Abstract : Geometric mechanics is a research branch of modern differential geometric formulation for Lagrangian and Hamiltonian systems in mechanics, including related dynamical systems theory such as Hamiltonian bifurcations. Through the breakthrough of the famous symplectic reduction by Marsden and Weinstein and others, the scope of the field has expanded from mathematics and physics towards numerical and data sciences. This minisymposium describes the overviews on the main streams of geometric mechanics, including geometric methods in fluids and thermodynamics, and bridges many contemporary related topics such as dynamical systems, numerical simulations, information geometry, and data sciences from the view point of geometric mechanics.
[05487] Feedback Integrators for Mechanical Systems with Holonomic Constraints
Format : Talk at Waseda University
Author(s) :
Joris Vankerschaver (Ghent University Global Campus)
Dong Eui Chang (Korea Advanced Institute of Science and Technology)
Matthew Perlmutter (Universidade Federal de Minas Gerais)
Abstract : We present a straightforward method for the numerical integration of the equations of motion of mechanical systems with holonomic constraints, to produce numerical trajectories that remain in the constraint set and preserve the values of constrained quantities. Our method only requires changes to the vector field and can be used in conjunction with any numerical integration scheme. This talk will describe the theoretical foundations of the method and compare its performance with other integrators.
[04689] Geometric Integrators for Neural Symplectic Forms
Format : Talk at Waseda University
Author(s) :
Yuhan Chen (Kobe University)
Takashi Matsubara (Osaka University)
Takaharu Yaguchi (Kobe University)
Abstract : The neural symplectic form is a deep physical model for general Hamiltonian systems in arbitrary coordinates. A primal application of deep physical models is physical simulations; however, when general numerical integrators are used for discretization, the physical properties are destroyed. Structure-preserving numerical methods are effective to address this problem. Typical integrators are symplectic integrators, which can be derived as variational integrators. In this study, we show that variational integrators are available for neural symplectic forms.
[04691] Structure-Preserving Learning for GENERIC systems
Format : Talk at Waseda University
Author(s) :
Baige Xu (Kobe University)
Yuhan Chen (Kobe University)
Takashi Matsubara (Osaka University)
Takaharu Yaguchi (Kobe University)
Abstract : GENERIC (general equation for the non-equilibrium reversible-irreversible coupling) formulation is a key theory of non-equilibrium thermodynamics, with systems described by it having a unique geometric structure. We propose a neural network model that infers the equations of motion from observed data while preserving this geometric structure by applying the neural symplectic forms and introducing an equivalence relation between the models.
[05507] A discretization of Dirac structures and Lagrange-Dirac dynamical systems
Format : Talk at Waseda University
Author(s) :
Hiroaki Yoshimura (Waseda University)
Linyu Peng (Keio University)
Abstract : Various physical systems such as circuits, nonholonomic systems, as well as nonequilibrium thermodynamic systems can be formulated as a Lagrange-Dirac dynamical systems, in which an induced Dirac structure that is constructed by the canonical two-form and a distribution plays an essential role. In this talk, we show a discretization of such an induced Dirac structure and then we demonstrate how the associated discrete Lagrange-Dirac systems can be developed ,consistently with the discrete Lagrange-d’Alembert principle.
[03436] Noether’s conservation laws via the modified formal Lagrangians
Format : Talk at Waseda University
Author(s) :
Linyu Peng (Keio University)
Abstract : Noether’s theorem establishes a one-to-one correspondence between variational symmetries and conservation laws of variational differential equations. In this talk, we extend Noether’s theorem to general differential equations by defining the modified formal Lagrangians. This allows us to construct conservation laws of non-variational differential equations using their symmetries. Worked examples will be provided.
[05483] Harmonic exponential families on homogeneous spaces
Format : Talk at Waseda University
Author(s) :
Koichi Tojo (RIKEN AIP)
Taro Yoshino (The University of Tokyo)
Abstract : Exponential families play an significant role in the field of information geometry and are useful in Bayesian inference. Widely used families of probability measures, such as normal and gamma distributions can be considered as exponential families on homogeneous spaces with symmetry. Based on this observation, we presented a method to construct exponential families with symmetry using representation theory. In this talk, we will explain the method and its properties, illustrating them with examples.
[04618] Symmetries and bifurcations of resonant periodic orbits in perturbed Rayleigh-Bénard convection
Format : Talk at Waseda University
Author(s) :
Masahito Watanabe (Waseda University)
Hiroaki Yoshimura (Waseda University)
Abstract : Rayleigh-Bénard convection is natural convection that appears in a fluid layer with heated bottom and cooled top planes. When Rayleigh number is set just above a critical number, velocity fields of Rayleigh-Bénard convection may oscillate slightly. In such oscillatory convection both stable and chaotic fluid transport may occur. In this talk we explore the global structures of periodic fluid transport in two-dimensional perturbed Rayleigh-Bénard convection in the perspectives of resonance, symmetry, and bifurcation.
[05475] Geometric models in hydrodynamics
Format : Talk at Waseda University
Author(s) :
Tudor Stefan Ratiu (Shanghai Jiao Tong UniversityShanghai Jiao Tong University )
Abstract : In this talk the classical geometric formulation of
hydrodynamics will be extended by the use of a momentum map
with values in Cheeger-Simons differential characters. It will be
shown that this extended momentum map admits topological
conservation laws. Clebsch variables will be introduced for which
the helicity takes integer values.
