Answer Happy

Harmonic oscillator. The system 0 الي : = [] لیا۔ 0 is called a harmonic oscillator. (a) Find the eigenvalues, resolvent, and state transition matrix for the harmonic oscillator. Express x(t) in terms of x(0). (b) Sketch the vector field of the harmonic oscillator. (c) The state trajectories describe circular orbits, i. e., ||*(t)|| is constant. Verify this fact using the solution from part (a). (d) You may remember that circular motion (in a plane) is characterized by the velocity vector being orthogonal to the position vector. Verify that this holds for any trajectory of the harmonic oscillator. Use only the differential equation; do not use the explicit solution you found in part (a).