A massless inextensible string passes over a pulley which is a fixed distance above the floor. A bunch of bananas of mass m is attached to one end of the string. A monkey of mass M is initially at the other end B. The monkey climbs the string, and her displacement d(t) with respect to the end B is a given function of time. The system is initially at rest, so that the initial conditions are d(0) = 0, d(0) = 0. (a) Introduce suitable generalized coordinates, write the transformation equations, and calculate the lagrangian of the system in terms of these coordinates. Show that the equation governing the height z(t) of the monkey above the floor is (M+m)2-md = (m - M)g. (b) Find the hamiltonian H and Hamilton's equations. Rederive the equation for 2 from Hamilton's equations. (c) Is H = E=T +V? Is H conserved? Is E conserved? Justify briefly. (d) Integrate the equation to find z(t) as a function of t. In the special case that m = M, show that the bananas and the operation between them is constant.

A massless inextensible string passes over a pulley which is a fixed distance above the floor. A bunch of bananas of mass m is attached to one end of the string. A monkey of mass M is initially at the other end B. The monkey climbs the string, and her displacement d(t) with respect to the end B is a given function of time. The system is initially at rest, so that the initial conditions are d(0) = 0, d(0) = 0. (a) Introduce suitable generalized coordinates, write the transformation equations, and calculate the lagrangian of the system in terms of these coordinates. Show that the equation governing the height z(t) of the monkey above the floor is (M+m)2-md = (m - M)g. (b) Find the hamiltonian H and Hamilton's equations. Rederive the equation for 2 from Hamilton's equations. (c) Is H = E=T +V? Is H conserved? Is E conserved? Justify briefly. (d) Integrate the equation to find z(t) as a function of t. In the special case that m = M, show that the bananas and the operation between them is constant.