Divergence of a Gaussian thermostat
= Divergence of a Gaussian thermostat
{title2=$\operatorname{div}_{\mu}F=-\langle E,v\rangle$}
For $F=X+\lambda V$, the canonical fields $X,V$ preserve <Liouville volume of a surface geodesic flow>, so the divergence is $V\lambda$. For a <Gaussian thermostat>, $\lambda=\langle E,iv\rangle$ and $V(iv)=-v$, giving the stated sign. If $E=\nabla f$, then the density $e^{f\circ\pi}$ makes this divergence zero.