In the previous problem show that the conserved Noether charge associated with the symmetry 6.197 is indeed the angular momentum \(|\mathbf{r} \times \mu \mathbf{v}|\), which is naturally entirely in the \(z\) direction.

Data from previous problem

For the two-body central-force problem with a Newtonian potential, the effective two dimensional orbit dynamics can be described by the Lagrangian

1 L ). = 1 (i + 1 3 ) + 1 = 1 4 (x + j). k 1 k == r (6.195) =24 where k > 0 and we have chosen to use Cartesian coordinates. (a) Show that the equations of motion become x px=-k- * (x + y)3/2 14 = k- y (x + y2)3/2 (6.196) (b) Consider the rotation dx=ay, dy-ax, St=0 for small a. Show that this is a symmetry of the action. (6.197)