Separation of Motions of Many-Body Systems into Dynamically Independent Parts by Projection onto Equilibrium Varieties in Phase Space. II

Abstract

We continue the development started in the preceding paper, in which we treated the many-body problem by separating the motion into an oscillatory part $\delta {\mathbf{x}}^i$, $\delta {\mathbf{p}}_i$, and a nonoscillatory part ${\mathbf{X}}^i$, ${\mathbf{P}}_i$, the latter being obtained by a noncanonical transformation from ${\mathbf{x}}^i$, ${\mathbf{p}}_i$ which is just so tailored as to project out the oscillatory features from ${\mathbf{x}}^i$, ${\mathbf{p}}_i$, and thereby "projecting" ${\mathbf{x}}^i$, ${\mathbf{p}}_i$ onto the equilibrium variety $J_{\mathrm{k}}({\mathbf{x}}^i,{\mathbf{p}}_i)=0\mathrm{}$ (where $J_{\mathrm{k}}$ is the oscillatory action variable) in phase space. In this paper, we first develop a condensed notation in phase space which facilitates calculations. With the aid of this notation, we then give our results a simple geometrical interpretation in phase space by introducing a certain canonically invariant metrical tensor $O_{\mathrm{ij}}$. This tensor (which is antisymmetric) does not yield the usual orthogonal or pseudo-orthogonal metric, but rather, what is called a "symplectic metric" (i.e., invariant to the symplectic group of transformations). One then sees that the projections that we make are "orthogonal," in the symplectic sense, to the equilibrium varieties. Likewise, one can see quite generally, that the entire canonical formalism, including the Poisson brackets and the Hamiltonian equations of motion, reduces to simple geometrical relations in phase space, the form of which is suggestive for possible further developments, especially with regard to the treatment in higher approximations. We apply our ideas to the electron gas, and illustrate the dynamics of the plasma with the aid of a comparison with a simple two-dimensional model, possessing all the essential features described above. In this way, we are able to understand many of the basic features of the plasma motions, in terms of concepts such as the generalization of the notion of centrifugal force and Coriolis force to phase space. By going over to a local geodetic frame in the equilibrium variety, we are led in a natural way to the concept of a set of "quasiparticles" for the plasma. If the number of collective oscillatory coordinates is $s$, then there will be $3N-s$ of these "quasiparticle" coordinates. The latter do not represent any of the actual original particles out of which the system is constituted, but rather, they represent effective pulse-like distributions of charge, which move together in a correlated way so as to resemble an actual particle in many respects.