arXiv subjects

E. Piña

Publications and source records attributed to E. Piña.

8 recordsLinked to original sources

arXiv2018

Newtonian Few-Body Problem Central Configurations with Gravitational Charges of both Signs

The Newtonian n-Body Problem is modified assuming positive inertial masses but different sign for the interacting force which is assumed with the possibility of two different signs for the gravitational masses, according to the prescription two masses with same sign attract one to the other, two masses of different sign repel one to the other. As in electrostatics the signed mass is called charge. The inertial mass is always positive. The two body problem behaves as the similar Coulomb problem of charged particles with two equal charges. The solution is a central configuration with almost same behavior that the Newton two-body problem for hyperbolic orbits. The 3-Body problem was found with collinear solutions. The four body case of charged central configurations has only the planar [1] and collinear solutions.

physics.gen-ph↗

arXiv2012

Table of Families of Alternating Knots with their Conway's Function

A table of the families of alternating knots formed by conways is presented. The Conway's function is shown with the use of linear algebra in terms of natural numbers, called conways, that represent the number of crossings along a direction, as it was used by J. Conway for the classification of knots. Colored figures and tangles show the parts of the knots or tangles with a definite handedness: all the colored parts of the knot family are associated to a particular orientation. For example all the colored conways have a right hand screw thread, and all the white conways have the opposite handedness. Figures for six conways were colored with two different colors for forty two families in order to show the dissection of the knot in two tangles corresponding to a particular factorization of the Conway's function. The Conway's function of each family is expressed as the internal product of two vectors corresponding to each of two colored family 2-tangles, and with a full factorization which is not unique.

math.GN↗

arXiv2012

Algebra of Families of Alternating Knots and Links

Families of alternating knots (links) and tangles are studied using as building block the conway defined as the twisting of two strands. The regular representation of knots assumes the projection has the minimal number of overpassings, and the minimal number of conways. The continued fraction associated to rational knots is represented by gaussian brackets and products of 2-dimensional matrices. This gives birth to an algebra of rational knots and tangles which is easily generalized to alternating knots. A collection of 65 families of prime alternating knots with one to six conways is found. Eleven families with six conways show peculiar behavior not present in families with a lower or equal number of conways.

math.GN↗

arXiv2011

Computing collinear 4-Body Problem central configurations with given masses

An interesting description of a collinear configuration of four particles is found in terms of two spherical coordinates. An algorithm to compute the four coordinates of particles of a collinear Four-Body central configuration is presented by using an orthocentric tetrahedron, which edge lengths are function of given masses. Each mass is placed at the corresponding vertex of the tetrahedron. The center of mass (and orthocenter) of the tetrahedron is at the origin of coordinates. The initial position of the tetrahedron is placed with two pairs of vertices each in a coordinate plan, the lines joining any pair of them parallel to a coordinate axis, the center of masses of each and the center of mass of the four on one coordinate axis. From this original position the tetrahedron is rotated by two angles around the center of mass until the direction of configuration coincides with one axis of coordinates. The four coordinates of the vertices of the tetrahedron along this direction determine the central configuration by finding the two angles corresponding to it. The twelve possible configurations predicted by Moulton's theorem are computed for a particular mass choice.

math-ph↗

arXiv2009

Canonical Formalism in Special Relativity

A covariant Hamiltonian description was introduced in the dynamics of charges and electromagnetic interaction. By a canonical transformation this Hamiltonian formalism was transformed to obtain the Dirac generators for any form of relativistic dynamics, as coefficients of a first degree polynomial in the ten translation and rotation velocities of the Poincaré transformation. The Currie's world line conditions were generalized to any form of the dynamics. The explicit relation between the covariant field variables and the more usual 3-dimensional Fourier variables was derived.

math-ph↗

arXiv2009

New coordinates for the Four-Body problem

A new coordinate system is defined for the Four-Body dynamical problem with general masses, having as its origin of coordinates the center of mass. The transformation from the inertial coordinate system involves a combination of a rotation to the principal axis of inertia, followed by three changes of scale leading the principal moments of inertia to yield a body with three equal moments of inertia, and finally a second rotation that leaves unaltered the equal moments of inertia. These three transformations yield a mass-dependent rigid orthogonal tetrahedron of constant volume in the inertial coordinates. Each of those three linear transformations is a function of three coordinates that produce the nine degrees of freedom of the Four-Body problem, in a coordinate system with the center of mass as origin. The relation between the well known equilateral tetrahedron solution of the gravitational Four-Body problem and the new coordinates is exhibited, and the plane case of central configurations with four different masses is computed numerically in these coordinates.

math-ph↗

arXiv2009

Central configurations for the planar Newtonian Four-Body problem

The plane case of central configurations with four different masses is analyzed theoretically and is computed numerically. We follow Dziobek's approach to four body central configurations with a direct implicit method of our own in which the fundamental quantities are the quotient of the directed area divided by the corresponding mass and a new simple numerical algorithm is developed to construct general four body central configurations. This tool is applied 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.

math-ph↗

arXiv2007

Conway classification of alternating knots

The alternating knots, links and twists projected on the $S_2$ sphere were identified with the phase space of a Hamiltonian dynamic system of one degree of freedom. The saddles of the system correspond to the crossings, the edges correspond to the stable and unstable manifolds connecting the saddles. Each face is then oriented in one of two different senses determined by the direction of these manifolds. This correspondence can be also realized between the knot and the Poincaré section of a two degrees of freedom integrable dynamical system. The crossings corresponding to unstable orbits, and the faces to foliated torus, around a stable orbit. The associated matrix to that connected graph was decomposed in two permutations. The separation was shown unique for knots not for links. The characteristic polynomial corresponding to some knot, link or twist families was explicitly computed in terms of Chebyschev polynomials. A classification of rational knots was formulated in terms of the first derivative of the polynomial of a knot computed in $x=2$, equal to the number of crossings of the knot multiplying the same number used previously by Conway for tabulation of knot properties. This leads to a classification of knots exemplified for the families having up to five ribbons. We subdivide the families of $N$ ribbons in subfamilies related to the prime knots of $N$ crossings.

math.GT↗