Classical Dynamics of Particles and Systems · Problem 7-35

Central-Force Motion — Marion & Thornton

Physics — Mechanics · liouville-theorem-phase-space

Problem

Four particles are directed upward in a uniform gravitational field with the following initial conditions:

\begin{aligned} &\textbf{(1)}\; z(0) = z_0; \quad &p_z(0) = p_0\\ &\textbf{(2)}\; z(0) = z_0 + \Delta z_0; \quad &p_z(0) = p_0\\ &\textbf{(3)}\; z(0) = z_0; \quad &p_z(0) = p_0 + \Delta p_0\\ &\textbf{(4)}\; z(0) = z_0 + \Delta z_0; \quad &p_z(0) = p_0 + \Delta p_0 \end{aligned}

Show by direct calculation that the representative points corresponding to these particles always define an area in phase space equal to

\Delta z_0\,\Delta p_0

. Sketch the phase paths, and show for several times

t > 0

the shape of the region whose area remains constant.