Eduardo Piña

Central configurations for the planar Newtonian four-body problem

Planar central configurations of four different masses are analyzed theoretically and computed numerically. We follow Dziobek’s approach to four-body central configurations with a straightforward implicit (in the masses and distances) method of our own in which the fundamental quantities are each the quotient of a directed area divided by the corresponding mass. We apply a new simple numerical algorithm to construct general four-body central configurations. We use this tool to obtain new properties of the symmetric and non-symmetric central configurations. The explicit continuous connection between three-body and four-body central configurations where one of the four masses approaches zero is clarified. Some cases of coorbital 1+3 problems are also considered.

The mass action law in extended irreversible thermodynamics

Abstract The non-linear relationship that holds between the chemical flux and the chemical affinity for chemical reactions taking place amongst ideal gases or solutions in homogeneous phases is shown to belong to the framework of extended non-equilibrium thermodynamics. Its generalization to include non-ideal systems, as well as other features, are also discussed.

Properties of New Coordinates for the General Three-Body Problem

This is an analysis of the features of the new coordinate system given by the principal axes of inertia, as determined by Euler angles, and twodistances related to the inertia principal moments and an auxiliar angleas coordinates, for studying the general three-body problem, interactingthrough gravitational forces.The reduction of order is performed in these new coordinates by using the angular velocity vector or the Euler angles.The Eulerian case of collinear motion is revisited from our own perspective.The value of the auxiliar angle is computed for the Sun–Earth–Moon system.

The three-body problem with an inverse square law potential

We study the motion of three masses in a plane interacting with a central potential proportional to 1/r2 using the coordinates introduced recently by Pina. We show that this problem with four degrees of freedom (three angles and a distance related to the inertia moment of the system in these coordinates) is partially separable, and can be reduced to a problem with two degrees of freedom (two angles) with a new constant of motion. We find a symmetry of reflection (an involution) for this system and we use the symmetry lines to find periodic orbits in the angular coordinates. These orbits will not be periodic in general on the whole phase space because the coordinate of distance type grows as t when t→∞ and it is unbounded. However, if the inertia moment of the system remains constant, they will be periodic on the whole phase space.

PERIODIC ORBITS OF AN ELECTRIC CHARGE IN A MAGNETIC DIPOLE FIELD

Periodic orbits in the Stormer problem are studied using the symmetry lines of the Poincaré map introduced by De Vogelaere. Many known facts are explained by mean of these lines. The dynamics of four special symmetry lines when the Stormer parameter γ1 changes is presented, and we obtain a clear global view of the structure of the simple periodic orbits and their bifurcations, including the asymmetrical ones. New asymmetrical multiple periodic orbits are obtained.

A new solution to the lagrange's case of the three‐body problem

The motion of three particles, interacting by gravitational forces, is studied in a new coordinate system given by the principal axes of inertia, as determined by Euler angles, and using the inertia principal moments and an auxiliar angle as coordinates.The solution to the particular Lagrange case of the three‐body problem is reviewed and solved in these new coordinates.

On the simple Michaelis-Menten mechanism for chemical reactions

Several mathematical properties associated with the simple Michaelis-Menten mechanism for enzymatic reactions are proven. In particular it is shown that the usual interpretation of the slope of the experimental Michaelis-Menten rate law in terms of the reaction constants of the mechanism can be obtained, in the approximation in which the total concentration of the enzyme is small compared with the Michaelis-Menten constant, independently of the ratio between the total initial concentrations of the enzyme and substrate. Furthermore, the ratio of the total concentration of the enzyme to the Michaelis-Menten constant allows for the elimination of a fast variable in a singular perturbation method, yielding the Michaelis-Menten rate law as a first order approximation.

On the thermodynamic basis of the affinity decay rate

In the past five years exhaustive studies in chemical reactions have lead to an empirical equation describing how isothermal–isometric homogeneous reactions evolve towards equilibrium independently of their particular mechanism or rate law. Such an equation expresses the time rate of change of the chemical affinity as a linear function of the inverse of time. In this paper we show that by invoking the local equilibrium hypothesis one may provide, a time evolution equation for the chemical affinity that is uniquely given by the solution of the particular rate law of the reaction considered. Consequently such an equation is not of the same functional form for all reactions. On the other hand, integration of Dalton’s law under specific initial conditions, together with the local equilibrium assumption and the ideality requirement for the reacting species, exhibits a unique inverse time decay for the chemical affinity. This explains the good fitting of the inverse in time dependence of the chemical affinity w...

Thermodynamics of Radiation Modes.

We study the equilibrium thermodynamics of the electromagnetic radiation in a cavity of a given volume and temperature. We found three levels of description, the thermodynamics of one mode, the thermodynamics of the distribution of frequencies in a band by summing over the frequencies in it and the global thermodynamics by summing over all the frequencies. One equation relating frequency and volume is used to define the thermodynamics of one mode, and to explain the mystery of the frequency-dependent quantities having a similar behaviour to the non-frequency-dependent quantities for some thermodynamic equations and different behaviour for others. Besides, this frequency–volume relation is used to count the number of modes in a band of frequency.

Coexistence properties of the Redlich?Kwong model for a fluid

It is shown in this paper that for pure substances obeying the van der Waals (VdW) equation of state (EOS) or Redlich?Kwong (RK) type equations of state below the critical temperature, it is possible to obtain with the help of the Maxwell equal areas rule the vapour pressure curve, the difference between the molar volume of the vapour and of the liquid, and the difference between the molar entropies of vapour and liquid; each property is obtained in the first step as a dimensionless expression, function of a dimensionless temperature. Any particular property of a particular fluid obeying VdW EOS or one type of RK EOS can be obtained from the corresponding dimensionless expression by multiplying it with a dimension unit which is a function of the particular temperature-dependent parameter a(T) of the EOS considered for the fluid. In the cases of VdW and the original RK (EOS), the dimensionless functions are unique for all the pure fluids that obey the EOS under consideration. A comparison of VdW, RK and RK?Soave (SRK) predictions with experimental vapour pressure data is exemplified for carbon dioxide.