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
