Hiroaki Yoshimura

Reduction of Dirac structures and the Hamilton-Pontryagin principle

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 Hamiltons 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.

Dirac structures and the legendre transformation for implicit lagrangian and hamiltonian systems

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.

Stokes-Dirac structures through reduction of infinite-dimensional Dirac structures

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 Hamilton-Pontryagin principle and multi-Dirac structures for classical field theories

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, Maxwells equations, and elastostatics.

Dirac structures in vakonomic mechanics

Abstract 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 T Q × 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 ( T Q ⊕ T ∗ Q ) × V ∗ . Associated with this variational principle, we establish a Dirac structure on ( T Q ⊕ T ∗ Q ) × 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 × 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.

On the geometry of multi-Dirac structures and Gerstenhaber algebras

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.

Interconnection of Lagrange‐Dirac Dynamical Systems for Electric Circuits

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.

A free energy Lagrangian variational formulation of the Navier-Stokes-Fourier system

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 \cite{GBYo2016b} 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.

Variational discretization of the nonequilibrium thermodynamics of simple systems

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.

Lunar capture trajectories in the four-body problem

In this paper, we develop the low energy trajectory design of a spacecraft from the Earth to the Moon in the context of the Planar Restricted Four Body Problem, which may be approximately modeled by coupling two Planar Restricted Three Body Problems, i.e., the Sun-(Earth+Moon)-Spacecraft and the Earth-Moon-Spacecraft systems. The principal idea lies in the so-called tube dynamics, which may patch invariant manifolds of the two PR3BPs to obtain the required lunar capture trajectories. We first model the S-E-M-S/C system by the coupled Planar Restricted Circular Three Body Problem (PRC3BP) and we also analyze the case, in particular, where the E-M-S/C system is modeled by the Planar Restricted Elliptic Three Body Problem (PRE3BP), since the E-M system has the nonnegligible eccentricity. Finally, we develop the low energy lunar capture trajectory in the coupled PRC3B-PRE3BP in comparison with the case of the coupled PRC3BP.