Question 8.6: A projectile moving on the vertical xy-plane in a uniform gr......
A projectile moving on the vertical xy-plane in a uniform gravitational field is described by the Hamiltonian H = (p^{2}_{x} +p^{2}_{y} )/2m + mgy. Show that F = y −tp_{y}/m−gt²/2 and G = x−tp_{x}/m are constants of the motion and obtain new constants of the motion by means of Poisson’s theorem.
We have
\frac{dF}{dt}=\left\{F,H\right\}+ \frac{\partial F}{\partial t}=\left\{y,H\right\} – \frac{t}{m}\left\{p_{y},H\right\}-\frac{p_{y}}{m}-gt =\frac{p_{y}}{m}-\frac{t}{m} \left(-mg\right)- \frac{p_{y}}{m}-gt =0 (8.118)
and, similarly,
\frac{dG}{dt}=\left\{G,H\right\}+ \frac{\partial G}{\partial t}=\left\{x,H\right\} – \frac{t}{m}\left\{p_{x},H\right\}-\frac{p_{x}}{m} =\frac{p_{x}}{m}-0-\frac{p_{x}}{m} =0. (8.119)
The Poisson bracket {F,G} ≡ 0 gives a trivial constant of the motion. But H is a constant of the motion since it does not explicitly depend on time, so
M = {F,H} = {y,H} −\frac{t}{m}\left\{p_{y},H\right\}=\frac{p_{y}}{m}+gt (8.120)
as well as
N = {G,H} = {x,H} −\frac{t}{m}\left\{p_{x},H\right\}=\frac{p_{x}}{m} (8.121)
are new constants of the motion. Note that the five constants of the motion F, G, M, N and H cannot be independent. (Why not?)