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Problem 30: 2D harmonic Oscillator (6 pts Setup the Hamilton-Jacobi Differntial equation in cartesian coordinates for the 2-dimensional harmonic oscillator and solve it. Find x(t) and y(t)

Answer 1

Hamiltonian for 2-D HO: Hamilton Jacobi equation: 1(as(x, y,t) 1 1 (as(x, y,t) 2(み, as(x, y,t) 1 Since there is no explicit time dependence in Hamiltonian, as at as at as at Now using separation of variables, let's assume S(x,y.t)X(x) +Y)-Et Substituting this, we get

LHS only depends on "x" and RHS on "y". Both sides can be equal only if both are equal to some constant, say, α This integral can be solved using: Here a? = 2@g/k So,

Then for y, we have 2 ky® + (Y')--e-α,-@y 2 Solution is given by: Sin 2 Where b