We consider a class of system of N interacting particles in any dimension—the potential includes a quadratic pair potential and an arbitrary translation-invariant position-dependent potential that is homogeneous of degree −2. The group SU(1,2)¯ (1̄,1̄) is the dynamical group for the Hamiltonian. We illustrate the significance of the Casimir operator in relation to the separation of variables method; obtain a series of eigenfunctions that transform under the unitary irreducible representations of SU(1,2)¯ (1̄,1̄) labeled by the ground state energy; indicate the structure of arbitrary eigenfunctions; and specify when the complete energy spectrum is linear. We treat N-body examples which include two- and three-body forces. For N identical particles in one dimension interacting with a quadratic pair potential and an inverse square pair potential, we exhibit a series of eigenfunctions characterized by four quantum numbers. These eigenfunctions reduce to the complete set of eigenfunctions for five particles. We indicate how a complete set of eigenfunctions for N particles are obtained.

Large convective electric fields of the order of 10 mV/m (sometimes as high as 22 mV/m) are observed at rotational magnetopause discontinuities. These observations were made with the long cylindrical (179-m base line) probes carried on the ISEE 1 satellite. These electric field observations yield convective velocity magnitudes, V*= IE x B/B21, of the order of 150 km/s. In this V* format some of theseobservations are similar to the high speed plasma velocity observations that were made at the magnetopause with the plasma experiment carried on the ISEE 1 satellite. It is shown that, for many of these magnetopause crossings, there exists a special moving coordinate system where the observed electric fields vanish. Such a unique reference system is often used in theoretical studies of magnetic discontinuities. This special coordinate system does not move at the local plasma velocity but moves instead at a velocity intermediate between the convective velocity and the local Alfv6n velocity. It is used here as a diagnostic tool for the experimental investigation of rotational discontinuities at the magnetopause.

This letter presents a detailed experimental analysis of the electric fields and electron flow velocities at a rotational magnetopause discontinuity. It is shown for this event that the magnetosheath plasma velocity is sharply decelerated on entry into the magnetosphere in very good agreement with a theoretical magnetohydrodynamic fit. This type of event is remarkable because the quantity E · J is negative during the magnetopause crossing. This case represents one of two types of valid solution to the MHD boundary conditions. It differs from previously published examples of reconnective magnetopause geometries in that energy is being transferred locally from kinetic energy into electromagnetic energy at the time of these measurements.

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