In this paper, we develop variational integrators for the nonequilibrium thermodynamics of simple closed systems. These integrators are obtained by a discretization of the Lagrangian variational formulation of nonequilibrium thermodynamics developed in (Gay-Balmaz and Yoshimura 2017a J. Geom. Phys. part I 111 169-93
Gay-Balmaz and Yoshimura 2017b J. Geom. Phys. part II 111 194-212) and thus extend the variational integrators of Lagrangian mechanics, to include irreversible processes. In the continuous setting, we derive the structure preserving property of the flow of such systems. This property is an extension of the symplectic property of the flow of the Euler-Lagrange equations. In the discrete setting, we show that the discrete flow solution of our numerical scheme verifies a discrete version of this property. We also present the regularity conditions which ensure the existence of the discrete flow. We finally illustrate our discrete variational schemes with the implementation of an example of a simple and closed system.

We propose a variational formulation for the nonequilibrium thermodynamics of discrete open systems, i.e., discrete systems which can exchange mass and heat with the exterior. Our approach is based on a general variational formulation for systems with time-dependent nonlinear nonholonomic constraints and time-dependent Lagrangian. For discrete open systems, the time-dependent nonlinear constraint is associated with the rate of internal entropy production of the system. We show that this constraint on the solution curve systematically yields a constraint on the variations to be used in the action functional. The proposed variational formulation is intrinsic and provides the same structure for a wide class of discrete open systems. We illustrate our theory by presenting examples of open systems experiencing mechanical interactions, as well as internal diffusion, internal heat transfer, and their cross-effects. Our approach yields a systematic way to derive the complete evolution equations for the open systems, including the expression of the internal entropy production of the system, independently on its complexity. It might be especially useful for the study of the nonequilibrium thermodynamics of biophysical systems.

Dirac structures are geometric objects that generalize both Poisson structures and presymplectic structures on manifolds. They naturally appear in the formulation of constrained mechanical systems. In this paper, we show that the evolution equations for nonequilibrium thermodynamics admit an intrinsic formulation in terms of Dirac structures, both on the Lagrangian and the Hamiltonian settings. In the absence of irreversible processes, these Dirac structures reduce to canonical Dirac structures associated with canonical symplectic forms on phase spaces. Our geometric formulation of nonequilibrium thermodynamic thus consistently extends the geometric formulation of mechanics, to which it reduces in the absence of irreversible processes. The Dirac structures are associated with the variational formulation of nonequilibrium thermodynamics developed in the work of Gay-Balmaz and Yoshimura, J. Geom. Phys. 111, 169-193 (2017a) and are induced from a nonlinear nonholonomic constraint given by the expression of the entropy production of the system.

We study trajectory design for a spacecraft escaping from Mars in the context of spatial restricted three-body problem (SR3BP) by using tube dynamics. First, we compute the halo and quasi-halo orbits around L1of Sun-Mars system together with the associated invariant manifolds. Then we investigate whether the spacecraft with a certain energy is inside of the tube to connect the manifold with the Phobos orbit. We also perform grid search with the SR3BP with the same energy as the invariant manifold. Comparing numerical results between grid search and the tube dynamics, we show how to determine the propriety for the escape trajectory by using the tube dynamics. Consequently we clarify the way to judge whether the spacecrafts can escape or not in the spatial problem.

In this paper, we show a low energy Earth-Moon transfer in the context of the Sun-Earth-Moon spacecraft 4-body system. We consider the 4-body system as the coupled system of the Sun-Earth-spacecraft 3-body system perturbed by the Moon (which we call the Moon-perturbed system) and the Earth-Moon-spacecraft 3-body system perturbed by the Sun (which we call the Sun-perturbed system). In both perturbed systems, analogs of the stable and unstable manifolds are computed numerically by using the notion of Lagrangian coherent structures, wherein the stable and unstable manifolds play the role of separating orbits into transit and non-transit orbits. We obtain a family of non-transit orbits departing from a low Earth orbit in the Moon-perturbed system, and a family of transit orbits arriving into a low lunar orbit in the Sun-perturbed system. Finally, we show that we can construct a low energy transfer from the Earth to the Moon by choosing appropriate trajectories from both families and patching these trajectories with a maneuver. (C) 2017 COSPAR. Published by Elsevier Ltd. All rights reserved.

Part I of this paper introduced a Lagrangian variational formulation for nonequilibrium thermodynamics of discrete systems. This variational formulation extends Hamilton's principle to allow the inclusion of irreversible processes in the dynamics. The irreversibility is encoded into a nonlinear nonholonomic constraint given by the expression of entropy production associated to all the irreversible processes involved.
In Part II, we develop this formulation for the case of continuum systems by extending the setting of Part I to infinite dimensional nonholonomic Lagrangian systems. The variational formulation is naturally expressed in the material representation, while its spatial version is obtained via a nonholonomic Lagrangian reduction by symmetry. The theory is illustrated with the examples of a viscous heat conducting fluid and its multicomponent extension including chemical reactions and mass transfer. (C) 2016 Elsevier B.V. All rights reserved.

In this paper, we present a Lagrangian variational formulation for nonequilibrium thermodynamics. This formulation is an extension of Hamilton's principle of classical mechanics that allows the inclusion of irreversible phenomena. The irreversibility is encoded into a nonlinear phenomenological constraint given by the expression of the entropy production associated to all the irreversible processes involved. From a mathematical point of view, our variational formulation may be regarded as a generalization to nonequilibrium thermodynamics of the Lagrange-d'Alembert principle used in nonlinear nonholonomic mechanics, where the conventional Lagrange-d'Alembert principle cannot be applied since the nonlinear phenomenological constraint and its associated variational constraint must be treated separately. In our approach, to deal with the nonlinear nonholonomic constraint, we introduce a variable called the thermodynamic displacement associated to each irreversible process. This allows us to systematically define the corresponding variational constraint. In Part I, our variational theory is illustrated with various examples of discrete systems such as mechanical systems with friction, matter transfer, electric circuits, chemical reactions, and diffusion across membranes. In Part II of the present paper, we will extend our variational formulation of discrete systems to the case of continuum systems. (C) 2016 Elsevier B.V. All rights reserved.

In this paper, we make a generalization of Routh's reduction method for Lagrangian systems with symmetry to the case where not any regularity condition is imposed on the Lagrangian. First, we show how implicit Lagrange-Routh equations can be obtained from the Hamilton-Pontryagin principle, by making use of an anholonomic frame, and how these equations can be reduced. To do this, we keep the momentum constraint implicit throughout and we make use of a Routhian function defined on a certain submanifold of the Pontryagin bundle. Then, we show how the reduced implicit Lagrange-Routh equations can be described in the context of dynamical systems associated to Dirac structures, in which we fully utilize a symmetry reduction procedure for implicit Hamiltonian systems with symmetry. (C) 2016 Elsevier B.V. All rights reserved.

In this paper, we investigate the low-energy escape trajectory design for a mission called Martian Moons eXplorer to achieve the world's first sample return from Martian moon. The hybrid usage of chemical and electric propulsion with combination of the three-body and two-body problems has come into consideration in order to seek a fast low-energy escape from Mars. We first study the needs of pre-departure sequence. Then, we determine the transition point from a low-energy three-body phase to a low-thrust two-body phase, in which the tube dynamics is employed for the low-energy three-body phase. We finally develop charts to reveal the relation between the velocity in Mars Escape Injection maneuver and the required time of flight.

A low energy transfer from the Earth to the Moon is proposed in the context of the 4-body Problem. We propose a new model by regarding the Sun-Earth-Moon-Spacecraft (S/C) 4-body system as the coupled system of the Sun-perturbed 3-body system and the Moon-perturbed 3-body system. In particular, we clarify the tube structures of invariant manifolds of the 4-body Problem by investigating the Lagrangian coherent structures of such a coupled 3-body system with perturbations. Lastly, we construct a low-energy transfer trajectory from the Earth to the Moon by patching two trajectories obtained from the perturbed systems at a Poincare section. We develop an optimal trajectory by minimizing the Delta-v at the Poincare section.

In this paper, we investigate the low-energy escape trajectory design for a mission called Martian Moons eXplorer to achieve the world's first sample return from Martian moon. The hybrid usage of chemical and electric propulsion with combination of the three-body and two-body problems has come into consideration in order to seek a fast low-energy escape from Mars. We first study the needs of pre-departure sequence. Then, we determine the transition point from a low-energy three-body phase to a low-thrust two-body phase, in which the tube dynamics is employed for the low-energy three-body phase. We filially develop charts to reveal the relation between the velocity in Mars Escape Injection maneuver and the required time of flight.

In this paper, we explore dynamics of the nonholonomic system called vakonomic mechanics in the context of Lagrange-Dirac dynamical systems using a Dirac structure and its associated Hamilton-Pontryagin variational principle. We first show the link between vakonomic mechanics and nonholonomic mechanics from the viewpoints of Dirac structures as well as Lagrangian submanifolds. Namely, we clarify that Lagrangian submanifold theory cannot represent nonholonomic mechanics properly, but vakonomic mechanics instead. Second, in order to represent vakonomic mechanics, we employ the space TQ x V*, where a vakonomic Lagrangian is defined from a given Lagrangian (possibly degenerate) subject to nonholonomic constraints. Then, we show how implicit vakonomic Euler-Lagrange equations can be formulated by the Hamilton-Pontryagin variational principle for the vakonomic Lagrangian on the extended Pontryagin bundle (TQ circle plus T*Q) x V*. Associated with this variational principle, we establish a Dirac structure on (TQ circle plus T*Q) x V* in order to define an intrinsic vakonomic Lagrange-Dirac system. Furthermore, we also establish another construction for the vakonomic Lagrange-Dirac system using a Dirac structure on T*Q x V*, where we introduce a vakonomic Dirac differential. Finally, we illustrate our theory of vakonomic Lagrange-Dirac systems by some examples such as the vakonomic skate and the vertical rolling coin. (C) 2015 Published by Elsevier B.V.

This paper develops the theory of Dirac reduction by symmetry for nonholonomic systems on Lie groups with broken symmetry. The reduction is carried out for the Dirac structures, as well as for the associated Lagrange-Dirac and Hamilton-Dirac dynamical systems. This reduction procedure is accompanied by reduction of the associated variational structures on both Lagrangian and Hamiltonian sides. The reduced dynamical systems obtained are called the implicit Euler-Poincare-Suslov equations with advected parameters and the implicit Euler-Poisson-Suslov equations with advected parameters. The theory is illustrated with the help of finite and infinite dimensional examples. It is shown that equations of motion for second order Rivlin-Ericksen fluids can be formulated as an infinite dimensional nonholonomic system in the framework of the present paper. (C) 2014 Elsevier Inc. All rights reserved.

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 procedure. In other words, we must understand interconnection of subsystems. Such an understanding has been already shown in the context of Hamiltonian systems on vector spaces via the port-Hamiltonian systems program, in which an interconnection may be 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 program of the port-Hamiltonian systems on vector spaces to the case of Lagrangian systems on manifolds and also 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 show some examples: a mass-spring mechanical systems, an electric circuit, and a nonholonomic mechanical system.

For the design of spacecraft transfer orbits in the Earth-Moon system, we need to model the dynamics in the context of the Planar Elliptic Restricted Three-Body Problem (PER3BP) since it has a non-negligible eccentricity. In this paper, we investigate invariant manifolds of the PER3BP by analyzing Lagrangian Coherent Structures (LCS). In particular, we show that the transfer orbits from the exterior realm to Moon can be developed by employing the geometric characteristics of the LCS obtained by long time numerical integrations.

For the sake of spacecraft mission design, it is indispensable to develop a low energy transfer of spacecrafts using very little fuel for interplanetary transport network. The Planar Circular Restricted Three-Body Problem (PCR3BP) has been a fundamental tool for the analysis of such a space mission design. In this paper, we explore stable and unstable invariant manifolds associated with the collinear Lagrange points L1, L2 of the PCR3BP, in which geometrical structures of the invariant manifolds are clarified on a Poincaré section. Further, we compute the Finite Time Lyapunov Exponent fields (FTLE fields) to obtain Lagrangian Coherent Structures (LCS) as the ridges of the FTLE fields. In particular, we compare the LCS with the invariant manifolds on the Poincare section from the viewpoint of the numerical integration times.

In this paper, we study symmetry reduction for a binary asteroid system modeled by a rigid body and a particle. In particular, we demonstrate how translational and rotational symmetry reduction appeared in the binary asteroid system can be carried out in the context of Dirac reduction by stages and with the associated reduction of implicit Hamiltonian systems. Then we investigate stability of relative equilibria of the asteroid pair and show stability regions by using the energy-momentum method. Lastly, we illustrate some numerical simulations for stable and unstable orbits near from relative equilibria of the Collinear and T configurations.

The analysis of the Planer Restricted Three-Body Problem has been quite essential for the sake of space mission design. There exists unstable periodic orbits called Lyapunov orbits, near Lagrange points together with the associated invariant manifolds called "tube". Making use of the tube is crucial to construct a robust and low energy trajectory. In order to analyze such a tube, we employ the monodromy matrix approach. However, it may induce some numerical inaccuracy. In this paper, we will show how the Finite Time Lyapunov Exponent can capture the tube to clarify the structure of the invariant manifolds in the PRTBP.

In this paper, we propose an efficient numerical scheme to compute sparse matrix inversions for Implicit Differential Algebraic Equations of large-scale nonlinear mechanical systems. We first formulate mechanical systems with constraints by Dirac structures and associated Lagrangian systems. Second, we show how to allocate &lt
i&gt
input-output relations&lt
/i&gt
to the variables in kinematical and dynamical relations appearing in DAEs by introducing an oriented bipartite graph. Then, we also show that the matrix inversion of Jacobian matrix associated to the kinematical and dynamical relations can be carried out by using the input-output relations and we explain solvability of the sparse Jacobian matrix inversion by using the bipartite graph. Finally, we propose an efficient symbolic generation algorithm to compute the sparse matrix inversion of the Jacobian matrix, and we demonstrate the validity in numerical efficiency by an example of the stanford manipulator. © 2013 Author(s).

The paper presents a modeling method of flexible multibody dynamics in the context of implicit Lagrangian systems. The main idea is based on a Diakoptical method originally developed by Kron, which reticulates required kinematical and dynamical relations into separate ones and then incorporate them into an interconnection of implicit Lagrangian systems. By using the idea, we first show how a flexible beam attached to a rigid base undergoing a large overall motion can be modeled as an interconnected implicit Lagrangian system, in which geometrically nonlinear couplings between flexible deformations and large overall motions are incorporated into a nonlinear finite element model. Lastly, some numerical results are shown in comparison with the model developed by Simo and Vu-Quoc.

We introduce a variational principle for field theories, referred to as the Hamilton-Pontryagin principle, and we show that the resulting field equations are the Euler-Lagrange equations in implicit form. Second, we introduce multi-Dirac structures as a graded analog of standard Dirac structures, and we show that the graph of a multisymplectic form determines a multi-Dirac structure. We then discuss the role of multi-Dirac structures in field theory by showing that the implicit Euler-Lagrange equations for fields obtained from the Hamilton-Pontryagin principle can be described intrinsically using multi-Dirac structures. Finally, we show a number of illustrative examples, including time-dependent mechanics, nonlinear scalar fields, Maxwell's equations, and elastostatics. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4731481]

We show the variational formulation of the Rayleigh-Plesset equation for single bubble dynamics in static water by the Hamilton-Pontryagin principle. Then, we illustrate the bifurcation phenomena associated to an amplitude of the bubble radius response as to the amplitude of the input pressure under forced oscillating pressure at constant temperature. Lastly, we show that there exists a stable state of the amplitude of bubble radius under the forced pressure with higher frequencies.

The exploration of cavitation bubbles has been one of the most important challenging research problems in fluid dynamics. Much effort has been done to elucidate how bubbles may rebound and shrink from the viewpoint of macroscopic level. However, the bubble nucleation and collapse have not been fully clarified since they must be essentially microscopic phenomena. Therefore, it is indispensable to study bubble dynamics from the microscopic viewpoint of molecular dynamics. In this paper, we will show the MD simulation of bubble dynamics using Lennard-Jones particles and soft-core particles as non-condensable gas, and then we will illustrate generation of a single bubble by repulsive force. We will also show how the single bubble can shrink and collapse for the cases with and without non-condensable gas in the bubble.

In a companion paper, we introduced a notion of multi-Dirac structures, a graded version of Dirac structures, and we discussed their relevance for classical field theories. In the current paper we focus on the geometry of multi-Dirac structures. After recalling the basic definitions, we introduce a graded multiplication and a multi-Courant bracket on the space of sections of a multi-Dirac structure, so that the space of sections has the structure of a Gerstenhaber algebra. We then show that the graph of a k-form on a manifold gives rise to a multi-Dirac structure and also that this multi-Dirac structure is integrable if and only if the corresponding form is closed. Finally, we show that the multi-Courant bracket endows a subset of the ring of differential forms with a graded Poisson bracket, and we relate this bracket to some of the multisymplectic brackets found in the literature. (C) 2011 Elsevier B.V. All rights reserved.

This paper develops a trajectory planning for a space robot, which enables us to simultaneously control its base attitude as well as the end effector trajectory. First, it is shown how the space robot dynamics can be formulated in the context of regular Lagrangian systems with holonomic constraints. Second, geometry of the space robot motion is explored; namely, it is shown how geometric phases corresponding to deviations of the base attitude are yielded in conjunction with the end effector motion. In our trajectory planning, it is demonstrated how the base attitude of the space robot can be controlled by the end effector in iteratively drawing complementary circles to reduce the geometric phase. Finally, we demonstrate the validity of our approach with numerical simulations.

There is much known on the port-Hamiltonian theory of interconnection of Dirac structures through shared variables. This interconnection is known as Composition of Dirac structures. In this paper, we will show an alternative interconnection of Dirac structures called Bowtie interconnection in the context of Lagrange-Dirac dynamical systems. In particular, we try to illustrate the following two things: Firstly, how composition of Dirac structures may be used in the Lagrangian theory of LC-circuits. Secondly, how composition of Dirac structures may be linked with bowtie interconnection.

We develop the Hamilton-Pontryagin principle for Lagrangians with advective parameters, which yields an implicit analogue of Euler-Poincare equations with advective parameters. Then, we derive the reduced Hamilton-Pontryagin principle and illustrate it with the example of incompressible ideal fluids, where the configuration space is given by the group of (volume preserving) diffeomorphisms. Incorporating pressure and momentum densities as Lagrange multipliers into the Hamilton-Pontryagin principle, we finally show that the dynamics of incompressible ideal fluids can be effectively formulated in the context of implicit Euler-Poincare equations.

In this paper, we study nonlinear dynamics of a small air bubble in water under forced oscillating pressure at a constant temperature. First, we briefly describe the mathematical model of Rayleigh-Plesset equations and then we consider the bifurcation phenomena appeared in bubble dynamics. We investigate how a small bubble may grow and collapse; namely, it is shown how it may cause chaotic phenomena by repeating period-doubling bifurcations as an amplitude of input pressure increases. Furthermore, we also examine their transitions to chaotic states associated to different frequencies. Finally, we show that there exists some stable state for the small bubble under forcing the pressure with very high frequencies.

Much concern has been focused upon the development of fish robots and underwater vehicles from the viewpoint of ocean probe and environmental study. However, in order to understand the propulsion mechanism of fish swimming, it is inevitable to fully utilize geometric notions such as Lie groups and principal bundles, which have not been familiar in engineering. In this paper, we will show a mechanical model for fish swimming, which consists of three rigid bodies in a two-dimensional perfect fluid and then will show how the fish can swim by making use of the action of special Euclidean groups and with Lagrangian reduction theory. Finally, we will illustrate some numerical simulations on fish propulsion.

Since the Galileo probe discovered the binary asteroid, i.e., "Ida" and "Dactyl" in 1993, much attention has been paid on the space mission design and analysis of the spacecraft approaching to binary astroids. In this study, we will explore dynamics of a binary asteroid to derive relative equilibria in the context of the cotangent bundle reduction. Then, we will show the stability regions of the relative equilibria by using the energy-momentum method.

Cavitation has been known as unsteady phenomena that may cause damage for hydraulic machinery such as erosion, while it is pointed out that the cavitation of micro bubbles may provide medical applications recently. Such a cavitation inception may be induced by the growth and collapse of small bubbles. The goal of this paper is to analyze single bubble dynamics in static water represented by the Rayleigh-Plesset equation. To do this, we will fist study how a single small air bubble may be behaved under the forced oscillating pressure at constant temperature. Then, we will illustrate the bifurcation phenomena associated to an amplitude of the bubble radius response as to the amplitude of the input pressure. Lastly, we will show that there exists a stable state of the amplitude of bubble radius under the forced pressure with higher frequencies.

We first study interconnection and Brayton-Moser equations for circuits, which are well-known as fundamental equations in reciprocal nonlinear circuits. Then, we focus on the FitzHugh-Nagumo model, which is a mathematical model of neural membrane and we show there exists a mixed-potential function. Finally, we demonstrate that the membrane potential dynamics can be effectively analyzed by using the Brayton-Moser equations.

This paper develops a discrete Hamiltonian system with holonomic constraints with Geometric Constraint Stabilization. It is first shown that constrained mechanical systems with nonconservative external forces can be formulated by using canonical symplectic structures in the context of Hamiltonian systems. Second, it is shown that discrete holonomic Hamiltonian systems can be developed via the discretization based on the Backward Differentiation Formula and also that geometric constraint stabilization can be incorporated into the discrete Hamiltonian systems. It is demonstrated that the proposed method enables one to stabilize constraint violations effectively in comparison with conventional methods such as Baumgarte Stabilization and Gear-Gupta-Leimkuhler Stabilization, together with an illustrative example of linkage mechanisms.

Previous constructions of Lagrangian mechanics for electric circuits have been found to diverge significantly from the standard Lagrangian mechanics of mechanical systems [1], [2]. The Lagrangian for a generic L-C circuit is degenerate, which prevents one from invoking the standard Euler-Lagrange equations [6]. Additionally, an interconnection of disconnected circuits places a Kirchhoff current constraint on the simultaneous dynamics of the two systems. This motivates us to develop the concept of interconnection for degenerate Lagrangian systems. Lagrange-Dirac Dynamical Systems (LDDS) have proven to be especially well suited for exactly such difficulties [8]. We provide a brief overview of LDDS following [6]. We then propose a means of interconnecting primitive subsystems by imposing an additional constraint. Finally, we demonstrate the interconnection theory by an example of L-C circuits.

We develop discrete Lagrangian systems with holonomic constraints by employing the discrete Lagrange-d'Alembert principle, which was originally proposed by [5, 6]. Especially, we focus on the class of discrete holonomic Lagrangian systems in the context of the index 2 model, i.e., discrete Lagrange-d'Alembert equations with velocity-level constraints, while the lower index formulation may induce constraint violations called drift-off phenomena. So we incorporate geometric constraint stabilization proposed by [7, 8] into the discrete holonomic Lagrangian systems in order to avoid the constraint violations. We demonstrate numerical validity in making use of discrete Lagrange-d'Alembert equations for the index 2 model of holonomic mechanical systems with an illustrative example of linkage mechanisms.

We consider the concept of Stokes-Dirac structures in boundary control theory proposed by van der Schaft and Maschke. We introduce Poisson reduction in this context and show how Stokes-Dirac structures can be derived through symmetry reduction from a canonical Dirac structure on the unreduced phase space. In this way, we recover not only the standard structure matrix of Stokes-Dirac structures, but also the typical non-canonical advection terms in (for instance) the Euler equation.

The authors' recent paper in Reports in Mathematical Physics develops Dirac reduction for cotangent bundles of Lie groups, which is called Lie-Dirac reduction. This procedure simultaneously includes Lagrangian, Hamiltonian, and a variational view of reduction. The goal of the present paper is to generalize Lie-Dirac reduction to the case of a general configuration manifold; we refer to this as Dirac cotangent bundle reduction. This reduction procedure encompasses, in particular, a reduction theory for Hamiltonian as well as implicit Lagrangian systems, including the case of degenerate Lagrangians.
First of all, we establish a reduction theory starting with the Hamilton-Pontryagin variational principle, which enables one to formulate an implicit analogue of the Lagrange-Poincare equations. To do this, we assume that a Lie group acts freely and properly on a configuration manifold, in which case there is an associated principal bundle and we choose a principal connection. Then, we develop a reduction theory for the canonical Dirac structure on the cotangent bundle to induce a gauged Dirac structure. Second, it is shown that by making use of the gauged Dirac structure, one obtains a reduction procedure for standard implicit Lagrangian systems, which is called Lagrange-Poincare-Dirac reduction. This procedure naturally induces the horizontal and vertical implicit Lagrange-Poincare equations, which are consistent with those derived from the reduced Hamilton-Pontryagin principle. Further, we develop the case in which a Hamiltonian is given (perhaps, but not necessarily, coming from a regular Lagrangian); namely, Hamilton-Poincare-Dirac reduction for the horizontal and vertical Hamilton-Poincare equations. We illustrate the reduction procedures by an example of a satellite with a rotor.
The present work is done in a way that is consistent with, and may be viewed as a specialization of the larger context of Dirac reduction, which allows for Dirac reduction by stages. This is explored in a paper in preparation by Cendra, Marsden, Ratiu and Yoshimura.

In this paper, we develop a geometric approach to constraint stabilization for holonomic mechanical systems in the context of Lagrangian formulation. We first show that holonomic mechanical systems, for the case in which a given Lagrangian is hyperregular, can be formulated by using the Lagrangian two-form, namely, a symplectic structure on the tangent bundle of a configuration manifold that is induced from the cotangent bundle via the Legendre transformation. Then, we present an idea of geometric constraint stabilization and we show that a holonomic Lagrangian system with geometric constraint stabilization can be formulated by the Lagrange-d'Alembert principle, together with its local coordinate expression for the sake of numerical computations. Finally, we illustrate the numerical verification that the proposed method enables to stabilize constraint violations effectively in comparison with the Baumgarte and Gear-Gupta-Leimkuhler methods together with an example of a linkage mechanism.

This paper presents a geometric approach to holonomic mechanical systems in the context of Lagrangian systems. First, it is shown how a standard regular Lagrangian system can be established in the context of symplectic geometry; namely, how a symplectic structure on the tangent bundle of a configuration manifold can be induced from the canonical symplectic structure on the cotangent bundle by using the Legendre transformation and also how a second-order Lagrangian vector field can be developed by an energy function associated to a given Lagrangian as well as the induced symplectic structure on the tangent bundle. Second, it is demonstrated that Lagrangian systems with holonomic constraints can be formulated in the context of the induced symplectic structure by combining constraint distributions with the second-order Lagrangian vector field. Further, it is shown how the standard Lagrangian system can be also understood in the context of an induced Dirac structure on the tangent bundle. Finally, a holonomic Lagrangian system is illustrated by a planar linkage system, together with a local expression of differential algebraic equations (DAE) of index 3.

This paper develops a reduction theory for Dirac structures that includes, in a unified way, reduction of both Lagrangian and Hamiltonian systems. It includes the reduction of variational principles and in particular, the Hamilton-Pontryagin variational principle. It also includes reduction theory for implicit Lagrangian systems that could be degenerate and have constraints.
In this paper we focus on the special case in which the configuration manifold is a Lie group G. In our earlier papers we established the link between the Hamilton-Pontryagin principle and Dirac structures. We begin the paper with the reduction of this principle. The traditional view of Poisson reduction in this case is to reduce T*G with its natural Poisson structure to g* with its Lie-Poisson structure. However, the basic step of reducing Hamilton's phase space principle already shows that it is important to use g circle plus g* for the reduced space, rather than just g*. In this way, our construction includes both Euler-Poincare as well as Lie-Poisson reduction. The geometry behind this procedure, which we call Lie-Dirac reduction starts with the standard (i.e., canonical) Dirac structure on T*G (which can be viewed either symplectically or from the Poisson viewpoint) and for each mu epsilon g*, produces a Dirac structure on g circle plus g*. This geometry then simultaneously supports both Euler-Poincare and Lie-Poisson reduction.
In the last part of the paper, we include nonholonomic constraints, and illustrate this construction with Suslov systems in nonholonomic mechanics, both from the Euler-Poincare and Lie-Poisson viewpoints.

This paper begins by recalling how a constraint distribution on a configuration manifold induces a Dirac structure together with an implicit Lagrangian system, a construction that is valid even for degenerate Lagrangians. In such degenerate cases, it is shown in this paper that an implicit Hamiltonian system can be constructed by using a generalized Legendre transformation, where the primary constraints are incorporated into a generalized Hamiltonian on the Pontryagin bundle. Some examples of degenerate Lagrangians for L-C circuits, nonholonomic systems, and point vortices illustrate the theory.

Part I of this paper introduced the notion of implicit Lagrangian systems and their geometric structure was explored in the context of Dirac structures. In this part, we develop the variational structure of implicit Lagrangian systems. Specifically, we show that the implicit Euler-Lagrange equations can be formulated using an extended variational principle of Hamilton called the Hamilton-Pontryagin principle. This vari- ational formulation incorporates, in a natural way, the generalized Legendre transfor- mation, which enables one to treat degenerate Lagrangian systems. The definition of this generalized Legendre transformation makes use of natural maps between iterated tangent and cotangent spaces. Then, we develop an extension of the classical Lagrange- d’Alembert principle called the Lagrange-d’Alembert-Pontryagin principle for implicit Lagrangian systems with constraints and external forces. A particularly interesting case is that of nonholonomic mechanical systems that can have both constraints and external forces. In addition, we define a constrained Dirac structure on the constraint momentum space, namely the image of the Legendre transformation (which, in the degenerate case, need not equal the whole cotangent bundle). We construct an im- plicit constrained Lagrangian system associated with this constrained Dirac structure by making use of an Ehresmann connection. Two examples, namely a vertical rolling disk on a plane and an L-C circuit are given to illustrate the results.

This paper develops the notion of implicit Lagrangian systems and presents some of their basic properties in the context of Dirac structures. This setting includes degenerate Lagrangian systems and systems with both holonomic and nonholonomic constraints, as well as networks of Lagrangian mechanical systems. The definition of implicit Lagrangian systems with a configuration space $Q$ makes use of Dirac structures on $T^*Q$ that are induced from a constraint distribution on $Q$ as well as natural symplectomorphisms between the spaces $T^*TQ$, $TT^*Q$, and $T^*T^*Q$. Two illustrative examples are presented; the first is a nonholonomic system, namely a vertical disk rolling on a plane and the second is an L-C circuit, a degenerate Lagrangian system with holonomic constraints.

The paper presents a multiport model of flexible multibody systems by analogy with a connection multiport in electrical circuit theory. First we introduce a concept of a fundamental pair, that is, a pair of a mechanical joint and its adjacent body to recognize the flexible multibody system as an interconnected system of such fundamental pairs. Second we employ a finite element model to describe flexible deformations associated with large overall motions using moving frames and we also model various kinematical and dynamical relations of the fundamental pair such as geometric nonlinear effects associated with the flexible deformations and kinematical constraints due to the mechanical joint by nonenergic multiports together with dual connection matrices. Finally it is shown that the interconnection of the nonenergic multiports with physical elements provides a multiport model of the fundamental pair and also that the equations of motion of the flexible multibody system can be systematically formulated by the present approach.

The paper presents a framework of dynamical formalism for flexible multibody systems specifically focusing upon how the network-theoretic concepts extensively work on structural understanding of complex nonlinear mechanical systems like the flexible multibody systems. The fundamental idea lies in nonenergicness which is called Tellegen's theorem in the electrical network theory, and it is shown how the kinematical and dynamical relations of the flexible multibody systems are modelled as structural relations by utilizing the nonenergicness. It is also demonstrated that such a structural understanding can provide us with a systematic way of formulating system equations of the flexible multibody systems.

A bond graph method of modelling multibody dynamics is demonstrated. Specifically, a symbolic generation scheme which fully utilizes the bond graph information is presented. It is also demonstrated that structural understanding and representation in bond graph theory is quite powerful for the modelling of such large scale systems, and that the non-energic multiport of junction structure, which is a multiport expression of the system structure, plays an important role, as first suggested by Paynter.
The principal part of the proposed symbolic scheme, that is, the elimination of excess variables, is done through tearing and interconnection in the sense of Kron using newly defined causal and causal coefficient arrays.