Researches

Main Research Topics

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(i) Dynamical Systems on Dirac Manifolds

Mechanical systems subject to nonholonomic constraints or degenerate Lagrangian systems — known collectively as constrained mechanical systems — evolve under geometric structures called Dirac structures, which generalize symplectic and Poisson structures. Our research investigates the geometric and variational structures of Lagrangian and Hamiltonian systems on Dirac manifolds. Applications include multibody systems, water molecules, and continua such as viscous fluids. Parts of this work have been carried out in collaboration with research institutions including the California Institute of Technology, École Normale Supérieure (ENS) in Paris, Ghent University, École Polytechnique Fédérale de Lausanne (EPFL), and Shanghai Jiao Tong University.
 

(ii) Structure-Preserving Modeling and Discrete Variational Formulations

For numerical simulations of conservative systems—such as planetary motion or molecular dynamics—numerical schemes with good energy-preserving properties, such as symplectic integrators, are essential. Symplectic integrators preserve the underlying symplectic structure, and when a Lagrangian is given, discrete variational integrators provide a systematic method for constructing such schemes. However, in nonholonomic systems, symplectic structure is no longer preserved, and new structure-preserving methods are required. Our work develops a new framework for structure-preserving modeling and discrete variational formulations based on discrete Dirac systems. Some aspects of this research are conducted in collaboration with Keio University.
 

(iii) Geometric Theory of Nonequilibrium Thermodynamics and Its Applications

We study variational formulations for nonequilibrium thermodynamic systems as an extension of Lagrangian mechanics based on Hamilton’s principle. On an extended thermodynamic configuration space, we incorporate nonlinear and nonholonomic constraints derived from entropy balance, along with associated variational constraints, and show that the evolution equations of nonequilibrium thermodynamic systems can be formulated using a generalized Lagrange–d’Alembert principle. Recent work focuses on developing a geometric theory for open nonequilibrium thermodynamic systems, in which both matter and heat exchange occur with the environment. We are also investigating discrete variational methods and geometric structures induced by Dirac formulations. This research is being conducted in collaboration with École Normale Supérieure (ENS) in Paris.

Research Topics for Undergraduate, Master’s, and Doctoral Theses

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The main research themes currently offered for thesis supervision are listed below. Students select related topics from these areas according to their interests, and appropriate research problems are assigned accordingly. Additional themes not described here in detail are also available.
 

(i) Chaotic Mixing and Lagrangian Coherent Structures in Rayleigh–Bénard Convection

Convection is a fluid phenomenon driven by differences in fluid temperature, which induce variations in density or surface tension and generate motion through buoyancy forces. Such processes play essential roles in large-scale geophysical phenomena, including ocean currents and atmospheric circulation. Our laboratory studies complex flow phenomena that emerge when small perturbations are introduced into Rayleigh–Bénard convection—a cellular flow pattern that forms when a thin fluid layer is subjected to a slight temperature difference between its top and bottom surfaces.
Although Eulerian observations reveal only minor fluctuations in the velocity field, a Lagrangian perspective uncovers complex transport phenomena known as chaotic mixing. In previous work, we modeled steady Rayleigh–Bénard convection as a near-integrable system, treating streamlines associated with steady convection cells as Hamiltonians under perturbed velocity fields. Through this approach, we elucidated transport mechanisms using Lagrangian Coherent Structures (LCS). Furthermore, by analyzing Poincaré maps, we demonstrated that stable regions composed of KAM tori and unstable regions forming a chaotic sea correspond topologically to the invariant structures identified as LCS in long-time integrations. Currently, we are developing a two-dimensional Rayleigh–Bénard convection experimental apparatus. Using this setup, we visualize flow fields with Particle Image Velocimetry (PIV) and identify Lagrangian Coherent Structures based on observed velocity data.
 

(ii) Unsteady Behavior of Multiphase Flows and Mechanisms of Shock-Wave Propagation

Cavitation is a long-studied unsteady phenomenon in multiphase flows consisting of liquid and vapor phases. It occurs when liquid flowing at very high speed locally drops below its saturated vapor pressure, leading to the rapid formation and collapse of numerous vapor bubbles. Alongside turbulence, it remains one of the major unresolved problems in fluid mechanics. In particular, cavitation occurs when fluid strikes the propellers of fluid machinery such as pumps or screws, causing noise, mechanical damage, and erosion. A well-known example is the failure of the H-II rocket No. 8, where cavitation in the liquid-hydrogen turbopump inducer damaged the impeller, leading to engine shutdown. On the other hand, cavitation has recently gained attention for positive applications, including medical jet scalpels and environmental purification. In our laboratory, we investigate cloud cavitation that occurs when water ablated by a laser—used in medical jet scalpels—is injected at high speed from a nozzle into a quiescent fluid. Experiments are conducted using a high-speed video camera (up to 1.3 million FPS), applying synchronized Schlieren and shadowgraph imaging to observe the unsteady behavior of bubble clouds and shock-wave propagation during bubble collapse. Complementary numerical studies are also performed using computational fluid dynamics (CFD).

 

(iii) Molecular Dynamics Study of Fundamental Processes in Cavitation

Cavitation is typically modeled from a macroscopic perspective as a multiphase fluid phenomenon governed by the Navier–Stokes equations. However, understanding the fundamental processes involved in cavitation inception requires a microscopic viewpoint at the molecular level.
In this research, we aim to elucidate the mechanism of bubble nucleation by numerically reproducing phase-transition processes using a monoatomic model in which intermolecular forces are described by the Lennard–Jones potential. Numerical experiments are conducted both for cases with and without non-condensable gases possessing soft-core potentials, allowing us to investigate their influence on cavitation inception. Through these simulations, we seek to uncover the fundamental processes underlying cavitation nucleation that remain insufficiently understood in conventional macroscopic models.

 

(iv) Dynamics of Multi-Body Systems and Trajectory Design

• Modeling and Numerical Analysis of Multi-Body Dynamics
  In applications involving industrial robots, vehicles, satellites with flexible antennas, space robots, and fish locomotion, modeling multi-body dynamics (MBD)—systems composed of many interconnected bodies—requires methods that explicitly account for the connection structure of the system. Our laboratory has been studying MBD modeling as an interconnected system and numerical analysis using large-scale sparse tableau methods since the late 1980s. Recent efforts focus on geometric approaches based on gauge theory and Dirac structures, structure-preserving modeling techniques derived from these frameworks, and numerical solution methods using discrete variational integrators.
• Trajectory Design for Spacecraft Using the Restricted Three-Body Problem
  For small spacecraft used in deep-space missions—such as the asteroid explorer Hayabusa—low-energy trajectory design is essential. Our laboratory has refined the “tube dynamics” method originally developed by Professor Jerrold E. Marsden’s group at the California Institute of Technology in collaboration with researchers at the Jet Propulsion Laboratory. We treat the planar circular restricted four-body problem as a perturbed version of the planar circular restricted three-body problem and extract invariant structures (tubes) as Lagrangian Coherent Structures (LCS), represented as ridges of the FTLE field. This approach enables a rigorous modeling of the four-body system as a concatenation of perturbed three-body subsystems. Using the improved tube-dynamics framework, we successfully designed spacecraft trajectories in the Earth–Moon–Sun–spacecraft four-body system, achieving transfers from low Earth orbit to low lunar orbit and establishing a method for identifying optimal trajectories. Current work aims to extend this theory to spatial four-body systems. Part of this research is conducted jointly with Professor Shane Ross at Virginia Tech.
• Nonlinear Control and Trajectory Generation for Nonholonomic Systems
  Vehicles such as mobile robots, bicycles, and automobiles—expected to play increasingly important roles in autonomous transportation, disaster response, nuclear facility operations, and planetary exploration—are mechanical systems governed by nonholonomic dynamics, defined by non-integrable constraint distributions on the configuration space.
  Our laboratory studies nonlinear control and trajectory generation for a two-wheeled mobile robot, focusing on posture stabilization and nonholonomic trajectory control using methods based on system passivity, such as artificial potential approaches and energy-shaping techniques.