Hamiltonian relativistic two-body problem: center of mass and orbit reconstruction (original) (raw)
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Il Nuovo Cimento B, 1978
Recent models of relativistic action at a distance through singular Lagrangians with multiplicative potentials, describing two-point bound states, arc re-examined. They are reformulated in such a way to be well suited to the study of extended bodies; we introduce a set of vierbeins, attached to the barycentric co-ordinates, which connect the Minkowski space with an inner relative space, and we define new relative co-ordinates in it. By using the irreducible representation theory of the Poincar6 group, we show that this relative space is the natural relativistic generalization of the nonrelativistic relative one. The nonrelativistic limit of these models is exhibited, by recovering the ~'ewtonian two-body problem with central forces.
A possible approach to the two-body relativistic problem
In this paper a model which describes a relativistic interaction between two point particles via an action at a distance is derived from a set of hypotheses on the relativistic dynamics. From this set of hypotheses a singular Lagrangian is obtained. The aim of this paper is to find a link between the singular-Lagrangian approach and other approaches to the relativistic dynamics of two particles. The connection of this Lagrangian model with the predictive approach of the relativistic mechanics is studied, by showing that it is possible to calculate the instantaneous forces, at least in principle. An explicit canonical transformation is given, such that a subset of the new canonical variables becomes free of constraints. In this way the instant form of the relativistic dynamics found by Bakamjan and Thomas and by Foldy is recovered.
Position variables in classical relativistic hamiltonian mechanics
Nuclear Physics B, 1979
We construct within the equal-time hamiltonian formalism of classical relativistic mechanics, Poincare invariant and covariant systems of two scalar particles interacting at a distance. Position variables are constructed in terms of the canonical variables of the theory by demanding that they transform under the Lorentz transformations as the space components of four-vectors. The possibility of identifying position variables with canonical coordinates in the center-of-momentum frame is shown. In that particular frame, equations of motion take a simple form and can be solved as in nonrelativistic mechanics. A velocity always smaller than that of light is ensured for each particle in the case of a large variety of relativistic potentials which reduce in the nonrelativistic limit to the usual central potentials. The present approach is similar to that adopted earlier by Pauri and Prosperi; it differs, however, by the additional requirement of manifest covariance of the underlying theory.
Circular orbits in classical relativistic two-body systems
Annals of Physics, 1970
Schild described the circular motion of two classical point charges interacting through their time-symmetric electromagnetic field. This work is compared here with quantum theories and generalized to include the interaction via massless and massive scalar fields. Energy and angular momentum for the bound system are computed. The approach to the potential theory limit, in which one of the particles is at rest, is studied carefully. The main purpose of this work is to gain insight into the nature of highly relativistic bound systems.
2002
In the Wigner-covariant rest-frame instant form of dynamics it is possible to develop a relativistic kinematics for the N-body problem which solves all the problems raised till now on this topic. The Wigner hyperplanes, orthogonal to the total timelike 4-momentum of any N-body configuration, define the intrinsic rest frame and realize the separation of the center-of-mass motion. The point chosen as origin of each Wigner hyperplane can be made to coincide with the covariant non-canonical Fokker-Pryce center of inertia. This is distinct from the canonical pseudo-vector describing the decoupled motion of the center of mass (having the same Euclidean covariance as the quantum Newton-Wigner 3-position operator) and the non-canonical pseudo-vector for the Møller center of energy. These are the only external notions of relativistic center of mass, definable only in terms of the external Poincaré group realization. Inside the Wigner hyperplane, an internal unfaithful realization of the Poincaré group is defined while the analogous three concepts of center of mass weakly coincide due to the first class constraints defining the rest frame (vanishing of the internal 3-momentum). This unique internal center of mass is consequently a gauge variable which, through a gauge fixing, can be localized atthe origin of the Wigner hyperplane. An adapted canonical basis of relative variables is found by means of the classical counterpart of the Gartenhaus-Schwartz transformation. The invariant mass of the N-body configuration is the Hamiltonian for the relative motions. In this framework we can introduce the same dynamical body frames, orientation-shape variables, spin frame and canonical spin bases for the rotational kinematics developed for the non-relativistic N-body problem.
Physical Review D, 2000
We extract all the invariants (i.e. all the functions which do not depend on the choice of phasespace coordinates) of the dynamics of two point-masses, at the third post-Newtonian (3PN) approximation of general relativity. We start by showing how a contact transformation can be used to reduce the 3PN higher-order Hamiltonian derived by Jaranowski and Schäfer [2] to an ordinary Hamiltonian. The dynamical invariants for general orbits (considered in the center-of-mass frame) are then extracted by computing the radial action variable prdr as a function of energy and angular momentum. The important case of circular orbits is given special consideration. We discuss in detail the plausible ranges of values of the two quantities ωstatic, ω kinetic which parametrize the existence of ambiguities in the regularization of some of the divergent integrals making up the Hamiltonian. The physical applications of the invariant functions derived here (e.g. to the determination of the location of the last stable circular orbit) are left to subsequent work.
2008
A complete exposition of the rest-frame instant form of dynamics for arbitrary isolated systems (particles, fields, strings, fluids)admitting a Lagrangian description is given. The starting point is the parametrized Minkowski theory describing the system in arbitrary admissible non-inertial frames in Minkowski space-time, which allows one to define the energy-momentum tensor of the system and to show the independence of the description from the clock synchronization convention and from the choice of the 3-coordinates. In the inertial rest frame the isolated system is seen as a decoupled non-covariant canonical external center of mass carrying a pole-dipole structure (the invariant mass MMM and the rest spin vecbarS{\vec {\bar S}}vecbarS of the system) and an external realization of the Poincare' group. Then an isolated system of positive-energy charged scalar articles plus an arbitrary electro-magnetic field in the radiation gauge is investigated as a classical background for defining relativistic atomic physics. The electric charges of the particles are Grassmann-valued to regularize the self-energies. The rest-frame conditions and their gauge-fixings (needed for the elimination of the internal 3-center of mass) are explicitly given. It is shown that there is a canonical transformation which allows one to describe the isolated system as a set of Coulomb-dressed charged particles interacting through a Coulomb plus Darwin potential plus a free transverse radiation field: these two subsystems are not mutually interacting and are interconnected only by the rest-frame conditions and the elimination of the internal 3-center of mass. Therefore in this framework with a fixed number of particles there is a way out from the Haag theorem,at least at the classical level.
Relativistic and separable classical Hamiltonian particle dynamics
Annals of Physics, 1981
We show within the Hamiltonian formalism the existence of classical relativistic mechanics of N scalar particles interacting at a distance which satisfies the requirements of Poincart invariance, separability, world-line invariance and Einstein causality. The line of approach which is adopted here uses the methods of the theory of systems with constraints applied to manifestly covariant systems of particles. The study is limited to the case of scalar interactions remaining weak in the whole phase space and vanishing at large space-like separation distances of the particles. Poincare invariance requires the inclusion of many-body, up to N-body, potentials. Separability requires the use of individual or two-body variables and the construction of the total interaction from basic two-body interactions. Position variables of the particles are constructed in terms of the canonical variables of the theory according to the world-line invariance condition and the subsidiary conditions of the non-relativistic limit and separability. Positivity constraints on the interaction masses squared of the particles ensure that the velocities of the latter remain always smaller than the velocity of light.