Asmall toy boat of mass m, with horizontal coordinate x, floats on water the vertical coordinate of the water at and time t, is given by y y(x, t) (we ignore the third cartesian coordinate) assume the boat is exactly at height y (10 points) show that the lagrangian is a. m m. (1y)i my'ja + 2 l 2 mgy, where y/x y', oy/ot j. b. (10 points) find the momentum p, conjugate to r, of the boat, from this lagrangian. write the time derivative in terms of this canonical momen- tum (10 points) find the time-dependent hamiltonian h(p, x, t) c. d. (5 points) if the surface of the water has a constant slope a, and moves to the right with velocity u, then y(x, t) = a(x - ut). write the hamiltonian you found in part c., for this special case of y(x, t) a(x - ut), find the generating func- (15 points) for the case tion of a canonical transformation from x andp to new variables x and p bringing the hamiltonian to the form e. p2 h(p, x, t) 2m +u(x, t), where m is a function constant (not equal to m, in general) and u(x, t) is some