Question

5. Consider the following time-dependent Lagrangian for a system with one degree of freedom , (10) where 8, m and k are fixed real constants greater than zero. (total 10 points) (a) Write down the Euler-Lagrange equation of motion for this system, and interpret the resulting equation in terms of a known physical system. (1 point) (b) Find Hamiltonian via Legendre transformation. (1 point) (c) Show that q(t) and the corresponding canonical momentum p(t) can be found as follows for given initial conditions, yo = 9(0) and po = p(0) (2 points), (11) q(t) = e-Bt/2 e=B1/2 (90 (coswt + 2 sin wt) + mo sin wt PO) = 2942 [-(mw + ) sin ut + pa (cos ut – 2 sinut)] (12) where w= - > 0. Remember that p(t) = 3. mj(t). Even if you do not know how to obtain above equations, you can continue to next questions. (d) Sketch the trajectory of the motion in (g(t), g(t)) and explain why the trajectory has the particular shape. (1 point) A rough sketch is good enough. (e) Sketch the trajectory of the motion in (q(t),p(t)) and explain why the trajectory has the particular shape. (1 point) A rough sketch is good enough. (1) Rewrite the solution in matrix form, and find M. (1 point). (O) = m()(%) (13) 0 1 (g) Verify that transformation matrix M satisfies MyMT = y, where y = 1 its meaning. (1 point) -10), and explain (h) Introduce a new coordinate Q = ept/2.q, and rewrite the Lagrangian in terms of Q and 0. You will get a term which is proportional to QQ and you can drop this term, since it is a total derivative term, QQ = Q. (1 point) (i) Find a continuous symmetry of the Lagrangian in these new variables, and derive the corre- sponding conserved quantity. (1 point)

Answer 1

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