Find an algebraic equation which must be satisfied by all possible equilibrium points am. Hint: All solutions. have the form {on = Xo. for some dimensionless scalar X- That scalar must satisfy a certain cubic equation. You don't need to solve that equation. The Hessian of L" may be shown to be are so aux as H-- ' :11..qu \"[31] I} + r'li'ffnlf o|3 |:c {1|2 _ 51.1 The eigenvectors of IHarn} : IflIXo] are :1 and any vectors orthogonal to :1. Find the eigenvalues associated with these eigenvectors by directly using that it 1: is an eigenvector and A is a corresponding eigenvalue, HI: = Are. You may leave X as arbitrary in your answers. Consider a particle with mass m. and charge q which can move in three dimensions and is attached to an ideal spring which is anchored to the origin. The spring has equilibrium length zero and spring constant If. There is also a charge Q which is fixed at the location a