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Why isn't the Gear predictor-corrector algorithm for integration of the equations of motion symplectic?

+ 12 like - 0 dislike
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Okumura et al., J. Chem. Phys. 2007 states that the Gear predictor-corrector integration scheme, used in particular in some molecular dynamics packages for the dynamics of rigid bodies using quaternions to represent molecular orientations, is not a symplectic scheme. My question is: how can one prove that? Does it follow from the fact that the Gear integrator is not time-reversible (and if so, how can one show that)? If not, how do you prove that an integration scheme is not symplectic?

This post has been migrated from (A51.SE)
asked Sep 14, 2011 in Theoretical Physics by F'x (175 points) [ no revision ]

1 Answer

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Take a look at the notes on lectures 1 and 2 of Geometric Numerical Integration found here. Quoting from Lecture 2

A numerical one-step method $y_{n+1} = \Phi_h(y_n)$ is called symplectic if, when applied to a Hamiltonian system, the discrete flow $y \mapsto \Phi_h(y)$ is a symplectic transformation for all sufficiently small step sizes.

From your link you have $$x(t+h) = x(t) + h \dot{x}(t) + h^2 \left\{\frac{3}{24}f(t+h) +\frac{10}{24}f(t) -\frac{1}{24}f(t-h) \right\}$$ and $$\dot{x}(t+h) = \frac{x(t+h) - x(t)}{h} + h \dot{x}(t) + h \left\{\frac{7}{24}f(t+h) +\frac{6}{24}f(t) -\frac{1}{24}f(t-h) \right\}$$

Now take $\omega(\xi,\eta) = \xi^T J \eta$ where $J = \left(\begin{array}{cc} 0 & \mathbb{I} \\ \mathbb{I} & 0 \end{array}\right)$. Then the integrator is symplectic if and only if $\omega(x(t),\dot{x}(t))=\omega(x(t+h),\dot{x}(t+h))$ for sufficiently small $h$.

All that you need to do is to fill in the values of $x(t+h)$ and $\dot{x}(t+h)$ from the integrator, and show that this condition does not hold.

This post has been migrated from (A51.SE)
answered Sep 15, 2011 by Joe Fitzsimons (3,555 points) [ no revision ]
Dear Joe Fitzsimons, you should insert a minus sign into the matrix $J$ representing the symplectic form $\omega$.

This post has been migrated from (A51.SE)

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