To make the content of the book more systematic, this book mainly briefs some related basic knowledge reported by other monographs and papers about geometric mechanics. The main content of this book is based on the last 20 years’ jobs of the authors. All physical processes can be formulated as the Hamiltonian form with the energy conservation law as well as the symplectic structure if all dissipative effects are ignored. On the one hand, the important status of the Hamiltonian mechanics is emphasized. On the other hand, a higher requirement is proposed for the numerical analysis on the Hamiltonian system, namely the results of the numerical analysis on the Hamiltonian system should reproduce the geometric properties of which, including the first integral, the symplectic structure as well as the energy conservation law.
Author(s): Weipeng Hu, Chuan Xiao, Zichen Deng
Publisher: Springer
Year: 2023
Language: English
Pages: 539
City: Singapore
Preface
Contents
About the Authors
1 Introduction
1.1 The Vitality of Geometric Mechanics
1.1.1 Start with the Euler Differential Scheme for the Linear Harmonic Oscillator
1.1.2 Investigation and Improvement of the Störmer–Verlet Scheme for the Mathematical Pendulum Model
1.2 From Lagrangian Mechanics to Hamiltonian Mechanics
1.2.1 Lagrangian Mechanics
1.2.2 Hamiltonian Mechanics
1.3 The Sole of Geometric Mechanics-Geometric Integration
References
2 Symplectic Methods for a Finite-Dimensional System
2.1 Mathematical Formulations for the Symplectic Method
2.2 Typical Symplectic Discretization Methods
2.2.1 Symplectic Runge–Kutta Methods
2.2.2 Splitting and Composition Methods
2.3 Applications of Symplectic Methods in Mechanics
2.3.1 Symplectic Precise Integration of Folding and Unfolding Processes of Undercarriage
2.3.2 Symplectic Runge–Kutta Method for Aerospace Dynamics Problems
References
3 Multi-symplectic Method for an Infinite-Dimensional Hamiltonian System
3.1 Start with the Wave Equation
3.2 Mathematical Foundation of Multi-symplectic Theory
3.2.1 Symplectic and Anti-symplectic Involutions and Reversibility
3.2.2 Decomposing Impulse and Energy Conservation
3.2.3 Multi-symplectic Structure and Conservation Laws
3.2.4 Hamiltonian Functionals
3.2.5 A More Popular Version of Multi-symplectic Theory
3.3 Typical Discretization Approaches for the Multi-symplectic Method
3.3.1 The Explicit Midpoint Scheme
3.3.2 The Euler Box Scheme
3.4 Applications of Multissymplectics in Wave Propagation
3.4.1 Multi-symplectic Method for the Membrane Free Vibration Equation
3.4.2 Multi-symplectic Method for Generalized (2 + 1)-Dimensional KdV–mKdV Equation
3.4.3 Multi-symplectic Runge–Kutta Methods for the Landau–Ginzburg–Higgs Equation
3.4.4 Multi-symplectic Runge–Kutta Methods for the Generalized Boussinesq Equation
3.4.5 Multi-symplectic Method to Simulate the Soliton Resonance of the (2 + 1)-Dimensional Boussinesq Equation
3.4.6 Multi-symplectic Method for Peakon–Antipeakon Collision of Quasi-Degasperis–Procesi Equation
3.4.7 Multi-symplectic Analysis of the Gaussian Solitary Wave Solution for the Logarithmic–KdV Equation
References
4 Dynamic Symmetry Breaking and Generalized Multi-symplectic Method for Non-conservative Systems
4.1 Introduction to Dynamic Symmetry Breaking
4.2 From a Multi-symplectic Integrator to a Generalized Multi-symplectic Integrator
4.3 Symmetry Breaking of an Infinite-Dimensional Dynamic System
4.4 Illustrating the Structure-Preserving Properties of the Generalized Multi-symplectic Method Applied in Wave Propagation
4.4.1 An Implicit Difference Scheme Focusing on the Local Conservation Properties of the Burgers Equation
4.4.2 Competition between Geometric Dispersion and Viscous Dissipation in Wave Propagation of the KdV-Burgers Equation
4.4.3 Generalized Multi-symplectic Discretization for the Compound KdV–Burgers Equation
4.4.4 Almost Structure-Preserving Analysis for Weakly Linear Damping Nonlinear Schrödinger Equation with Periodic Perturbation
References
5 Structure-Preserving Analysis of Impact Dynamic Systems
5.1 Introduction to Progress on Impact Dynamics
5.1.1 Tubes and Columns Under Axial Loading
5.1.2 Beams and Plates Under Transverse Loading
5.1.3 Sandwich Structures Under Impact or Blast Loading
5.1.4 Cellular Materials Under Impact Loading
5.2 Energy Loss in the Pulse Detonation Engine Due to Fuel Viscosity
5.3 Wave Propagation in a Non-homogeneous Centrosymmetric Damping Plate Subjected to Impact Series
5.4 Wave Propagation in a Non-homogeneous Asymmetric Circular Plate Subjected to Impact Series
References
6 Structure-Preserving Analysis of the Dynamics of Micro/Nano Systems
6.1 Chaos in an Embedded Single-Walled Carbon Nanotube
6.2 Energy Dissipation of the Damping Cantilevered Single-Walled Carbon Nanotube Oscillator
6.3 Chaos in Embedded Fluid-Conveying Single-Walled Carbon Nanotubes Under a Transverse Harmonic Load Series
6.4 Chaotic Region of Elastically Restrained Single-Walled Carbon Nanotubes
6.5 Axial Dynamic Buckling Analysis of Embedded Single-Walled Carbon Nanotube by a Complex Structure-Preserving Method
References
7 Structure-Preserving Analysis of Astrodynamics Systems
7.1 Coupling Dynamic Behaviors of a Spatial Flexible Beam with Weak Damping
7.2 Nonspherical Perturbation of the Dynamic Behaviors of a Spatial Flexible Damping Beam
7.3 Minimum Control Energy of the Spatial Beam with an Assumed Attitude Adjustment Target
7.4 Energy Dissipation/Transfer and Stable Attitude of the Spatial On-Orbit Tethered System
7.5 Internal Resonance of a Flexible Beam in a Spatial Tethered System
7.6 Coupling Dynamic Behaviors of a Flexible Stretching Hub-Beam System
7.7 Vibration and Elastic Wave Propagation in a Spatial Flexible Damping Panel Attached to Four Special Springs
References
