Consider a particle of mass m moving under the influence of a central potential \(V(r) = \frac{1}{2}k r^2 \ \), where \( r = \sqrt{x^2 + y^2} \ \). The Lagrangian for this system is given by:
\(L = \frac{1}{2} m (\dot{x}^2 + \dot{y}^2) - \frac{1}{2} k (x^2 + y^2) \ \)
We know that Noether's theorem links symmetries to conserved quantities. Assume that the system exhibits rotational symmetry around the origin. Use this information to determine the conserved quantity associated with this symmetry.
Calculate the conserved quantity (angular momentum) in this system if the initial conditions are:
1
\(3 \text{kg m}^2/\text{s} \ \)
2
\( 6 \text{kg m}^2/\text{s} \ \)
3
\( 9 \text{kg m}^2/\text{s} \ \)
4
\(5 \text{kg m}^2/\text{s} \ \)