The Routhian 133 of the Lagrangian, since the time would there be only a parameter which would not be involved in the transformation. Analogously to formula (40.6), the partial time derivatives of L and H are related by (40.8) PROBLEMS PROBLEM 1. Find the Hamiltonian for a single particle in Cartesian, cylindrical and spherical co-ordinates. SOLUTION. In Cartesian co-ordinates x, y, 2, in cylindrical co-ordinates r, , z, in spherical co-ordinates r, 0, , PROBLEM 2. Find the Hamiltonian for a particle in a uniformly rotating frame of reference. SOLUTION. Expressing the velocity V in the energy (39.11) in terms of the momentum p by (39.10), we have H = p2/2m-S rxp+U. PROBLEM 3. Find the Hamiltonian for a system comprising one particle of mass M and n particles each of mass m, excluding the motion of the centre of mass (see §13, Problem). SOLUTION. The energy E is obtained from the Lagrangian found in §13, Problem, by changing the sign of U. The generalised momenta are Pa = OL/OV Hence - = (mM/14) = = Substitution in E gives 41. The Routhian In some cases it is convenient, in changing to new variables, to replace only some, and not all, of the generalised velocities by momenta. The trans- formation is entirely similar to that given in 40. To simplify the formulae, let us at first suppose that there are only two co-ordinates q and E, say, and transform from the variables q, $, q, $ to q, $, p, & where P is the generalised momentum corresponding to the co- ordinate q. 134 The Canonical Equations