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

+ 12 like - 0 dislike
109 views

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)

+ 14 like - 0 dislike

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 (3,555 points)
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)

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOver$\varnothing$lowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.