First-order Results Research Articles

Relativistic satellite orbits: central body with higher zonal harmonics

Satellite orbits around a central body with arbitrary zonal harmonics are considered in a relativistic framework. Our starting point is the relativistic Celestial Mechanics based upon the first post-Newtonian approximation to Einstein’s theory of gravity as it has been formulated by Damour et al. (Phys Rev D 43:3273–3307, 1991; 45:1017–1044, 1992; 47:3124–3135, 1993; 49:618–635, 1994). Since effects of order \((\mathrm{GM}/c^2R) \times J_k\) with \(k \ge 2\) for the Earth are very small (of order \( 7 \times 10^{-10}\,\times \,J_k\)) we consider an axially symmetric body with arbitrary zonal harmonics and a static external gravitational field. In such a field the explicit \(J_k/c^2\)-terms (direct terms) in the equations of motion for the coordinate acceleration of a satellite are treated first with first-order perturbation theory. The derived perturbation theoretical results of first order have been checked by purely numerical integrations of the equations of motion. Additional terms of the same order result from the interaction of the Newtonian \(J_k\)-terms with the post-Newtonian Schwarzschild terms (relativistic terms related to the mass of the central body). These ‘mixed terms’ are treated by means of second-order perturbation theory based on the Lie-series method (Hori–Deprit method). Here we concentrate on the secular drifts of the ascending node \( \) and argument of the pericenter \( \). Finally orders of magnitude are given and discussed.

Perturbation theory for confined systems

We discuss the application of perturbation theory to a system of particles confined in a spherical box. A simple argument shows that the particles behave almost independently in sufficiently strong confinement. We choose the helium atom with a moving nucleus as a particular example and compare results of first order with those for the nucleus clamped at the center of the box. We provide a suitable explanation for some numerical results obtained recently by other authors.

Existentially Closed Models and Conservation Results in Bounded Arithmetic

We develop model-theoretic techniques to obtain conservation results for first order Bounded Arithmetic theories, based on a hierarchical version of the well-known notion of an existentially closed model. We focus on the classical Buss' theories Si2 and Ti2 and prove that they are ∀Σbi conservative over their inference rule counterparts, and ∃∀Σbi conservative over their parameter-free versions. A similar analysis of the Σbi-replacement scheme is also developed. The proof method is essentially the same for all the schemes we deal with and shows that these conservation results between schemes and inference rules do not depend on the specific combinatorial or arithmetical content of those schemes. We show that similar conservation results can be derived, in a very general setting, for every scheme enjoying some syntactical (or logical) properties common to both the induction and replacement schemes. Hence, previous conservation results for induction and replacement can be also obtained as corollaries of these more general results.

On the consistent description of quantum relaxation processes: The harmonic oscillator

The quantum-mechanical description of a relaxation process is critically reviewed. The initially reversible “universe” consists of a harmonic oscillator (system) interacting with N further harmonic oscillators (heat bath). The coupling is linear in both, system- and bath-coordinates. The underlying dynamical equations are exactly solvable. A reasonable quantum-mechanical treatment of relaxation processes should guarantee that for all times the canonical commutation relations deviate only in a higher order of the coupling constant from their reversible values. A hierarchy of descriptions with decreasing quantum-mechanical consistency is formulated and discussed: 1. (i) the exact irreversible solution in connection with a decoupling of system and bath at t = 0; 2. (ii) the exact irreversible solution in connection with a conditional thermal average at t = 0; 3. (iii) a semi-Markovian stochastic differential equation (SDE), where the dissipation is described by a single parameter while the fluctuation (i.e. the correlation function of the stochastic force includes the information on the continuous bath-spectrum; 4. (iv) a complete Markovian SDE; dissipation and fluctuation are characterized by one single parameter. For the Drude model the descriptions (i)–(iii) are equivalent in relation to consistent results of first order in the dissipative parameters. On the contrary, description (iv) is unsuitable in the quantum-mechanical regime.