umerically, there would no
need to do any more randomization per transition than generating a
random momentum vector. Instead, what is done in practice to account
for numerical errors during integration is to apply a Metropolis
acceptance step, where the probability of keeping the proposal
<span class="math inline">\((\rho^{*}, \theta^{*})\)</span> generated by transitioning from
<span class="math inline">\((\rho, \theta)\)</span> is</p>
<p><span class="math display">\[
\min \!
\left(
1,
\ \exp \! \left( H(