1. Consider the problem of a sliding wedge shaped block of mass M along the x axis without friction, on which another block of mass m slides down without friction. The angle of the wedge is 0 relative to the x axis. (see the Example.1 of the handout Examples-of-Lagrangians-. . . .pdf ). Given the Lagrangian in the form L=(m + M)if+ mi, + mija2cos0 - grz sinli, show that the construction of H requires the inversion of 2 x 2 matrix C such that pi = >, Cija;, and its inversion. Find C explicitly and C-1 using matrix methods. If you find the general problem too messy, you are allowed to use special values M = 2m, m = 15,0 =1/3, which simplify the inversion. ... [10] 2. Loosely (i.c. approximately) based on the above problem, we define a Hamiltonian with two degrees of freedom (i.c. two generalized coordinates 41, 92 and momenta p1, p2) H = Pi Pip2 cos e 2m1 2r2 1 9 92 sind, 2m3 where m3 = vmim2. (a) Write the Hamiltonian equations of motion for the two generalized coordinates and generalized momenta using the usual formulas pi = and q; = of . . . 5] (b) Show that one of the above is a constant of motion. Using that constant of motion, solve for the time dependence of the two coordinates for general initial conditions. . .. [10] (c) Show that H is a constant by plugging in the solution for the coor- dinates and velocities, and find its time independent value. This number can be taken as the energy for the particular solution in terms of the initial conditions used above. . . . 15]