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.