Application of Liouville's Theorem to Electron Orbits in the Earth's Magnetic Field

It is pointed out that in the application of Liouville's theorem to the problem of cosmic-ray intensities, Lemaitre and Vallarta have implicitly taken the electron momentum as that corresponding to a free particle. Calling this momentum $p^{\prime }$ the particle momentum, we have to realize that Liouville's theorem is usually based upon the Hamiltonian equations in which the momentum $p$ associated with an electron is not the same as $p^{\prime }$, but is connected with it by the relation $p=p^{\prime }+\frac{\mathrm{eU}}{c}$, where $U$ is the vector potential determining the magnetic field. The Hamiltonian equations are not valid in terms of momenta of the type $p^{\prime }$, and, it is not, therefore, clear that Liouville's theorem is valid when expressed in terms of these momenta. The object of the paper is to show that a theorem the equivalent of Liouville's theorem is, in fact, true in terms of the coordinates and the momenta $p^{\prime }$, so that the ultimate validity of the use of the theorem by Lemaitre and Vallarta is substantiated. It is to be observed, moreover, that the validity of this extended form of Liouville's theorem is true even in the presence of an electric field.