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

In two papers (of which this is the first) our central concern is to draw conclusions about the over-all dynamical properties of a many-body system. This is done without trying to solve the equations of motion, but rather, on the basis of our knowledge of oscillatory or collective variables (or more generally, from the existence of conservation rules and of the uniform constants of the motion). Our main result is that, corresponding to the collective coordinates (or the uniform constants of the motion) there exists a separation of the motions into two parts, one of which is collective or oscillatory, and regular, and the other of which is noncollective, nonoscillatory, and irregular. This separation is here obtained by a (canonically invariant) method of projections in phase space, from the actual phase point ${\mathbf{x}}^i$, ${\mathbf{p}}_i$ along a certain line, which is the direction of a "pure" de-excitation of an oscillation, down to a certain "projected point" ${\mathbf{X}}^i$, ${\mathbf{P}}_i$, which is the intersection of the line with an equilibrium subspace (or variety), the latter consisting of all the points in the phase space for which the collective excitation is zero. We apply this separation to an illustrative example consisting of a simple two-dimensional model, possessing all the essential features of the general problem under discussion. We obtain the results corresponding to the Bohm-Pines theory, as applied to this case, in a very simple way, without having to introduce supernumerary variables or subsidiary conditions (our results being generalized to the plasma case in the following paper). Instead of subsidiary conditions, we have a corresponding number of identities among the "projected motions" ${\mathbf{X}}^i$, ${\mathbf{P}}_i$, so that in effect, ${\mathbf{X}}^i$, ${\mathbf{P}}_i$, together with the collective oscillatory variables, span a space of $6N$ dimensions (where $N$ is the number of particles). This definition of the ${\mathbf{X}}^i$, ${\mathbf{P}}_i$ replaces the two canonical transformations of Bohm-Pines, and is equivalent to a certain noncanonical transformation, which removes the collective part of the motion. Our method may also be regarded as a systematic generalization of that of Tomonaga; firstly, being an extension of the latter's method from configuration space to phase space, and to collective variables that are momentum dependent, and secondly, being the development of a general separation method for arbitrary variables, which contains Tomonaga's Taylor expansion of the Hamiltonian as a special case. The projection method associates to each actual motion ${\mathbf{x}}^i\mathrm{}\left(t\right)\mathrm{}$, ${\mathbf{p}}_i\mathrm{}\left(t\right)\mathrm{}$ a unique equilibrium motion ${\mathbf{X}}^i\mathrm{}\left(t\right)\mathrm{}$, ${\mathbf{P}}_i\mathrm{}\left(t\right)\mathrm{}$, about which it oscillates. This association is such that, from the very way in which it is defined, the possibility of an indefinitely large increase of $\delta {\mathbf{x}}^i={\mathbf{x}}^i-{\mathbf{X}}^i$, $\delta {\mathbf{p}}_i={\mathbf{p}}_i-{\mathbf{P}}_i$ with time is avoided, so that the $\delta {\mathbf{x}}^i$, $\delta {\mathbf{p}}_i$, will oscillate stably in every order of approximation, without the need for special precautions to avoid secular terms, as is necessary in the usual perturbation treatments (e.g., in celestial mechanics).