Bounding the ground-state energy of a many-body system with the differential method

This paper promotes the differential method as a new fruitful strategy for estimating a ground-state energy of a many-body system. The case of an arbitrary number of attractive Coulombian particles is specifically studied and we make some favorable comparison of the differential method to the existing approaches that rely on variational principles. A bird's-eye view of the treatment of more general interactions is also given.Comment: version 1->2 (main revisions): subsection 2.2, equation (18), footnote 6 have been adde

Third post-Newtonian constrained canonical dynamics for binary point masses in harmonic coordinates

The conservative dynamics of two point masses given in harmonic coordinates up to the third post-Newtonian (3pN) order is treated within the framework of constrained canonical dynamics. A representation of the approximate Poincar\'e algebra is constructed with the aid of Dirac brackets. Uniqueness of the generators of the Poincar\'e group resp. the integrals of motion is achieved by imposing their action on the point mass coordinates to be identical with that of the usual infinitesimal Poincar\'e transformations. The second post-Coulombian approximation to the dynamics of two point charges as predicted by Feynman-Wheeler electrodynamics in Lorentz gauge is treated similarly.Comment: 42 pages, submitted to Phys. Rev.

Asymptotic conditions of motion for radiating charged particles

Approximate asymptotic conditions on the motion of compact, electrically charged particles are derived within the framework of general relativity using the Einstein- Infeld-Hoffmann (EIH) surface integral method. While superficially similar to the Abraham-Lorentz and Lorentz-Dirac (ALD) equations of motion, these conditions differ from them in several fundamental ways. They are not equations of motion in the usual sense but rather a set of conditions which these motions must obey in the asymptotic future of an initial value surface. In addition to being asymptotic, these conditions of motion are approximate and apply, as do the original EIH equations, only to slowly moving systems. Also, they do not admit the run- away solutions of these other equations. As in the original EIH work, they are integrability conditions gotten from integrating the empty-space (i.e., source free) Einstein-Maxwell equations of general relativity over closed two-surfaces surrounding the sources of the fields governed by these equations. No additional ad hoc assumptions, such as the form of a force law or the introduction of inertial reaction terms, needed to derive the ALD equations are required for this purpose. Nor is there a need for any of the infinite mass renormalizations that are required in deriving these other equations.Comment: 15 page

Kinetic theory of spatially inhomogeneous stellar systems without collective effects

We review and complete the kinetic theory of spatially inhomogeneous stellar systems when collective effects (dressing of the stars by their polarization cloud) are neglected. We start from the BBGKY hierarchy issued from the Liouville equation and consider an expansion in powers of 1/N in a proper thermodynamic limit. For $N\rightarrow +\infty$, we obtain the Vlasov equation describing the evolution of collisionless stellar systems like elliptical galaxies. At the order 1/N, we obtain a kinetic equation describing the evolution of collisional stellar systems like globular clusters. This equation does not suffer logarithmic divergences at large scales since spatial inhomogeneity is explicitly taken into account. Making a local approximation, and introducing an upper cut-off at the Jeans length, it reduces to the Vlasov-Landau equation which is the standard kinetic equation of stellar systems. Our approach provides a simple and pedagogical derivation of these important equations from the BBGKY hierarchy which is more rigorous for systems with long-range interactions than the two-body encounters theory. Making an adiabatic approximation, we write the generalized Landau equation in angle-action variables and obtain a Landau-type kinetic equation that is valid for fully inhomogeneous stellar systems and is free of divergences at large scales. This equation is less general than the Lenard Balescu-type kinetic equation recently derived by Heyvaerts (2010) since it neglects collective effects, but it is substantially simpler and could be useful as a first step. We discuss the evolution of the system as a whole and the relaxation of a test star in a bath of field stars. We derive the corresponding Fokker-Planck equation in angle-action variables and provide expressions for the diffusion coefficient and friction force

Reduction of the two-body dynamics to a one-body description in classical electrodynamics

We discuss the mapping of the conservative part of two-body electrodynamics onto that of a test charged particle moving in some external electromagnetic field, taking into account recoil effects and relativistic corrections up to second post-Coulombian order. Unlike the results recently obtained in general relativity, we find that in classical electrodynamics it is not possible to implement the matching without introducing external parameters in the effective electromagnetic field. Relaxing the assumption that the effective test particle moves in a flat spacetime provides a feasible way out.Comment: 20 pages, revtex; minor change