Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 1 / 30B / i / Solution

Write $z=(x,v)$ and $B(z)=(v,-\nabla V(x))$. The characteristic flow map ${\Phi }_t$ solves $\dot{z}=B(z)$. Its vector field is globally Lipschitz continuous, so Picard-Lindelöf theorem gives a unique global flow with inverse ${\Phi }_{-t}$. Also ${\mathrm{div}}_{z}B=0$.

The proposed Dirac measure is the unit mass at ${\Phi }_t(z_0)$. For every compactly supported smooth test function $\psi$,This is the weak formulation of $f_t+\mathrm{div}(Bf)=0$, equal to the Liouville equation because $\mathrm{div}B=0$. The initial unit mass is recovered at $t=0$. The characteristic equations are precisely $\dot{\hat{x}}=\hat{v}$, $\dot{\hat{v}}=-\nabla V(\hat{x})$; the time-derivative dots, lost in the converted text, are present in the PDF.