Simulating Hamiltonian DynamicsGeometric integrators are time-stepping methods, designed such that they exactly satisfy conservation laws, symmetries or symplectic properties of a system of differential equations. In this book the authors outline the principles of geometric integration and demonstrate how they can be applied to provide efficient numerical methods for simulating conservative models. Beginning from basic principles and continuing with discussions regarding the advantageous properties of such schemes, the book introduces methods for the N-body problem, systems with holonomic constraints, and rigid bodies. More advanced topics treated include high-order and variable stepsize methods, schemes for treating problems involving multiple time-scales, and applications to molecular dynamics and partial differential equations. The emphasis is on providing a unified theoretical framework as well as a practical guide for users. The inclusion of examples, background material and exercises enhance the usefulness of the book for self-instruction or as a text for a graduate course on the subject. |
Contents
Numerical methods | 15 |
5 | 25 |
6 | 27 |
Hamiltonian mechanics | 36 |
The modified equations | 105 |
3 | 119 |
Higherorder methods | 143 |
7 | 195 |
Highly oscillatory problems | 257 |
Highly oscillatory problems | 258 |
Molecular dynamics | 287 |
Molecular dynamics | 292 |
Hamiltonian PDEs | 316 |
Hamiltonian PDEs | 320 |
36 | 338 |
48 | 344 |
Common terms and phrases
adiabatic invariant angular momentum applied approach approximation backward error analysis canonical center of mass chapter composition methods computed conservation law consider constant constraints coordinate defined denote derived differential equations discretization discussion eigenvalues energy error equations of motion Euler Euler-B Euler's method example explicit exponential flow map formulation function geometric given Hamiltonian system Hence higher-order implicit midpoint initial conditions introduce iteration Kepler problem Lagrangian linear LTS method modified equations modified Hamiltonian molecular dynamics momenta multi-symplectic N-body nonlinear numerical method numerical solution obtain orbit parameters particle phase space planar potential energy qn+1 quaternions RATTLE RK methods rotation matrix Runge-Kutta methods scheme second-order Section simulation solve splitting method Störmer-Verlet method structure matrix Sundman transformation symmetric symplectic integration symplectic map symplectic method tensor timestep trajectory unconstrained system variable stepsize vector field velocity wedge product
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Page vi - elegance" in its "architectural," structural makeup. Ease in stating the problem, great difficulty in getting hold of it and in all attempts at approaching it, then again some very surprising twist by which the approach, or some part of the approach, becomes easy, etc. Also, if the deductions are lengthy or complicated, there should be some simple general principle involved, which "explains" the complications and detours, reduces the apparent arbitrariness to a few simple guiding motivations, etc.
Page vi - One expects a mathematical theorem or a mathematical theory not only to describe and to classify in a simple and elegant way numerous and a priori disparate special cases. One also expects "elegance" in its "architectural,