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Hamiltonian Map

Consider a 1-D Hamiltonian Map of the form

\begin{displaymath}
H(p,q) = {\textstyle{1\over 2}}p^2+V(q),
\end{displaymath} (1)

which satisfies Hamilton's Equations
$\displaystyle \dot q$ $\textstyle =$ $\displaystyle {\partial H\over\partial p}$ (2)
$\displaystyle \dot p$ $\textstyle =$ $\displaystyle -{\partial H\over\partial q}.$ (3)

Now, write
\begin{displaymath}
{\dot q}_i ={(q_{i+1}-q_i)\over\Delta t},
\end{displaymath} (4)

where
$\displaystyle q_i$ $\textstyle =$ $\displaystyle q(t)$ (5)
$\displaystyle q_{i+1}$ $\textstyle =$ $\displaystyle q(t+\Delta t).$ (6)

Then the equations of motion become
$\displaystyle q_{i+1}$ $\textstyle =$ $\displaystyle q_i+p_i\Delta t$ (7)
$\displaystyle p_{i+1}$ $\textstyle =$ $\displaystyle p_i-\Delta t\left({\partial V\over\partial q_i}\right)_{q=q_i}.$ (8)

Note that equations (7) and (8) are not Area-Preserving, since
\begin{displaymath}
{\partial(q_{i+1},p_{i+1})\over\partial(q_i,p_i)}
= \left\v...
...
= 1+(\Delta t)^2 {\partial^2 V\over\partial{q_i}^2} \not= 1.
\end{displaymath} (9)

However, if we take instead of (7) and (8),
$\displaystyle q_{i+1}$ $\textstyle =$ $\displaystyle q_i+p_i\Delta t$ (10)
$\displaystyle p_{i+1}$ $\textstyle =$ $\displaystyle p_i-\Delta t\left({\partial V\over\partial q_i}\right)_{q=q_{i+1}}$ (11)


$\displaystyle {\partial(q_{i+1},p_{i+1})\over\partial(q_i,p_i)}$ $\textstyle =$ $\displaystyle \left\vert\begin{array}{cc}1 & -\Delta t {\partial\over\partial q...
...al V\over \partial q}\right)}_{q=q_{i+1}}\\  \Delta t & 1\end{array}\right\vert$  
  $\textstyle =$ $\displaystyle 1+(\Delta t)^2 {\partial^2 V\over\partial{q_i}^2} = 1,$ (12)

which is Area-Preserving.




© 1996-9 Eric W. Weisstein
1999-05-25