汽轮机、转动机械振动实例

风花雪月

动力系统的保结构算法的现状与未来(一)


　　

对称的几何观点是力学研究中一个非常重要的视角，从基本的原理公式，到具体的应用，不论是早期的创立者Newton、Lagrange、Posiion、Jacobi和 Hamilton，还是后继续发展者Noether,、Rieman、Ruth、Kelvin、Cartan，特别是现代学者Arnold、Guillemin、Moser、Smal、Souriau、Sternberg、Marsden都一致强调了几何方法和力学研究的密不可分的关系，其中最重要的例子是Alexander Rowan Hamilton提出的力学定理，它使人们可以用更深的几何工具---微分流形来理解和研究刚体体系及太阳系等复杂系统的力学性质;可以用相应的Hamilton函数的对称性的概念来理解和研究诸如能量、线性动量与角动量等Hamilton系统的守恒性质。所以，结合对称的观点和几何的方法建立起了包括Lagrangian力学体系和Hamiltonian力学体系的几何力学的理论在当代科学技术的不断进步过程中起着非常重要的意义。

风花雪月

Geometric integration is the discipline concerned with numerical discretization methods that respect qualitative features of the underlying differential equations.In a sense, geometric integration attempts to unify two worlds: the qualitative world of pure mathematics, with an emphasis on geometry, and the quantitative world of numerical analysis. Typical themes in geometric integration are


　　


Symplectic methods for Hamiltonian ordinary differential equations;


　　

Multisymplectic methods and Moser-Veselov integrators for Hamiltonian partial differential equations;


　　

Lie-group methods, for differential equations evolving in Lie groups or on homogeneous manifolds;


　　

Methods for differential equations which conserve energy, volume, angular momentum or other invariants;


　　

Methods that respect laws of physics and the Stokes theorem by employing finiteelement discretization on topological complexes.


　　

Although geometric integration evolves very fast, there exist a number of excellent references which are a good place to start if you wish to learn more:
1. C.J. Budd & M.D. Piggott, "Geometric integration and its applications", Technical Report,University of Bath (2002).


　　

2. E. Hairer, Ch. Lubich & G. Wanner, "Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations", Springer Verlag, Berlin (2002).


　　

A great deal of useful material on geometric integration can be found at the website of the Geometric Integration Interest Group of FoCM.

心灯

Geometric integration is the discipline concerned with numerical discretization methods that respect qualitative features of the underlying differential equations.In a sense, geometric integration attempts to unify two worlds: the qualitative world of pure mathematics, with an emphasis on geometry, and the quantitative world of numerical analysis. Typical themes in geometric integration are


　　


Symplectic methods for Hamiltonian ordinary differential equations;


　　

Multisymplectic methods and Moser-Veselov integrators for Hamiltonian partial differential equations;


　　

Lie-group methods, for differential equations evolving in Lie groups or on homogeneous manifolds;


　　

Methods for differential equations which conserve energy, volume, angular momentum or other invariants;


　　

Methods that respect laws of physics and the Stokes theorem by employing finiteelement discretization on topological complexes.


　　

Although geometric integration evolves very fast, there exist a number of excellent references which are a good place to start if you wish to learn more:
1. C.J. Budd & M.D. Piggott, "Geometric integration and its applications", Technical Report,University of Bath (2002).


　　

2. E. Hairer, Ch. Lubich & G. Wanner, "Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations", Springer Verlag, Berlin (2002).


　　

A great deal of useful material on geometric integration can be found at the website of the Geometric Integration Interest Group of FoCM.