Robert Gibson and teaching philosophy of nature in 18th century Ireland

The dissemination of natural philosophy in the 18th-century, which was based primarily on Newton's pioneering work in mechanics, optics and astrophysics, is presented as seen through a remarkable textbook written by a little known Irish mathematics teacher, Robert Gibson. Later, he became the deputy surveyor general of Ireland (1752-1760).Comment: 15 pages, 10 figure

Stability and eccentricity of periodic orbit for two planets in a 1:1 resonance

The nonlinear stability domain of Lagrange's celebrated 1772 solution of a three-body problem is obtained numerically as a function of the masses of the bodies and the common eccentricity of their Keplerian orbits. This domain shows that this solution may be realized in extra-solar planetary systems similar to those that have been discovered recently with two Jupiter-size planets orbiting a solar-size star. For an exact 1:1 resonance, the Doppler shift variation in the emitted light would be the same as for stars which have only a single planetary companion. But it is more likely that in actual extra-solar planetary systems there are deviations from such a resonance, raising the interesting prospect that Lagrange's solution can be identified by an analysis of the observations. The existence of another stable 1:1 resonance solution which would have a more unambiguous Doppler shift signature is also discussed.Comment: 9 pages 11 figure

Theory and experiments on the ice-water front propagation in droplets freezing on a subzero surface

An approximate theory is presented describing the propagation of the ice-water front that develops in droplets of water that are deposited on a planar surface at a temperature below the melting point of ice. A calculation based on this theory is compared with our experimental observations of the time evolution of this front. The results of calculations of this front by Schultz et al7, obtained by integrating numerically the exact differential equations for this problem, were published graphically, but only for the time-dependent velocity of this front. Unfortunately, these theoretical results cannot be compared directly with our experimental observations. Our experiments were performed by freezing water droplets directly on a block of dry ice, and in order to examine the effects of the heat conductivity of a substrate during the freezing process, such droplets were also deposited on a glass plate and on a copper plate placed on dry ice. The temperature at the base of these droplets, and the dependence of the freezing time on their size was also investigated ex perimentally, and compared with our analytic approximation of the theory. Such experiment have not been published previously, and reveal that the usual assumption that the temperature at the base of the droplets is a constant, made in all previous theoretical papers on this subject, cannot be implemented in practice.Comment: 17 pages 8 figure

Is Bohm's interpretation of quantum mechanics consistent?

The supposed equivalence of the conventional interpretation of quantum mechanics with Bohm's interpretation is generally demonstrated only in the coordinate representation. It is shown, however, that in the momentum representation this equivalence is not valid.Comment: 7 page

Visiting Newton's Atelier before the Principia, 1679-1684

The manuscripts that presumably contained Newton's early development of the fundamental concepts that led to his Principia have been lost. A plausible reconstruction of this development is presened based on Newton's exchange of letters with Robert Hooke in 1679, with Edmund Halley in 1686, and on some clues in the diagram associated with Proposition1 in Book1 of the Principia that have been ignored in the past. The graphical method associated with this proposition leads to a rapidly convergent method to obtain orbital curves for central forces, and elucidates how Newton may have have been led to formulate some of his other propositions in the Principia

Kepler's Area Law in the Principia: Filling in some details in Newton's proof of Prop. 1

During the past 25 years there has been a controversy regarding the adequacy of Newton's proof of Prop. 1 in Book 1 of the {\it Principia}. This proposition is of central importance because its proof of Kepler's area law allowed Newton to introduce a geometric measure for time to solve problems in orbital dynamics in the {\it Principia}. It is shown here that the critics of Prop. 1 have misunderstood Newton's fundamental limit argument by neglecting to consider the justification for this limit which he gave in Lemma 3. We clarify the proof of Prop. 1 by filling in some details left out by Newton which show that his proof of this proposition was adequate and well grounded.Comment: 4 figure

Reply to C. Tsallis's "Comments on Critique of q-entropy for thermal statistics by M. Nauenberg"

In this note it is shown that in his ``Comments'', Tsallis did not point out any flaws in the main criticism of my paper, namely, that the nonextensive q-entropy formalism fails to satisfy the zeroth law of thermodynamics. Details are also given of a rigorous thermodynamic proof which demonstrates that, contrary to Tsallis's assertion, the application of a nonextensive formalism to black-body radiation does not lead to the well known $T^4$ Stefan-Boltzman law

Hooke's memorandum on the development of orbital dynamics

I discuss a memorandum entitled " A True state of the Case and Controversy between Sir Isaak Newton and Dr. Robert Hooke as the Priority of that Noble Hypothesis of motion of ye Planets about ye Sun as their Centers", where Hooke summarizes his life long contributions to the development of the physics of orbital motion and the universal theory of gravitatio

The Reception of Newton's Principia

Newton's Principia, when it appeared in 1687, was received with the greatest admiration, not only by the foremost mathematicians and astronomers in Europe, but also by philosophers like Voltaire and Locke and by members of the educated public. In this account I describe some of the controversies that it provoked, and the impact it had during the next century on the development of celestial mechanics, and the theory of gravitation.Comment: 27 page

Period Relation for the 2:1 resonance in the GJ876 planetary system

The recent radial velocity Keck data for GJ876 in Laughlin et al. (2005) is shown to be in good agreement, apart from a 6 m/s scatter, with a theoretical calculation (Nauenberg, 2002) based on orbital parameters from a fit to the earlier Keck data. The time variation of the periods of the inner and outer planets, which are locked in a near 2:1 resonance, are evaluated, and their mean values, Pi and Po, are shown to satisfy closely the relation Po/Pi=2+Po/P, where P is the period for the retrograde precesion of the common mean periastron of these planets.Comment: 5 pages 9 figure