Search

The notion of implicit port-Lagrangian systems for nonholonomic mechanics was proposed in Yoshimura & Marsden (2006a, J. Geom. Phys., 57, 133–156; 2006b, J. Geom. Phys., 57, 209–250; 2006c, Proc. of the 17th International Symposium on Mathematical Theory of Networks and Systems, Kyoto) as a Lagrangian analogue of implicit port-Hamiltonian systems. Such port-systems have an interconnection structure with ports through which power is exchanged with the exterior and which can be modeled by Dirac structures. In this paper, we present the notions of implicit port-Lagrangian systems and port-Dirac dynamical systems in nonequilibrium thermodynamics by generalizing the Dirac formulation to the case allowing irreversible processes, both for closed and open systems. Port-Dirac systems in nonequilibrium thermodynamics can be also deduced from a variational formulation of nonequilibrium thermodynamics for closed and open systems introduced in Gay-Balmaz & Yoshimura (2017a, J. Geom. Phys., 111, 169–193; 2018a, Entropy, 163, 1–26). This is a type of Lagrange–d’Alembert principle for the specific class of nonholonomic systems with nonlinear constraints of thermodynamic type, which are associated to the entropy production equation of the system. We illustrate our theory with some examples such as a cylinder-piston with ideal gas, an electric circuit with entropy production due to a resistor and an open piston with heat and matter exchange with the exterior.

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.

International audience; 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.

The two-phase flow void fraction is a critical parameter for characterising the pressure drop as well as heat and mass transfer capability of the working fluid within thermal systems, the accurate estimation of which drives heat exchanger design and control optimisation. A semitheoretical expression for the void fraction of two-phase flows, also applicable to small-sized channels, is obtained from an analytical study based on the principle of minimum entropy production and the introduction of empirical coefficients to be fitted to experimental data available in the open literature. These coefficients embody the importance of the simplified physical terms of this formulation while recovering the accuracy loss owing to nonlinear phenomena, heat and mass transfer, and three-dimensional effects. By accounting for surface tension, this model generalises previous theories and describes the influence of smaller-sized channels in terms of the stable void fraction. This mathematical framework can be used to summarise data covering different refrigerants, channel diameters, and operating conditions.

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 $TQ\times V^{\ast}$, 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 $(TQ\oplus T^{\ast}Q)\times V^{\ast}$. Associated with this variational principle, we establish a Dirac structure on $(TQ\oplus T^{\ast}Q)\times V^{\ast}$ to define an intrinsic vakonomic Lagrange-Dirac system. Furthermore, we establish another construction for the vakonomic Lagrange-Dirac system using a Dirac structure on $T^{\ast}Q\times V^{\ast}$, where we introduce a vakonomic Dirac differential. Lastly, we illustrate our theory of vakonomic Lagrange-Dirac systems by some examples such as the vakonomic skate and the vertical rolling coin., Comments are welcome

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-Poincar\'e-Suslov equations with advected parameters and the implicit Lie-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.

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.

As a high specific speed pump, the contra-rotating axial flow pump distinguishes itself in a rear rotor rotating in the opposite direction of the front rotor, which remarkably contributes to the energy conversion, the reduction of the pump size, better hydraulic and cavitation performances. However, with two rotors rotating reversely, the significant interaction between blade rows was observed in our prototype contra-rotating rotors, which highly affected the pump performance compared with the conventional axial flow pumps. Consequently, a new type of rear rotor was designed by the rotational speed optimization methodology with some additional considerations, aiming at better cavitation performance, the reduction of blade rows interaction and the secondary flow suppression. The new rear rotor showed a satisfactory performance at the design flow rate but an unfavorable positive slope of the head — flow rate curve in the partial flow rate range less than 40% of the design flow rate, which should be avoided for the reliability of pump-pipe systems. In the present research, to understand the internal flow field of new rear rotor and its relation to the performances at the partial flow rates, the velocity distributions at the inlets and outlets of the rotors are firstly investigated. Then, the boundary layer flows on rotor surfaces, which clearly reflect the secondary flow inside the rotors, are analyzed through the limiting streamline observations using the multi-color oil-film method. Finally, the unsteady numerical simulations are carried out to understand the complicated internal flow structures in the rotors.

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 input-output relations 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.

We present a variational formulation for the Navier–Stokes–Fourier system based on a free energy Lagrangian. This formulation is a systematic infinite-dimensional extension of the variational approach to the thermodynamics of discrete systems using the free energy, which complements the Lagrangian variational formulation using the internal energy developed in [F. Gay-Balmaz and H. Yoshimura, A Lagrangian variational formulation for nonequilibrium thermodynamics, Part II: Continuum systems, J. Geom. Phys. 111 (2017) 194–212] as one employs temperature, rather than entropy, as an independent variable. The variational derivation is first expressed in the material (or Lagrangian) representation, from which the spatial (or Eulerian) representation is deduced. The variational framework is intrinsically written in a differential-geometric form that allows the treatment of the Navier–Stokes–Fourier system on Riemannian manifolds.

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^* ⊕ 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 µ є g^*, produces a Dirac structure on g^* ⊕ 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.

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 variational formulation incorporates, in a natural way, the generalized Legendre transformation, 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 implicit 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.

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.