Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 8D (Classical Dynamics)

For smooth functions $f,g$ on the six-dimensional phase space with canonical coordinates $(q_i,p_i)$, the Poisson bracket is

Hamilton's equations and the multivariable chain rule giveThus the Hamiltonian conservation law gives

For the phase-space dilation generator $D=q\cdot p$,Therefore the condition $\{f,D\}=0$ is the first-order equation

If $f(\lambda q,{\lambda }^{-1}p)=f(q,p)$ for every positive real $\lambda$, differentiate with respect to $\lambda$ at $\lambda =1$. The chain rule gives exactlyso every smooth scale-invariant $f$ satisfies the constraint. The same conclusion for negative $\lambda$ follows wherever the stated invariance is defined; differentiation only needs the positive component containing $\lambda =1$.