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.
Joris Vankerschaver (Ghent University Global Campus)
Masahito Watanabe (Waseda University)
Baige Xu (Kobe University)
Talks in Minisymposium :
[03436] Noether’s conservation laws via the modified formal Lagrangians
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.
[04618] Symmetries and bifurcations of resonant periodic orbits in perturbed Rayleigh-Bénard convection
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.
[04689] Geometric Integrators for Neural Symplectic Forms
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
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.
