Question

22. For the point transformation in a system of two degrees of freedom, find the most general transformation equations for Pi and P2 consistent with the over- all transformation being canonical. Show that with a particular choice for Py and P2 the Hamiltonian -(PI-P2)" + p2 + (q1+q2)2 H241 can be transformed to one in which both 01 and 02 are ignorable. By this means solve the problem and obtain expressions for gi, 42, PI, and p2 as functions of time and their initial values.

Answer 1

Using the relation for a point transformation Q_1 = partialdifferential F_2/partialdifferential P_1 = q_1^2 Q_2 = partialdifferential F_2/partialdifferential P_2 = q_1 + q_2 The generating function must be F_2 = q_1^2 P_1 + (q_1 + q_2)P_2 The momentum coordinates are P_1 = partialdifferential F_2/partialdifferential q_1 = 2q_1 P_1 + p_2 p_2 = partialdifferential F_2/partialdifferential q_2 = p_2 solving for p_1 and p_2 we found the must general transformation we get p_2 = p_2 p_1 = p_1 - p_2/2q_1 Therefore the is given by H = p_1 ^2 + p_2 + q_2^2 but whether we choose p_2 = p_2 + (q_1 + q_2)^2 H = p_1^2 + p_2 the doesnot depend upon Q_1 and Q_2 Now solve this we get p_1 = -partialdifferential H/partialdifferential q_1 = 0 materialimplication p_1 = a p_2 = -partialdifferential H/partialdifferential q_2 = 0 materialimplication p_2 = b \r\n Q_1 = partialdifferential H/partialdifferential p_1 = 2p_1 materialimplication Q_1 = 2p_1 + c Q_2 = partialdifferential H/partialdifferential p_2 = 1 materialimplication Q_2 = t + d Where a b c d are constants ie, initial values replacing with old coordinates