## Symplectic Geometric Algorithms for Hamiltonian Systems"Symplectic Geometric Algorithms for Hamiltonian Systems" will be useful not only for numerical analysts, but also for those in theoretical physics, computational chemistry, celestial mechanics, etc. The book generalizes and develops the generating function and Hamilton-Jacobi equation theory from the perspective of the symplectic geometry and symplectic algebra. It will be a useful resource for engineers and scientists in the fields of quantum theory, astrophysics, atomic and molecular dynamics, climate prediction, oil exploration, etc. Therefore a systematic research and development of numerical methodology for Hamiltonian systems is well motivated. Were it successful, it would imply wide-ranging applications. |

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### Contents

1 | |

Chapter 1 Preliminaries of Differentiable Manifolds | 39 |

Chapter 2 Symplectic Algebra and Geometry Preliminaries | 113 |

Chapter 3 Hamiltonian Mechanics and Symplectic Geometry | 165 |

Chapter 4 Symplectic Difference Schemes for Hamiltonian Systems | 187 |

Chapter 5 The Generating Function Method | 213 |

Chapter 6 The Calculus of Generating Functions and Formal Energy | 248 |

Chapter 7 Symplectic RungeKutta Methods | 277 |

Chapter 10 VolumePreserving Methods for SourceFree Systems | 443 |

Chapter 11 Contact Algorithms for Contact Dynamical Systems | 477 |

Chapter 12 Poisson Bracket and LiePoisson Schemes | 498 |

Chapter 13 KAM Theorem of Symplectic Algorithms | 549 |

Chapter 14 LeeVariational Integrator | 581 |

Chapter 15 Structure Preserving Schemes for Birkhoff Systems | 617 |

Chapter 16 Multisymplectic and Variational Integrators | 641 |

Symbol | 662 |

### Other editions - View all

Symplectic Geometric Algorithms for Hamiltonian Systems Kang Feng,Mengzhao Qin No preview available - 2010 |

### Common terms and phrases

analytic approximation called canonical coefficients Comput conservation law construct coordinate corresponding Darboux Darboux transformation defined Definition denoted derivative diffeomorphism differentiable manifold differential equations discrete dynamical systems energy equivalent Euler scheme Example exists formula GL(n Hairer Hamilton–Jacobi equation Hamiltonian function Hamiltonian system invariant Jacobian labeled Lagrangian Lemma Lie algebra Lie–Poisson linear M. Z. Qin Math Mathematics mechanics momentum multisymplectic non-singular numerical methods obtain operator order conditions order scheme P-tree phase flow phase space Poisson Poisson bracket polynomial preserves Proof properties Proposition prove R–K method rooted tree Runge–Kutta methods satisfies scalar self-adjoint smooth source-free system step subspace sufficiently small symmetric symplectic algorithms symplectic difference schemes symplectic geometry symplectic integrators symplectic manifold symplectic mapping symplectic matrix symplectic schemes symplectic structure tangent Theorem tk+1 transformation tree of order variables vector field