Gans J, Shalloway D
Biophysics Program, Department of Molecular Biology and Genetics, Cornell University, Ithaca, New York 14853, USA.
Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 2000 Apr;61(4 Pt B):4587-92. doi: 10.1103/physreve.61.4587.
It is often assumed, when interpreting the discrete trajectory computed by a symplectic numerical integrator of Hamilton's equations in Cartesian coordinates, that velocity is equal to the momentum divided by the physical mass. However, the "shadow Hamiltonian" which is almost exactly solved by the symplectic integrator will, in general, induce a nonlinear relationship between velocity and momentum. For the (symplectic) momentum- and midpoint-momentum-Verlet algorithms, the "shadow mass" that relates velocity and momentum is momentum independent only for a quadratic potential and, even in this case, differs from the physical mass. Thus, naively assuming the standard velocity-momentum relationship leads to inconsistencies and unnecessarily inaccurate estimates of velocity-dependent quantities. As examples, we calculate the shadow Hamiltonians for the momentum- and midpoint-momentum-Verlet solutions of the multidimensional harmonic oscillator, and show how their velocity-momentum relationships depend on the time step. Of practical importance is the conclusion that, to gain the full advantage of symplecticity, velocities derived from interpolated positions, rather than conventional velocity-Verlet velocities, should be used to compute physical properties.
在笛卡尔坐标系中解释由哈密顿方程的辛数值积分器计算出的离散轨迹时，人们常常假定速度等于动量除以物理质量。然而，辛积分器几乎能精确求解的“影子哈密顿量”通常会在速度和动量之间引入非线性关系。对于（辛）动量-Verlet算法和中点动量-Verlet算法，关联速度和动量的“影子质量”仅在二次势的情况下与动量无关，即便如此，它仍与物理质量不同。因此，天真地假定标准的速度-动量关系会导致不一致性，并对与速度相关的量产生不必要的不准确估计。例如，我们计算多维谐振子的动量-Verlet解和中点动量-Verlet解的影子哈密顿量，并展示它们的速度-动量关系如何依赖于时间步长。具有实际重要性的结论是，为了充分利用辛性，应使用从插值位置导出的速度而非传统的速度-Verlet速度来计算物理性质。
