5. Consider the following time-dependent Lagrangian for a system with one degree of freedom q, c=...

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5. Consider the following time-dependent Lagrangian for a system with one degree of freedom q, c=" (Gme -). (10) where 3, 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) (e) Show that q(t) and the corresponding canonical momentum p(t) can be found as follows for given initial conditions, go = g(0) and po = p(0) (2 points), 9(t) = e/2 sinwt+ Cos wt + sin (11) )sin at + p (cos ut - sin ur) (12) p(t) = t/2 sin wt 2 4 where w = - 20. Remember that p(t) = + má(t). Even if you do not know how to obtain above equations, you can continne to next questions. %3! (d) Sketch the trajectory of the motion in (q(t), 4(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 (g(t). p(1)) and explain why the trajectory has the particular shape. (1 point) A rough sketch is good enough. (f) Rewrite the solution in matrix form, and find M. (1 point). M((). - M(t) Po (13) p(t) (g) Verify that transformation matrix M satisfies MyMT -7, where y = and explain -1 0 its meaning. (1 point) (h) Introduce a new coordinate Q = e/2 q, and rewrite the Lagrangian in terms of Q and Q. You will get a term which is proportional to QQ and you can drop this term, since it is a total derivative term, QQ -10. (1 point) (i) Find a contimmous symmetry of the Lagrangian in these new variables, and derive the corre- sponding conserved quantity. (1 point) 5. Consider the following time-dependent Lagrangian for a system with one degree of freedom q, c=" (Gme -). (10) where 3, 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) (e) Show that q(t) and the corresponding canonical momentum p(t) can be found as follows for given initial conditions, go = g(0) and po = p(0) (2 points), 9(t) = e/2 sinwt+ Cos wt + sin (11) )sin at + p (cos ut - sin ur) (12) p(t) = t/2 sin wt 2 4 where w = - 20. Remember that p(t) = + má(t). Even if you do not know how to obtain above equations, you can continne to next questions. %3! (d) Sketch the trajectory of the motion in (q(t), 4(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 (g(t). p(1)) and explain why the trajectory has the particular shape. (1 point) A rough sketch is good enough. (f) Rewrite the solution in matrix form, and find M. (1 point). M((). - M(t) Po (13) p(t) (g) Verify that transformation matrix M satisfies MyMT -7, where y = and explain -1 0 its meaning. (1 point) (h) Introduce a new coordinate Q = e/2 q, and rewrite the Lagrangian in terms of Q and Q. You will get a term which is proportional to QQ and you can drop this term, since it is a total derivative term, QQ -10. (1 point) (i) Find a contimmous symmetry of the Lagrangian in these new variables, and derive the corre- sponding conserved quantity. (1 point)

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