Simulating Hamiltonian Dynamics

Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 7
About geometric integration......Page 11
An emphasis on methods......Page 12
How to use this book......Page 13
Computer software......Page 15
Notation......Page 16
Acknowledgements......Page 18
1 Introduction......Page 19
1.1 N-body problems......Page 20
1.2 Problems and applications......Page 21
1.3 Constrained dynamics......Page 24
1.4 Exercises......Page 26
2 Numerical methods......Page 29
2.1.1 Derivation of one-step methods......Page 31
2.1.2 Error analysis......Page 33
2.2 Numerical example: the Lennard–Jones oscillator......Page 36
2.3 Higher-order methods......Page 38
2.4 Runge–Kutta methods......Page 40
2.5 Partitioned Runge–Kutta methods......Page 43
2.6 Stability and eigenvalues......Page 45
2.7 Exercises......Page 50
3.1 Canonical and noncanonical Hamiltonian systems......Page 54
3.2.1 Linear systems......Page 57
3.2.2 Single-degree-of-freedom problems......Page 58
3.2.3 Central forces......Page 59
3.2.4 Charged particle in a magnetic field......Page 60
3.2.5 Lagrange’s equation......Page 61
3.3 First integrals......Page 62
3.4 The flow map and variational equations......Page 66
3.5 Symplectic maps and Hamiltonian flow maps......Page 70
3.5.1 One-degree-of-freedom systems......Page 73
3.5.2 The symplectic structure of phase space......Page 74
3.6 Differential forms and the wedge product......Page 79
3.7 Exercises......Page 84
4 Geometric integrators......Page 88
4.1 Symplectic maps and methods......Page 92
4.2 Construction of symplectic methods by Hamiltonian splitting......Page 94
4.2.1 Separable Hamiltonian systems......Page 96
4.2.2 A second-order splitting method......Page 98
4.3 Time-reversal symmetry and reversible discretizations......Page 99
4.3.1 Time-reversible maps......Page 100
4.3.2 Linear-reversible maps......Page 101
4.3.3 Time-reversible methods by symmetric composition......Page 102
4.4.1 Preservation of first integrals by splitting methods......Page 105
4.4.2 Implicit midpoint preserves quadratic first integrals......Page 108
4.5.1 Application to N-body systems: a molecular dynamics model problem......Page 109
4.5.2 Particle in a magnetic field......Page 112
Numerical experiment......Page 114
4.5.3 Weakly coupled systems......Page 115
4.6 Exercises......Page 117
5 The modified equations......Page 123
5.1 Forward v. backward error analysis......Page 125
5.1.1 Linear systems......Page 128
5.1.2 The nearby Hamiltonian......Page 132
5.2 The modified equations......Page 135
5.2.1 Asymptotic expansion of the modified equations......Page 136
5.2.2 Conservation of energy for symplectic methods......Page 138
Integrable systems......Page 141
Hyperbolic systems......Page 142
Adiabatic invariants......Page 144
5.3 Geometric integration and modified equations......Page 147
5.4 Modified equations for composition methods......Page 151
5.5 Exercises......Page 157
6 Higher-order methods......Page 160
6.1 Construction of higher-order methods......Page 161
6.2 Composition methods......Page 162
6.2.1 Composition methods for separable Hamiltonian systems......Page 163
6.2.2 Composition methods based on second-order symmetric methods......Page 165
6.2.3 Post-processing of composition methods......Page 166
6.3 Runge–Kutta methods......Page 167
6.3.1 Implicit Runge–Kutta methods......Page 168
6.3.2 Partitioned Runge–Kutta methods......Page 173
6.4 Generating functions......Page 177
6.5.1 Arenstorf orbits......Page 179
6.5.2 Solar system......Page 181
6.6 Exercises......Page 183
7 Constrained mechanical systems......Page 187
7.1 N-body systems with holonomic constraints......Page 188
7.2.1 Direct discretization: SHAKE and RATTLE......Page 191
7.2.2 Implementation......Page 196
7.2.3 Numerical experiment......Page 199
7.3 Transition to Hamiltonian mechanics......Page 202
7.4 The symplectic structure with constraints......Page 204
7.5 Direct symplectic discretization......Page 206
7.5.1 Second-order methods......Page 207
7.5.2 Higher-order methods......Page 208
7.6.1 Parametrization of manifolds – local charts......Page 209
7.6.2 The Hamiltonian case......Page 210
7.6.3 Numerical methods based on local charts......Page 212
7.7 Exercises......Page 213
8 Rigid body dynamics......Page 217
8.1 Rigid bodies as constrained systems......Page 219
8.1.1 Hamiltonian formulation......Page 222
8.1.2 Linear and planar bodies......Page 224
8.1.3 Symplectic discretization using SHAKE......Page 225
8.1.4 Numerical experiment: a symmetric top......Page 226
8.2 Angular momentum and the inertia tensor......Page 228
8.3 The Euler equations of rigid body motion......Page 230
8.3.1 Symplectic discretization of the Euler equations......Page 234
8.3.2 Numerical experiment: the Lagrangian top......Page 236
8.3.3 Integrable discretization: RATTLE and the scheme of MOSER AND VESELOV......Page 239
8.4 Order 4 and order 6 variants of RATTLE for the free rigid body......Page 241
8.5 Freely moving rigid bodies......Page 242
8.6.1 Euler angles......Page 246
8.6.2 Quaternions......Page 247
8.7 Exercises......Page 248
9 Adaptive geometric integrators......Page 252
9.1 Sundman and Poincaré transformations......Page 253
9.2 Reversible variable stepsize integration......Page 256
9.2.1 Local error control as a time transformation......Page 260
9.2.2 Semi-explicit methods based on generalized leapfrog......Page 261
9.2.3 Differentiating the control......Page 262
9.2.4 The Adaptive Verlet method......Page 263
9.3.1 Arclength parameterization......Page 264
9.3.2 Rescaling for the N-body problem......Page 265
9.3.3 Stepsize bounds......Page 266
9.4 Backward error analysis......Page 267
9.5 Generalized reversible adaptive methods......Page 269
9.5.1 Switching......Page 270
9.6 Poincaré transformations......Page 271
9.7 Exercises......Page 273
10 Highly oscillatory problems......Page 275
10.1.1 A single oscillatory degree of freedom......Page 276
10.1.2 Slow-fast systems......Page 279
10.1.3 Adiabatic invariants......Page 281
A linear model problem......Page 283
A nonlinear model problem......Page 284
10.2 Averaging and reduced equations......Page 287
10.3 The reversible averaging (RA) method......Page 289
A linear test problem......Page 291
A nonliner model problem......Page 293
10.4 The mollified impulse (MOLLY) method......Page 294
10.5 Multiple frequency systems......Page 297
10.6 Exercises......Page 298
11 Molecular dynamics......Page 305
11.1 From liquids to biopolymers......Page 308
11.2 Statistical mechanics from MD trajectories......Page 314
11.2.1 Ensemble computations......Page 316
11.3 Dynamical formulations for the NVT, NPT and other ensembles......Page 317
11.3.1 Coordinate transformations: the separated form......Page 320
11.3.2 Time-reparameterization and the Nosé–Hoover method......Page 321
11.4 A symplectic approach: the Nosé–Poincaré method......Page 323
11.4.1 Generalized baths......Page 326
11.4.2 Simulation in other ensembles......Page 329
11.5 Exercises......Page 331
12.1.1 The nonlinear wave equation......Page 334
12.1.2 Soliton solutions......Page 337
12.1.3 The two-dimensional rotating shallow-water equations......Page 338
12.1.4 Noncanonical Hamiltonian wave equations......Page 342
12.2 Symplectic discretizations......Page 343
12.2.1 Grid-based methods......Page 344
12.2.2 Particle-based methods......Page 348
12.3 Multi-symplectic PDEs......Page 353
12.3.1 Conservation laws......Page 355
12.3.2 Traveling waves and dispersion......Page 357
12.3.3 Multi-symplectic imtegrators......Page 359
Euler box scheme......Page 360
Preissman box scheme......Page 362
Discrete variational methods......Page 365
12.3.4 Numerical dispersion and soliton solutions......Page 367
12.3.5 Summary......Page 369
12.4 Exercises......Page 370
References......Page 375
Index......Page 392