Publications by François Gay-Balmaz

A multisymplectic integrator for elastodynamic frictionless impact problems

Mathieu Desbrun, Tudor Ratiu, François Gay-Balmaz, François Marie Alain Demoures, Alejandro Marcos Aragon

We present a structure preserving numerical algorithm for the collision of elastic bodies. Our integrator is derived from a discrete version of the field-theoretic (multisymplectic) variational description of nonsmooth Lagrangian continuum mechanics, combi ...

Elsevier2017

The Geometric Nature of the Flaschka Transformation

Tudor Ratiu, François Gay-Balmaz

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 semi ...

Springer Verlag2017

Multisymplectic variational integrators and space/time symplecticity

Tudor Ratiu, François Gay-Balmaz, François Marie Alain Demoures

Multisymplectic variational integrators are structure-preserving numerical schemes especially designed for PDEs derived from covariant spacetime Hamilton principles. The goal of this paper is to study the properties of the temporal and spatial discrete evo ...

World Scientific Publ Co Pte Ltd2016

Multisymplectic Variational Integrators For Nonsmooth Lagrangian Continuum Mechanics

Tudor Ratiu, François Gay-Balmaz, François Marie Alain Demoures

This paper develops the theory of multisymplectic variational integrators for nonsmooth continuum mechanics with constraints. Typical problems are the impact of an elastic body on a rigid plate or the collision of two elastic bodies. The integrators are ob ...

Cambridge Univ Press2016

Discrete variational Lie group formulation of geometrically exact beams dynamics

Tudor Ratiu, François Gay-Balmaz, Yves Weinand, François Marie Alain Demoures

The goal of this paper is to derive a structure preserving integrator for geometrically exact beam dynamics, by using a Lie group variational integrator. Both spatial and temporal discretization are implemented in a geometry preserving manner. The resultin ...

Springer2015

Lagrangian Reductions and Integrable Systems in Condensed Matter

Tudor Ratiu, François Gay-Balmaz

We consider a general approach for the process of Lagrangian and Hamiltonian reduction by symmetries in chiral gauge models. This approach is used to show the complete integrability of several one dimensional texture equations arising in liquid Helium phas ...

Springer2015

The geometry of the universal Teichmuller space and the Euler-Weil-Petersson equation

Tudor Ratiu, François Gay-Balmaz

On the identity component of the universal Teichmuller space endowed with the Takhtajan-Teo topology, the geodesics of the Weil Petersson metric are shown to exist for all time. This component is naturally a subgroup of the quasisymmetric homeomorphisms of ...

Elsevier2015

Multisymplectic Lie group variational integrator for a geometrically exact beam in R3

Tudor Ratiu, François Gay-Balmaz, François Marie Alain Demoures

In this paper we develop, study, and test a Lie group multisymplectic integra- tor for geometrically exact beams based on the covariant Lagrangian formulation. We exploit the multisymplectic character of the integrator to analyze the energy and momentum ma ...

Elsevier2014

Integrable G-strands on semisimple Lie groups

Tudor Ratiu, François Gay-Balmaz

The present paper derives systems of partial differential equations that admit a quadratic zero curvature representation for an arbitrary real semisimple Lie algebra. It also determines the general form of Hamilton's principles and Hamiltonians for these s ...

Iop Publishing Ltd2014

Equivalent Theories of Liquid Crystal Dynamics

Tudor Ratiu, François Gay-Balmaz, Cesare Tronci

There are two competing descriptions of nematic liquid crystal dynamics: the Ericksen-Leslie director theory and the Eringen micropolar approach. Up to this day, these two descriptions have remained distinct in spite of several attempts to show that the mi ...

Springer Verlag2013

Exact geometric theory of dendronized polymer dynamics

Tudor Ratiu, François Gay-Balmaz

Dendronized polymers consist of an elastic backbone with a set of iterated branch structures (dendrimers) attached at every base point of the backbone. The conformations of such molecules depend on the elastic deformation of the backbone and the branches, ...

2012

Invariant Higher-Order Variational Problems II

Tudor Ratiu, François Gay-Balmaz

Motivated by applications in computational anatomy, we consider a second-order problem in the calculus of variations on object manifolds that are acted upon by Lie groups of smooth invertible transformations. This problem leads to solution curves known as ...

Springer Verlag2012

Reduced Variational Formulations in Free Boundary Continuum Mechanics

Tudor Ratiu, François Gay-Balmaz

We present the material, spatial, and convective representations for elasticity and fluids with a free boundary from the Lagrangian reduction point of view, using the material and spatial symmetries of these systems. The associated constrained variational ...

Springer-Verlag2012

Invariant Higher-Order Variational Problems

Tudor Ratiu, François Gay-Balmaz

We investigate higher-order geometric k-splines for template matching on Lie groups. This is motivated by the need to apply diffeomorphic template matching to a series of images, e.g., in longitudinal studies of Computational Anatomy. Our approach formulat ...

2012

Euler-Poincare Approaches to Nematodynamics

Tudor Ratiu, François Gay-Balmaz, Cesare Tronci

Nematodynamics is the orientation dynamics of flowless liquid-crystals. We show how Euler-Poincar, reduction produces a unifying framework for various theories, including Ericksen-Leslie, Luhiller-Rey, and Eringen's micropolar theory. In particular, we sho ...

Springer Netherlands2012

Higher order Lagrange-Poincar, and Hamilton-Poincar, reductions

Tudor Ratiu, François Gay-Balmaz

Motivated by the problem of longitudinal data assimilation, e.g., in the registration of a sequence of images, we develop the higher-order framework for Lagrangian and Hamiltonian reduction by symmetry in geometric mechanics. In particular, we obtain the r ...

2011

Geometry of nonabelian charged fluids

Tudor Ratiu, François Gay-Balmaz

The goal of this paper is to derive the Hamiltonian structure of polarized and magnetized Euler-Maxwell fluids by reduction of the canonical symplectic form on phase space, and to generalize the dynamics to the nonabelian case. The Hamiltonian function we ...

2011

The Momentum Map Representation of Images

Tudor Ratiu, François Gay-Balmaz, Martins Bruveris

This paper discusses the mathematical framework for designing methods of Large Deformation Diffeomorphic Matching (LDM) for image registration in computational anatomy. After reviewing the geometrical framework of LDM image registration methods, we prove a ...

Springer Verlag2011

The helicity and vorticity of liquid-crystal flows

François Gay-Balmaz, Cesare Tronci

We present explicit expressions of the helicity conservation in nematic liquid-crystal flows, for both the Ericksen-Leslie and Landau-de Gennes theories. This is done by using a minimal coupling argument that leads to an Euler-like equation for a modified ...

2011

Flexible beam in R3 under large overall motions and Asynchronous Variational Integrators

Tudor Ratiu, François Gay-Balmaz, Yves Weinand, François Marie Alain Demoures

The theory of discrete variational mechanics has its roots in the optimal control literature of the 1960's. The past ten years have seen a major development of discrete variational mechanics and corresponding numerical integrators, due largely to pioneerin ...

IABSE ETH Zurich2011