**3.1 Spherically symmetric potential - Conservation laws** **3.2 Conservation Laws** Lagrangian for a body moving in spherically symmetric potential is given by $$ L = \frac 1 2 \mu \dot{\vec{r}^2} - V(r) \hspace{5em} (8) $$ where we have used the notation \( r = | \vec x |. \) The Lagrangian does not contain time explicitly, hence we obtain energy conservation $$ \frac 1 2 \mu \dot{\vec{x}^2} + V(r) = E \hspace{2em} (constant). \hspace{5em} (9) $$ The Lagrangian is also invariant under rotations about any axis and in particular about the coordinate axes. This gives us conservation of angular momentum. Thus we have $$ \vec L = \mu \vec x \times \vec v = constant \;\; of \;\; motion \hspace{5em} (10) $$ **3.3 Orbits lie in a plane** We shall now make use of the conservation laws to give solution of motion in a spherically symmetric potential to quadratures. Since \( \vec L = \mu \vec x \times \vec v