A particle moving in a horizontal plane is attached to the origin by a spring with spring constant k and zero natural length. a) write the Lagrangian in polar coordinates r, 9 b) nd the generalized momenta, and write the Hamiltonian in terms of the generalized coordinates and momenta (eliminate the generalized velocities from your expression for H, trade them for generalized momenta). c) Using Hamilton's equations and the Hamiltonian above, write the four Hamilton's equations of motion 309 midterm spring 2022.nb I d) Identify any ignorable coordinates and the conserved quantities associated with them