# François Gay-BalmazEcole Normale Supérieure de Paris | ENS

François Gay-Balmaz

CNRS mathematics

## About

173

Publications

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Citations since 2017

## Publications

Publications (173)

Respecting the laws of thermodynamics is crucial for ensuring that numerical simulations of dynamical systems deliver physically relevant results. In this paper, we construct a structure-preserving and thermodynamically consistent finite element method and time-stepping scheme for heat conducting viscous fluids. The method is deduced by discretizin...

We present a geometric variational discretization of nonlinear elasticity in 2D and 3D in the Lagrangian description. A main step in our construction is the definition of discrete deformation gradients and discrete Cauchy–Green deformation tensors, which allows for the development of a general discrete geometric setting for frame indifferent isotro...

Mixed quantum-classical spin systems have been proposed in spin chain theory, organic chemistry, and, more recently, spintronics. However, current models of quantum-classical dynamics beyond mean-field approximations typically suffer from long-standing consistency issues, and in some cases invalidate Heisenberg's uncertainty principle. Here, we pre...

It has been a long standing question how to extend the canonical Poisson bracket formulation from classical mechanics to classical field theories, in a completely general, intrinsic, and canonical way. In this paper, we provide an answer to this question by presenting a new completely canonical bracket formulation of Hamiltonian Classical Field The...

Motivated by compartmental analysis in engineering and biophysical systems, we present a variational framework for the nonequilibrium thermodynamics of systems involving both distributed and discrete (finite dimensional) subsystems by specifically using the ideas of interconnected systems. We focus on the process of non-isothermal diffusion and sho...

In this paper, we develop a Hamiltonian variational formulation for the nonequilibrium thermodynamics of simple adiabatically closed systems that is an extension of Hamilton's phase space principle in mechanics. We introduce the Hamilton-d'Alembert principle for thermodynamic systems by considering nonlinear nonholonomic constraints of thermodynami...

Many applications of porous media research involves high pressures and, correspondingly, exchange of thermal energy between the fluid and the matrix. While the system is relatively well understood for the case of nonmoving porous media, the situation when the elastic matrix can move and deform, is much more complex. In this paper, we derive the equ...

In this paper, we develop a Hamiltonian variational formulation for the nonequilibrium thermodynamics of simple adiabatically closed systems that is an extension of Hamilton's phase space principle in mechanics. We introduce the Hamilton-d'Alembert principle for thermodynamic systems by considering nonlinear nonholonomic constraints of thermodynami...

We establish a Lagrangian variational framework for general relativistic continuum theories that permits the development of the process of Lagrangian reduction by symmetry in the relativistic context. Starting with a continuum version of the Hamilton principle for the relativistic particle, we deduce two classes of reduced variational principles th...

This paper presents the variational discretization of the compressible Navier-Stokes-Fourier system, in which the viscosity and the heat conduction terms are handled within the variational approach to nonequilibrium thermodynamics as developed by one of the authors. In a first part, we review the variational framework for the Navier-Stokes-Fourier...

We deal with the reversible dynamics of coupled quantum and classical systems. Based on a recent proposal by the authors, we exploit the theory of hybrid quantum–classical wavefunctions to devise a closure model for the coupled dynamics in which both the quantum density matrix and the classical Liouville distribution retain their initial positive s...

A gauge-invariant wave equation for the dynamics of hybrid quantum-classical systems is formulated by combining the variational setting of Lagrangian paths in continuum theories with Koopman wavefunctions in classical mechanics. We identify gauge transformations with unobservable phase factors in the classical phase-space and we introduce gauge inv...

Numerical models of weather and climate critically depend on the long-term stability of integrators for systems of hyperbolic conservation laws. While such stability is often obtained from (physical or numerical) dissipation terms, physical fidelity of such simulations also depends on properly preserving conserved quantities, such as energy, of the...

We construct a structure-preserving finite element method and time-stepping scheme for inhomogeneous, incompressible magnetohydrodynamics (MHD). The method preserves energy, cross-helicity (when the fluid density is constant), magnetic helicity, mass, total squared density, pointwise incompressibility, and the constraint divB=0 to machine precision...

We construct a structure-preserving finite element method and time-stepping scheme for compressible barotropic magnetohydrodynamics both in the ideal and resistive cases, and in the presence of viscosity. The method is deduced from the geometric variational formulation of the equations. It preserves the balance laws governing the evolution of total...

We present a geometric variational discretization of nonlinear elasticity in 2D and 3D in the Lagrangian description. A main step in our construction is the definition of discrete deformation gradients and discrete Cauchy-Green deformation tensors, which allows for the development of a general discrete geometric setting for frame indifferent isotro...

We deal with the reversible dynamics of coupled quantum and classical systems. Based on a recent proposal by the authors, we exploit the theory of hybrid quantum-classical wavefunctions to devise a closure model for the coupled dynamics in which both the quantum density matrix and the classical Liouville distribution retain their initial positive s...

We review the variational formulation of nonequilibrium thermodynamics as an extension of the Hamilton principle and the Lagrange-d’Alembert principle of classical mechanics. We focus on the case of open systems that include the power exchange due to heat and matter transfer, with special emphasis on reacting systems which are very important in bio...

8 pages, 1 figure. To appear in Lecture Notes in Comput. Sci

This paper presents (Lagrangian) variational formulations for single and multicomponent semi-compressible fluids with both reversible (entropy-conserving) and irreversible (entropy-generating) processes. Semi-compressible fluids are useful in describing low-Mach dynamics, since they are soundproof . These models find wide use in many areas of fluid...

Many applications of porous media research involves high pressures and, correspondingly, exchange of thermal energy between the fluid and the matrix. While the system is relatively well understood for the case of non-moving porous media, the situation when the elastic matrix can move and deform, is much more complex. In this paper we derive the equ...

Many parts of biological organisms are comprised of deformable porous media. The biological media is both pliable enough to deform in response to an outside force and can deform by itself using the work of an embedded muscle. For example, the recent work (Ludeman et al., 2014) has demonstrated interesting 'sneezing' dynamics of a freshwater sponge,...

Numerical models of weather and climate critically depend on long-term stability of integrators for systems of hyperbolic conservation laws. While such stability is often obtained from (physical or numerical) dissipation terms, physical fidelity of such simulations also depends on properly preserving conserved quantities, such as energy, of the sys...

Many parts of biological organisms are comprised of deformable porous media. The biological media is both pliable enough to deform in response to an outside force and can deform by itself using the work of an embedded muscle. For example, the recent work (Ludeman et al., 2014) has demonstrated interesting ‘sneezing’ dynamics of a freshwater sponge,...

We present the Dirac structures and the associated Dirac system formulations for non-simple thermodynamic systems by focusing upon the cases that include irreversible processes due to friction and heat conduction. These systems are called non-simple since they involve several entropy variables. We review the variational formulation of the evolution...

Based on Koopman’s theory of classical wavefunctions in phase space, we present the Koopman-van Hove (KvH) formulation of classical mechanics as well as some of its properties. In particular, we show how the associated classical Liouville density arises as a momentum map associated to the unitary action of strict contact transformations on classica...

Based on the Koopman-van Hove (KvH) formulation of classical mechanics introduced in Part I, we formulate a Hamiltonian model for hybrid quantum-classical systems. This is obtained by writing the KvH wave equation for two classical particles and applying canonical quantization to one of them. We illustrate several geometric properties of the model...

We present a structure preserving discretization of the fundamental spacetime geometric structures of fluid mechanics in the Lagrangian description in 2D and 3D. Based on this, multisymplectic variational integrators are developed for barotropic and incompressible fluid models, which satisfy a discrete version of Noether theorem. We show how the ge...

In this paper, we make a review of our recent development of Dirac structures and the associated variational formulation for nonequilibrium thermodynamics (see, [15, 16]). We specifically focus on the case of simple and open systems, in which the thermodynamic state is represented by one single entropy and the transfer of matter and heat with the e...

We construct a structure-preserving finite element method and time-stepping scheme for compressible barotropic magnetohydrodynamics (MHD) both in the ideal and resistive cases, and in the presence of viscosity. The method is deduced from the geometric variational formulation of the equations. It preserves the balance laws governing the evolution of...

Based on the Koopman-van Hove (KvH) formulation of classical mechanics introduced in Part I, we formulate a Hamiltonian model for hybrid quantum-classical systems. This is obtained by writing the KvH wave equation for two classical particles and applying canonical quantization to one of them. We illustrate several geometric properties of the model...

Based on Koopman's theory of classical wavefunctions in phase space, we present the Koopman-van Hove (KvH) formulation of classical mechanics as well as some of its properties. In particular, we show how the associated classical Liouville density arises as a momentum map associated to the unitary action of strict contact transformations on classica...

Backbone curves are usually obtained by analyzing nonlinear oscillators (e.g., the Duffing equation) whereas the backbones of linear oscillators are incidental by-products of such studies. Nevertheless, we focus on the harmonic oscillator, a linear, homogeneous constant-coefficient second-order ordinary differential equation. The backbone of the si...

We present a structure preserving discretization of the fundamental spacetime geometric structures of fluid mechanics in the Lagrangian description in 2D and 3D. Based on this, multisymplectic variational integrators are developed for barotropic and incompressible fluid models, which satisfy a discrete version of Noether theorem. We show how the ge...

This paper presents (Lagrangian) variational formulations for single and multicomponent semi-compressible fluids with both reversible (entropy-conserving) and irreversible (entropy-generating) processes. Semi-compressible fluids are useful in describing \textit{low-Mach} dynamics, since they are \textit{soundproof}. These models find wide use in ma...

A family of mem-models, including the mem-dashpots, mem-springs, and most recently, mem-inerters, is emerging as a new and powerful way of capturing complex nonlinear behaviors of materials and systems under various types of dynamic loads involving different frequency, amplitude, and loading histories (e.g., hysteresis). Under the framework of nonl...

We present an overview of finite element variational integrators for compressible and incompressible fluids with variable density. The numerical schemes are derived by discretizing, in a structure preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles. Given a triangulation on...

The paper presents a port-Dirac formulation for thermodynamics of non-simple systems, in which we consider a non-simple system whose thermodynamic states may be represented by several entropy variables. Here we regard such a non-simple system as an interconnected system with ports that can be represented by a port-Dirac system in the context of Dir...

We make a review of the variational formulation of nonequilibrium thermodynamics as an extension of the Hamilton principle and the Lagrange-d’Alembert principle of classical mechanics. We then focus on the case of open systems that include the power exchange due to heat and matter transfer, with special emphasis on reacting systems which are very i...

We construct a structure-preserving finite element method and time-stepping scheme for inhomogeneous, incompressible magnetohydrodynamics (MHD). The method preserves energy, cross-helicity (when the fluid density is constant), magnetic helicity, mass, total squared density, pointwise incompressibility, and the constraint $\operatorname{div} B = 0$...

We present a finite element variational integrator for compressible flows. The numerical scheme is derived by discretizing, in a structure-preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles. Given a triangulation on the fluid domain, the discrete group of diffeomorphisms is...

This paper extends the Madelung–Bohm formulation of quantum mechanics to describe the time-reversible interaction of classical and quantum systems. The symplectic geometry of the Madelung transform leads to identifying hybrid quantum–classical Lagrangian paths extending the Bohmian trajectories from standard quantum theory. As the classical symplec...

We derive the equations of motion for the dynamics of porous medium, filled with incompressible fluid. We use a variational approach with a Lagrangian written as the sum of terms representing the kinetic and potential energy of the elastic matrix, and the kinetic energy of the fluid, coupled through the constraint of incompressibility. As an illust...

Dirac structures are geometric objects that generalize Poisson structures and presymplectic structures on manifolds. They naturally appear in the formulation of constrained mechanical systems and play an essential role in structuring a dynamical system through the energy flow between its subsystems and elements. In this paper, we show that the evol...

A variational formulation for nonequilibrium thermodynamics was recently proposed in Gay-Balmaz and Yoshimura (2017a, 2017b) for both discrete and continuum systems. This formulation extends the Hamilton principle of classical mechanics to include irreversible processes. In this paper, we show that this variational formulation yields a constructive...

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....

We derive the equations of motion for the dynamics of a porous media filled with an incompressible fluid. We use a variational approach with a Lagrangian written as the sum of terms representing the kinetic and potential energy of the elastic matrix, and the kinetic energy of the fluid, coupled through the constraint of incompressibility. As an ill...

In this paper, we describe and exploit a geometric framework for Gibbs probability densities and the associated concepts in statistical mechanics, which unifies several earlier works on the subject, including Souriau’s symplectic model of statistical mechanics, its polysymplectic extension, Koszul model, and approaches developed in quantum informat...

We construct a finite element discretization and time-stepping scheme for the incompressible Euler equations with variable density that exactly preserves total mass, total squared density, total energy, and pointwise incompressibility. The method uses Raviart-Thomas or Brezzi-Douglas-Marini finite elements to approximate the velocity and discontinu...

In this paper we describe and exploit a geometric framework for Gibbs probability densities and the associated concepts in statistical mechanics, which unifies several earlier works on the subject, including Souriau's symplectic model of statistical mechanics, its polysymplectic extension, Koszul model, and approaches developed in quantum informati...

The anelastic and pseudo-incompressible equations are two well-known soundproof approximations of compressible flows useful for both theoretical and numerical analysis in meteorology, atmospheric science, and ocean studies. In this paper, we derive and test structure-preserving numerical schemes for these two systems. The derivations are based on a...

We construct a finite element discretization and time-stepping scheme for the incompressible Euler equations with variable density that exactly preserves total mass, total squared density, total energy, and pointwise incompressibility. The method uses Raviart-Thomas or Brezzi-Douglas-Marini finite elements to approximate the velocity and discontinu...

We present a finite element variational integrator for compressible flows. The numerical scheme is derived by discretizing, in a structure preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles. Given a triangulation on the fluid domain, the discrete group of diffeomorphisms is...

Upon revisiting the Hamiltonian structure of classical
wavefunctions in Koopman–von Neumann theory,
this paper addresses the long-standing problem
of formulating a dynamical theory of classical–
quantum coupling. The proposed model not only
describes the influence of a classical system onto
a quantum one, but also the reverse effect—
the quantum ba...

In this paper, we describe the coadjoint orbits of the group of volume preserving diffeomorphisms associated to the motion of codimension one singular membranes in ideal fluids, i.e. filaments in 2D, membranes in 3D. We show that they can be identified with a certain class of nonlinear Grassmannian of submanifolds endowed with closed 1-form of a gi...

A variational formulation for nonequilibrium thermodynamics was recently proposed in [7, 8] for both discrete and continuum systems. This formulation extends the Hamilton principle of classical mechanics to include irreversible processes. In this paper, we show that this variational formulation yields a constructive and systematic way to derive fro...

Dirac structures are geometric objects that generalize Poisson structures and presymplectic structures on manifolds. They naturally appear in the formulation of constrained mechanical systems and play an essential role in the understanding of the interrelations between system elements in implicit dynamical systems. In this paper, we show how nonequ...

We introduce a geometric variational discretization framework for geophysical flow models. The numerical scheme is obtained by discretizing, in a structure-preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles. Being based on a discrete version of the Euler-Poincaré variationa...

Dirac structures are geometric objects that generalize Poisson structures and presymplectic structures on manifolds. They naturally appear in the formulation of constrained mechanical systems and play an essential role in structuring a dynamical system through the energy flow between its subsystems and elements. In this paper, we show that the evol...

This paper extends the Madelung-Bohm formulation of quantum mechanics to describe the time-reversible interaction of classical and quantum systems. The symplectic geometry of the Madelung transform leads to identifying hybrid classical-quantum Lagrangian paths extending the Bohmian trajectories from standard quantum theory. As the classical symplec...

In this paper, we introduce the notion of port-Lagrangian systems in nonequilibrium thermodynamics, which is constructed by generalizing the notion of port-Lagrangian systems for nonholonomic mechanics proposed in Yoshimura and Marsden [2006c], where the notion of interconnections is described in terms of Dirac structures. The notion of port-Lagran...

A variational principle is derived for two-dimensional incompressible rotational fluid flow with a free surface in a moving vessel when both the vessel and fluid motion are to be determined. The fluid is represented by a stream function and the vessel motion is represented by a path in the planar Euclidean group. Novelties in the formulation includ...

A variational formulation for nonequilibrium thermodynamics was recently proposed in \cite{GBYo2017a,GBYo2017b} for both discrete and continuum systems. This formulation extends the Hamilton principle of classical mechanics to include irreversible processes. In this paper, we show that this variational formulation yields a constructive and systemat...

We present a theory for the three-dimensional evolution of tubes with expandable walls conveying fluid. Our theory can accommodate arbitrary deformations of the tube, arbitrary elasticity of the walls, and both compressible and incompressible flows inside the tube. We also present the theory of propagation of shock waves in such tubes and derive th...

arXiv admin note: text overlap with arXiv:1802.05711

Irreversible processes play a major role in the description and prediction of atmospheric dynamics. In this paper, we present a variational derivation for moist atmospheric dynamics with rain process and subject to the irreversible processes of viscosity, heat conduction, diffusion, and phase transition. This derivation is based on a general variat...

We develop a variational integrator for the shallow‐water equations on a rotating sphere. The variational integrator is built around a discretization of the continuous Euler–Poincaré reduction framework for Eulerian hydrodynamics. We describe the discretization of the continuous Euler–Poincaré equations on arbitrary simplicial meshes. Standard nume...

Choosing state variables in a state-space representation of a nonlinear dynamical system is a nonunique procedure for a given input–output relationship and therefore a potentially challenging task. It can be even more challenging when there are piecewise-defined restoring forces, as in bilinear hysteresis or Bouc–Wen models, which are just two of m...

In this paper, we survey our recent results on the variational formulation of nonequilibrium thermodynamics for the finite-dimensional case of discrete systems, as well as for the infinite-dimensional case of continuum systems. Starting with the fundamental variational principle of classical mechanics, namely, Hamilton’s principle, we show, with th...

The equations of reversible (inviscid, adiabatic) fluid dynamics have a well-known variational formulation based on Hamilton's principle and the Lagrangian, to which is associated a Hamiltonian formulation that involves a Poisson bracket structure. These variational and bracket structures underlie many of the most basic principles that we know abou...

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...

A variational principle is derived for two-dimensional incompressible rotational fluid flow with a free surface in a moving vessel when both the vessel and fluid motion are to be determined. The fluid is represented by a stream function and the vessel motion is represented by a path in the planar Euclidean group. Novelties in the formulation includ...

We develop a variational integrator for the shallow-water equations on a rotating sphere. The variational integrator is built around a discretization of the continuous Euler-Poincar\'{e} reduction framework for Eulerian hydrodynamics. We describe the discretization of the continuous Euler-Poincar\'{e} equations on arbitrary simplicial meshes. Stand...

This paper presents the continuous and discrete variational formulations of simple thermodynamical systems whose configuration space is a (finite dimensional) Lie group. We follow the variational approach to nonequilibrium thermodynamics developed in \cite{GBYo2017a,GBYo2017b}, as well as its discrete counterpart whose foundations have been laid in...

We review opportunities for stochastic geometric mechanics to incorporate observed data into variational principles, in order to derive data-driven nonlinear dynamical models of effects on the variability of computationally resolvable scales of fluid motion, due to unresolvable, small, rapid scales of fluid motion.

Inspired by spatiotemporal observations from satellites of the trajectories of objects drifting near the surface of the ocean in the National Oceanic and Atmospheric Administration's "Global Drifter Program", this paper develops data-driven stochastic models of geophysical fluid dynamics (GFD) with non-stationary spatial correlations representing t...

We present a theory for the three-dimensional evolution of tubes with expandable walls conveying fluid. Our theory can accommodate arbitrary deformations of the tube, arbitrary elasticity of the walls, and both compressible and incompressible flows inside the tube. We also present the theory of propagation of shock waves in such tubes and derive th...

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...

After revisiting the Koopman-von Neumann theory of classical mechanics from a Hamiltonian viewpoint, a new expression of the Liouville density in terms of classical wavefunctions is exploited to formulate a dynamical theory for the interaction between classical and quantum systems.

We present a variational formulation for nonequilibrium thermodynamics which extends the Hamilton principle of mechanics to include irreversible processes. The variational formulation is based on the introduction of the concept of thermodynamic displacement. This concept makes possible the definition of a nonlinear nonholonomic constraint given by...

In this paper, it is shown that the evolution equations for nonequilibrium thermodynamics admit an intrinsic formulation in terms of Dirac structures, both on the Lagrangian and the Hamiltonian settings. The Dirac structures are constructed on the Pontryagin bundle P = TQ ⊕ T⁎Q, where Q = Q × ℝ is the thermodynamic configuration manifold. In partic...

Certain kinematic optimal control problems (the Clebsch problems) and their connection to classical integrable systems are considered. In particular, the rigid body problem and its rank 2k counterparts, the geodesic flows on Stiefel manifolds and their connection with the work of Moser, flows on symmetric matrices, and the Toda flows are studied.

This paper presents a geometric variational discretization of compressible fluid dynamics. The numerical scheme is obtained by discretizing, in a structure preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles. Our framework applies to irregular mesh discretizations in 2D and...

We propose two types of stochastic extensions of nonholonomic constraints for mechanical systems. Our approach relies on a stochastic extension of the Lagrange-d’Alembert framework. We consider in details the case of invariant nonholonomic systems on the group of rotations and on the special Euclidean group. Based on this, we then develop two types...

We analyze the symplectic structure of the coadjoint orbits of Lie groups with Lie algebras that contain admissible ideals. Such ideals were introduced by Pukanszky to investigate the global symplectic structure of simply connected coadjoint orbits of connected, simply connected, solvable Lie groups. Using the theory of symplectic reduction of cota...

In this paper, we show that the evolution equations for nonequilibrium thermodynamics can be formulated in terms of Dirac structures on the Pontryagin bundle \(\mathsf {P} =T \mathsf {Q}\oplus T ^* \mathsf {Q} \), where \(\mathsf {Q}=Q \times \mathbb {R}\) denotes the thermodynamic configuration manifold. In particular, we extend the use of Dirac s...

In this paper, we present a variational formulation for heat conducting viscous fluids, which extends the Hamilton principle of continuum mechanics to include irreversible processes. This formulation follows from the general variational description of nonequilibrium thermodynamics introduced in [3, 4] for discrete and continuum systems. It relies o...

We study the linear stability of elastic collapsible tubes conveying fluid, when the equilibrium configuration of the tube is helical. A particular case of such tubes, commonly encountered in applications, is represented by quarter- or semi-circular tubular joints used at pipe's turning points. The stability theory for pipes with non-straight equil...

We propose two types of stochastic extensions of nonholonomic constraints for mechanical systems. Our approach relies on a stochastic extension of the Lagrange-d'Alembert framework. We consider in details the case of invariant nonholonomic systems on the group of rotations and on the special Euclidean group. Based on this, we then develop two types...

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 d...

We call attention to a dual-pair concept for modeling hysteresis involving instantaneous switching: Speciﬁcally, there are two input–output pairs for each hysteresis model under one speciﬁc input, namely a differential pair and an integral pair. Currently in engineering mechanics, only one pair is being recognized and utilized, not the other. Where...

We show that the Flaschka map, originally introduced to analyze the dynamics of the integrable Toda lattice system, is the inverse of a momentum map. We discuss the geometrical setting of the map and apply it to the generalized Toda lattice systems on semisimple Lie algebras, the rigid body system on Toda orbits, and to coadjoint orbits of semidire...

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 equa- tions for nonequilibrium thermodynamics admit an intrinsic formulation in terms of Dirac structures, bot...

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 \cite{GBYo2016a}, and thus extend the variational integrators of Lagrangian mechanics, to include...

## Projects

Projects (2)

This is a new family of hysteresis models which can facilitate constitutive modeling, system identification and control, and deepen the understanding of complex material behaviors and system responses under various types of dynamic loads with different frequency, amplitude, and loading history. Dr. Leon Chua, who postulated “the missing memristor”, a fourth basic circuit element, also developed the theory of memristive systems which was later extended to include memcapacitors and meminductors, thereby introducing a new model class called “mem-models”. By applying well-known mechanical-electrical system analogies, the mathematics of mem-models has been transferred to the setting of engineering mechanics, resulting in mechanical counterparts of memristors, memcapacitors, and more. This transfer is nontrivial, yielding for example, a new concept and state variable called “absement”, the time integral of deformation. We focus on “mem-springs”, which are fundamentally new in treating diverse stress-strain behaviors. We inject a hybrid dynamical system viewpoint to ensure physically meaningful behaviors of mem-springs, and connect mem-models to classical theories including continuum damage mechanics, viscoelasticity, and Preisach model, revealing the great potential of mem-models.