[UPSC CSE 2026] Central Force Motion and Gravitation (Q62, 6+9=15M)

Consider a particle of mass m in two dimensions experiencing a central force \vec{F} = -k\vec{r}, where k is a positive constant and \vec{r} is the radius vector of the particle relative to the force center.

(i) What is the angular momentum \vec{J} of the particle relative to the force center? Show that \vec{J} is conserved.

(ii) Write down the system of equations of motion in two dimensions in polar coordinates. Reduce this system to a one-equation problem and find the equation for the effective potential energy U_{\text{eff}}.