Classical Mechanics · Newtonian mechanics and conservation laws
A discrete spatial symmetry does not create angular-momentum conservation
Problem
A unit-mass particle moves in $U(x,y)=\alpha(x^4+y^4)$ with $\alpha=2.00\,\mathrm{J/m^4}$. The potential is invariant under a quarter-turn and under reflections. Determine whether $L_z$ is conserved by computing $\dot L_z$ at $(x,y)=(1.00,2.00)\,\mathrm m$. Show that energy is nevertheless conserved, and identify the special lines on which the instantaneous torque vanishes. Explain why a finite symmetry group does not supply a continuous conservation law.
Hint
Differentiate the quartic potential to obtain both force components.
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