 andP 2 = f 2 ,v 2 . Needless to say, equation (6.1) can be understood as the defining condition for an interconnection Dirac structure.
Primitive Lagrangian Systems. Let us consider how dynamics of the disconnected subsystems can be formulated in the context of Lagrange-Dirac dynamical systems.
The configuration space of the subsystem 1 may be given by Q 1 = R × R with local coordinates (x 1 , x 2 ), while the configuration space of the subsystem 2 is Q 2 = R × R with local coordinates (x 2 , x 3 ). We can naturally define the canonical Dirac structures D 1 ∈ (T * Q 1 ) and D 2 ∈ (T * Q 2 ) in this example. For Primitive System 1, the Lagrangian L 1 : T Q 1 → R is given by while the Lagrangian L 2 : T Q 2 → R for Primitive System 2 is given by Then, we can define the generalized energy Though the original system has no external force, each disconnected primitive system has an interconnection constraint force at the interactive boundary. When viewing each system separately, the constraint force acts as an external force. This is because tearing always yields constraint forces at the boundaries associated with the disconnected primitive systems, as shown in Fig. 6.2 The constraint force fields at the boundaries F c1 : . Similarly, let X 2 : T Q 2 ⊕ T * Q 2 → T T * Q 2 be the partial vector field, which is defined at each point (p 2 = 0, p 3 = m 3 v 3 ) ∈ P 2 = FL(T Q 2 ) as X 2 (x 2 , x 3 ,v 2 , v 3 ,p 2 , p 3 ) = (x 2 , x 3 ,p 2 , p 3 ,ẋ 2 ,ẋ 3 ,ṗ 2 ,ṗ 3 ), together with the consistency conditionṗ 2 = 0.
Primitive System 1: We can formulate dynamics of Primitive System 1 in the context of the Lagrange-Dirac dynamical system (E 1 , D 1 , X 1 , F 1 ) as which may be given in coordinates bẏ and with p 1 = m 1 v 1 and p 2 = m 2 v 2 .
Primitive System 2: Similarly, we can also formulate dynamics of Primitive System 2 in the context of the Lagrange-Dirac dynamical system (E 2 , D 2 , X 2 , F 2 ) as which may be given in coordinates bẏ together withp 2 = 0 and p 3 = m 3 v 3 as well asṗ 2 = 0.
Recall that each primitive system is physically independent of the other, which implies that there exists no energetic interaction between them. Next, let us see how the disconnected primitive systems can be interconnected through a Dirac structure.
Interconnection of Distinct Dirac Structures. Let Q = Q 1 × Q 2 = R × R × R × R be an extended configuration space with local coordinates x = (x 1 , x 2 ,x 2 , x 3 ). Recall that the Dirac sum of the Dirac structures is given by D 1 ⊕ D 2 on T * Q. Now, the constraint distribution due to the interconnection is given by where ω Q = dx 2 − dx 2 is a one-form on Q. On the other hand, the annihilator ∆ It follows from this codistribution that f 2 = −f 2 , f 1 = 0 and f 3 = 0. Hence, we obtain the conditions for the interconnection as in equation (6.1); namely, f 2 +f 2 = 0 and v 2 =v 2 .
Recall that the distribution ∆ int on T * Q is obtained by ∆ int = (T π Q ) −1 (∆ c ) ⊂ T T * Q, and hence the Dirac structure D int can be defined as in (4.2). Further, recall that the interconnected Dirac structure D on T * Q is given by Interconnected Lagrange-Dirac Systems. Now, let us see how two primitive Lagrange-Dirac systems, namely, (E L1 , D 1 , X 1 , F 1 ) and (E L2 , D 2 , X 2 , F 2 ) can be interconnected to be a Lagrange-Dirac dynamical system. Define the Lagrangian L : T Q → R for the interconnected system by L = L 1 + L 2 , and hence the generalized energy is given by where the consistency conditionṗ 2 = 0 holds.
Finally, the interconnected Lagrange-Dirac system is given by (E L , D, X), which satisfies, for each (x, v, p), and with (x, p) ∈ P = FL(∆).
Thus, we can obtain the dynamics of the interconnected Lagrange-Dirac system.

(II) Example: Electric Circuits
Consider the electric circuit depicted in Figure 6.3, where R denotes a resistor, L an inductor, and C a capacitor.
As in Figure 6.4, we decompose the circuit into the disconnected two circuits, each of which we shall call "Primitive Circuit 1" and "Primitive Circuit 2", in which S 1 and S 2 denote external ports. In order to reconstruct the original circuit in Figure 6.3, the external ports may be connected by equating currents across each. Primitive Circuit 1. The configuration manifold for circuit 1 is denoted by Q 1 = R 3 with local coordinates q 1 = (q R , q L , q S1 ), where q R , q L and q S1 are the charge fluxes through the resistor R, inductor L and port S 1 respectively. There exists the KCL constraint distribution ∆ Q1 ⊂ T Q 1 in Circuit 1, which is given by for each q 1 = (q R , q L , q S1 ) ∈ Q 1 , where v 1 = (v R , v L , v S1 ) denotes the current vector at each q 1 , while the KVL constraint is given by its annihilator ∆ • Q1 . Then, we can naturally define the induced Dirac structure D 1 on T * Q 1 from ∆ Q1 as before.
The Lagrangian for Circuit 1, namely, L 1 on T Q 1 is given by which is degenerate. Define the generalized energy E L1 on T Q 1 ⊕ T * Q 1 by E L1 (q 1 , v 1 , p 1 ) = p 1 , v 1 − L 1 (q 1 , v 1 ) on T Q 1 ⊕ T * Q 1 . Circuit 1 also has the external force field due to the resistor and the external port, F 1 : T Q 1 ⊕ T * Q 1 → R as So, we can set up the Lagrange-Dirac dynamical system (E L1 , D 1 , X 1 , F 1 ).
Primitive Circuit 2. The configuration manifold for Circuit 2 is Q 2 = R 2 with local coordinates q 2 = (q S2 , q C ), where q S2 is the charge flux through the port S 2 and q C is the charge stored in the capacitor. The KCL constraint distribution is given by, for each q 2 , and so the KVL space is given by its annihilator ∆ • 2 (q 2 ). This gives us the Dirac structure D 2 on T * Q 2 . Set the Lagrangian L 2 : T Q 2 → R for Circuit 2 as and set the generalized energy E L2 (q 2 , v 2 , p 2 ) = p 2 , v 2 −L 2 (q 2 , v 2 ) on T Q 2 ⊕T * Q 2 . Circuit 2 has the external force field due to the resistor and the external port S 2 , namely, F 2 : Then, we can formulate the Lagrange-Dirac dynamical system (E L2 , D 2 , X 2 , F 2 ).
The Interconnection Dirac Structure Set Q = Q 1 × Q 2 . The interconnection constraint is given by By the tangent lift ∆ int = T π −1 Q (∆ 12 ), we can define the interconnection Dirac structure as D int = ∆ int ⊕ ∆ • int , which is denoted, in local coordinates, by The Interconnected System. Now, we can define the interconnected Dirac structure by which may be viewed as the induced Dirac structure from the constraint space which is given, in coordinates (q 1 , q 2 ) = (q R , q L , q S1 , q S2 , q C ), by Set the Lagrangian for the interconnected system as L = L 1 + L 2 and the external force field F = F 1 ⊕ F 2 : T Q ⊕ T * Q → T * Q. Set also E L = E L1 + E L2 . Let X = X 1 ⊕ X 2 : T Q⊕T * Q → T T * Q be a partial vector field. The interconnected Lagrange-Dirac dynamical system is given by the quadruple (E L , D, X, F ), which satisfies where P = FL(∆).
(III) Example: A Rolling Ball on Rotating Tables Consider the mechanical system depicted in Fig. 6.5, where there are rotating tables via a gear and a ball is rolling on one of the tables without slipping. We assume that there is no loss of energy and non-slipping in the gear and also that the external torque is constant. Let I i be moments of inertia for the table i = 1, 2. Let us decompose the system into distinct three systems; (1) a rotating (small) table, (2) a rotating (large) table, and (3) a rolling ball on a table.
Primitive System 1. For Primitive System 1 the configuration manifold is the circle, Q 1 = S 1 . The Lagrangian is the rotational energy of the system. L 1 (s 1 ,ṡ 1 ) = I 1 2ṡ 2 1 , and set the generalized energy E L1 on T Q 1 ⊕T * Q 1 . We employ the canonical Dirac structure on T * Q 1 as There is a constant torque acting on Primitive System 1, given by a map F 1 : T Q 1 ⊕T * Q 1 → T * Q 1 defined as F (s 1 , v s1 , p s1 ) = τ ds 1 , where τ is constant in R. Primitive System 2. For Primitive System 2, the configuration manifold is also the circle, Q 2 = S 1 and the Lagrangian is again the rotational energy L 2 (s 2 ,ṡ 2 ) = I 2 2ṡ 2 2 and set the generalized energy E L2 on T Q 2 ⊕ T * Q 2 . Again, we have the canonical Dirac structure D 2 = {(ṡ 2 ,ṗ s2 , α s2 , w s2 ) |ṡ 2 = w s2 ,ṗ s2 + α s2 = 0}.
Primitive System 3. Primitive System 3 is a rolling sphere of uniform density and radius 1.The sphere moves in space by changing its position and orientation relative to a reference configuration. The configuration manifold is given by the special Euclidean group Q 3 = SE (3), which we parametrize as (R, u) where R ∈ SO(3), u ∈ R 3 . Following Marsden and Ratiu [1999], let β be the set of points of the sphere in the reference configuration. For configuration (R, u) ∈ Q 3 , a point x ∈ β is transformed into R 3 by the action (R, u) · x = (R · x) + u. The Lagrangian is given by the kinetic energy as where Ṙ x +u 2 = x TṘTṘ x + 2x TṘTu +u 2 . We use body coordinates such that the center of the sphere in the reference configuration is at the origin so that β xdx = 0. Substituting these relations, the above Lagrangian is to be Setting m 3 = β ρdx = 4 3 πρ and noting that β x i x j dx = 0 when i = j, one finally obtains L 3 = m 3 2 tr(Ṙ TṘ ) +u 2 and with the generalized energy E L3 on T Q 3 ⊕ T * Q 3 . Since the motion along the z-direction is constrained so that the ball does not leave the plane of table 2, we get the holonomic constraint This yields the induced Dirac structure D 3 = {(δR, δu, δp R , δp u , α R , α u , w R , w u ) ∈ T T * Q 3 ⊕ T * T * Q 3 | δu 3 = 0, δu = w u , δR = w R , δp R + α R = 0, δp u + α u = λdz for some λ ∈ R}.
Interconnection Constraints. Let Q = Q 1 × Q 2 × Q 3 . In order to interconnection the three primitive systems, we need to impose the constraints due to the non slip conditions. First, consider the mapping Φ(v s ) = s −1 v s for v s ∈ T s S 1 . The map Φ sends the tangent fiber T s S 1 to the Lie-algebra s = T e S 1 ≈ R, where e is the identity element on S 1 viewed as a group. From this point on, we do not explicitly invoke Φ but use it to interpret T S 1 as S 1 × R. The interconnection constraint between System 1 and System 2 is given by ∆ c,1 = {(ṡ 1 ,ṡ 2 ,Ṙ,u) ∈ T Q |ṡ 1 +ṡ 2 = 0} and with its annihilator ∆ • c,1 = span(ω 1 ) where ω 1 = ds 1 − ds 2 . This constraint ensures that the gears rotate (without slipping) at the same speed in opposite directions.
Next, we consider the interconnection constraint between Systems 2 and 3. Note that the velocity of a point located at the bottom of the sphere is given by Note also that a point rotating on the gear of system 2 with the axle taken to be the origin has velocity So the non slip condition between System 2 and 3 is given by ∆ c,2 = {(ṡ 1 ,ṡ 2 ,Ṙ,u) ∈ T Q | i · (−ṘR T · k +u) = −ṡ 2 · u 2 , j · (−ṘR T · k +ẇ) =ṡ 2 · u 1 }, where i, j, k are the basis on R 3 . Setting the interconnection constraint distribution ∆ c = ∆ c,1 ∩ ∆ c,2 , let ∆ int := (T π Q ) −1 (∆ c ) and with its annihilator ∆ • int . Then, the interconnection Dirac structure is given by The Interconnected Lagrange-Dirac System. The Dirac structure for the interconnected system is given by D = (D 1 ⊕ D 2 ⊕ D 3 ) D int and we use the Lagrangian L = L 1 + L 2 + L 3 . The dynamics of the system are implied by the constraint: where X : T T * Q ⊕ T * T * Q → T T * Q is a partial vector field, and E L : T Q ⊕ T * Q → R is the generalized energy of L. This gives us the interconnected Lagrangian system (E L , D, X). By Theorem 4.9, these dynamics use the constraint distribution ∆ = (T Q 1 ⊕ T Q 2 ⊕ ∆ 3 ) ∩ ∆ c , and the canonical two-form on T * Q restricted to ∆ as the Dirac two-form. Noting the annihilator is given by ∆ • = ∆ • 3 + ∆ • c,1 + ∆ • c,2 , we finally obtain the dynamics aṡ q = v ∈ ∆,ṗ − ∂L ∂q ∈ ∆ • , p = ∂L ∂v .

Conclusions and Future Works
In this paper, we provided a notion of interconnection for Dirac structures and their associated Lagrange-Dirac dynamical systems. The principal idea lies in the fact that the interconnection of distinct Dirac structures itself is a Dirac structure. We also clarified the interconnection of the associated Lagrange-Dirac dynamical systems with the variational structure by the Lagrange-d'Alembert-Pontryagin principle. To do this, we developed the bowtie product, , which serves us well in the case of interconnection by constraints. We then showed how the bowtie product can effectively carry the Lagrange-d'Alembert-Pontryagin principle to the interconnected system. Lastly, we demonstrated our theory with some illustrative examples of a mass-spring mechanical system, an interconnection in circuits, and a non-holonomic mechanical system.
• The use of more general interconnection Dirac structures with nontrivial Dirac twoforms such as the one given with symplectic gyrators (in this paper, we restrict D int to the form ∆ int ⊕ ∆ • int ). As to examples of such general interconnections in physical systems, see for instance Wyatt and Chua [1977]; Yoshimura [1995] and Maschke, van der Schaft and Breedveld [1995].
• The integrability condition for the bowtie product of Dirac structures. As to the integrability condition for Dirac structures, see Dorfman [1993] and Dalsmo and van der Schaft [1998]. The link with symplectic categories (see Weinstein [2009]).
• Discrete version of interconnected Lagrange-Dirac mechanics; namely, discretizing the Hamilton-Pontryagin principle one arrives at a discrete mechanics version of Dirac structures (see Bou-Rabee and  and Leok and Ohsawa [2010]).
A Appendix: The Bowtie Product and the Dirac Tensor Product We will show the equivalence between the bowtie product and the Dirac tensor product developed by Gualtieri [2007], where the definition of the tensor product is given as follows.
Definition A.1. Let D a , D b ∈ (P ). Let d : P → P × P be the diagonal embedding in P × P . The tensor product of D a and D b is defined as where K = {(0, 0)}⊕{(−ξ, ξ) | ξ ∈ T * P } ⊂ T (P × P )⊕T * (P × P ) and K ⊥ is the orthogonal complement in T (P × P ) ⊕ T * (P × P ).
Proposition A.2. The bowtie product and the tensor product of D a and D b are equivalent: where the notation ≡ implies the fact that these structures are isomorphic.
Proof. First note that K ⊥ = {(v, v)} ⊕ T * (P × P ) so that K + K ⊥ = {(v, v)} ⊕ T * (P × P ). Then we see that where we chose the element (v, v, α, 0) as the representative for the equivalence class [(v, v, α, α ) Splitting the expression D a ⊕ D b into D a and D b , we obtain we see that the assignment φ : (v, v, α, 0) → (v, α) is a bijection and also that this is just the definition of D a D b .