Solution Of Triangles Research Articles

Numerical estimates for stability domains of Lagrangian solutions to the restricted three-body problem

The geometric parameters of stability domains of Lagrangian solutions to the classical restricted three-body problem are quantitatively estimated. It is shown that these domains are ellipsoid-like plane figures stretched along the tangent to the circle that passes through the Lagrangian triangle solutions. A heuristic algorithm is proposed for determining the maximum size of these domains of attraction.

Linear Stability of the Lagrangian Triangle Solutions for Quasihomogeneous Potentials

In this paper, we study the linear stability of the relative equilibria for homogeneous and quasihomogeneous potentials. First, in the case the potential is a homogeneous function of degree −a, we find that any relative equilibrium of the n-body problem with a>2 is spectrally unstable. We also find a similar condition in the quasihomogeneous case. Then we consider the case of three bodies and we study the stability of the equilateral triangle relative equilibria. In the case of homogeneous potentials we recover the classical result obtained by Routh in a simpler way. In the case of quasihomogeneous potentials we find a generalization of Routh inequality and we show that, for certain values of the masses, the stability of the relative equilibria depends on the size of the configuration.

Linear Stability of the Elliptic Lagrangian Triangle Solutions in the Three-Body Problem

This paper concerns the linear stability of the well-known periodic orbits of Lagrange in the three-body problem. Given any three masses, there exists a family of periodic solutions for which each body is at the vertex of an equilateral triangle and travels along an elliptic Kepler orbit. Reductions are performed to derive equations which determine the linear stability of the periodic solutions. These equations depend on two parameters – the eccentricity e of the orbit and the mass parameter β=27(m1m2+m1m3+m2m3)/(m1+m2+m3)2. A combination of numerical and analytic methods is used to find the regions of stability in the βe-plane. In particular, using perturbation techniques it is rigorously proven that there are mass values where the truly elliptic orbits are linearly stable even though the circular orbits are not.

On the equilateral triangle solution tothe three-body problem

The N-body problem has no general solution for morethan two particles. Various restricted solutions such as the equilateral trianglesolutions of the three-body problem can,however, be found. Here, we discuss howto find and understand such restricted solutions. We dothis using generalized coordinates and the Lagrangianmethod. We note that the restricted solutions can beviewed as solutions arising from the imposition ofconstraints that do not require any constraint forces.We also note that an N-body problem leads to a centralforce problem provided the distances between theparticles are equal. In three dimensions this allows theparticles to be at the vertices of an equilateraltriangle or a regular tetrahedron. We presentcorresponding restricted solutions to the three- andthe four-body problems, respectively.

Spherical Trigonometry in the Astronomy of the Medieval Kerala School

Although the methods of plane trigonometry became the cornerstone of classical Indian mathematical astronomy, the corresponding techniques for exact solution of triangles on the sphere’s surface seem never to have been independently developed within this tradition. Numerous rules nevertheless appear in Sanskrit texts for finding the great-circle arcs representing various astronomical quantities; these were presumably derived not by spherics per se but from plane triangles inside the sphere or from analemmatic projections, and were supplemented by approximate formulas assuming small spherical triangles to be plane.

On the instability of lagrange solutions in the three-body problem

We consider the relation between the Lyapunov instability of Lagrange equilateral triangle solutions and their orbital instability. We present a theorem on the orbital instability of Lagrange solutions. This theorem is extended to the planarn-body problem.

A Quick Solution of Triangle Counting

A triangle is new if it contains a triangle from the bottom row of the diagram. The number of new triangles is Tn+ ?Tn. A triangle is crusty if it contains a triangle from the shaded 'crust'. All triangles are either nablas (V) or deltas (Ai). The number of new but not crusty triangles is clearly Tn -1 T-2The new, crusty deltas have their top apices beaded. There are 2 n + 1 of them. The new, crusty nablas have their bottom apices .:circled. There are n of them. Thus

Similarity Solution of Two-Dimensional Point Vortices

Theory of the similarity solution of two-dimensional point vortices in an unbounded region is presented. To get the solution, the time dependence of the solution for all vortex positions are assumed to be the same. It is shown that the system either rotates rigidly or collapses according to the initial position of vortices. Then numerical calculations are made especially for the system of three vortices. It is found that the rigid rotation corresponds to the regular triangle solution or the straight line solution, and that irrespective of the strengths of three vortices, the regular triangle solution always exists while the straight line solution is obtained by solving a certain cubic equation.

Dynamical motions of charged particles in Newtonian and post-Newtonian approximation

This paper contains a derivation of the equations of motion of the system of three mutually interacting charged particles. We give the general Hamiltonian up to the first post-Newtonian approximation and we discuss the very special cases that allow exact solutions,i.e. the equilateral triangle and collinear solutions.

Measure of Degenerate Relative Equilibria. I

Relative equilibria of three positive masses always correspond to the well known central configurations found by Euler and Lagrange-the collinear and equilateral triangle solutions of the three body problem. The name equilibrium suggests the fact that each of the masses appears at rest in a rotating barycentric coordinate system. By identifying any two configurations provided that one can be made congruent to the other by a rotation followed by a scalar multiplication, we define a relative equilibria class. Wintner [7] proves that there are only five relative equillibria classes in this case. The three Euler classes and the two Lagrange classes are always nondegenerate [3]. In [2] we showed how a degeneracy arises in the four body problem. In the plane E2 we place three unit masses at the vertices of an equilateral triangle with center of mass at the origin. We place at the origin an arbitrary fourth positive mass, mi. It follows easily for all values of m4 that this configuration is a relative equilibrium. We find that the relative equilibria class to which this configuration belongs is degenerate when m4 equals a unique positive mass ma* < 1.

Direct Methods of Latitude and Longitude Determination by Mini-Computer

In celestial navigation, the advent of the hand held or mini-computer brought forth a plethora of methods for the solution of spherical triangles. The azimuths and altitudes thus derived have not only eliminated the need for bulky tables but also have speeded up the determination of the data needed for the graphic construction of intersecting lines of position from which the “fix” is derived. In this conventional type of solution, the corrected sextant altitude is compared with the computed altitude then, from the assumed position, the line of position is constructed perpendicular to the azimuth of the body observed and either towards the body or away from it depending upon whether the observed altitude is greater or less, respectively, than that which has been computed. Extremely sophisticated systems are being developed which combine special computer systems directly with the sextant. However, the present paper relates only to the minicomputers currently on the market and their apparent ultimate utility in the celestial navigation field—the direct display of both latitude and longitude without recourse to graphic methods.

A note on relative motion in the general three-body problem

It is shown that the equations of the general three-body problem take on a very symmetric form when one considers only their relative positions, rather than position vectors relative to some given coordinate system. From these equations one quickly surmises some well known classical properties of the three-body problem such as the first integrals and the equilateral triangle solutions. Some new Lagrangians with relative coordinates are also obtained. Numerical integration of the new equations of motion is about 10 percent faster than with barycentric or heliocentric coordinates.

Rifting History of the Woodlark Basin in the Southwest Pacific

Marine geophysical evidence has been obtained for the rate and history of sea-floor spreading in the Woodlark Basin. The eastern part of this basin is presently separating from the Australian Plate at over 4 cm per yr in a northerly direction. The western part of the basin is not presently spreading. This spreading rift marks the southern boundary of the Solomons Plate which is bounded by subduction zones in the north and east (the New Britain and northern Solomons Trenches, respectively) and in the west by a combination strike-slip rifting (dip-slip) boundary in eastern Papua (New Guinea). A vector triangle solution near the Solomons Trench-Woodlark Rift triple point gives underthrusting of the Solomons Plate beneath the northern Solomon Trench in a northeasterly direction at about 11 cm per yr. The Woodlark Basin began opening as a sphenochasm, with a pole near the tip of eastern Papua about 20 m.y. B.P. This was caused by left-lateral shear in the region induced by a change in the relative motion pole of the Australia and Pacific Plates. The basin opened only a few degrees at this time, then stopped. Rifting in the entire basin resumed about 3 m.y. B.P., based on magnetic anomaly data. About 1 m.y. B. P., the spreading center in the western basin shifted to the Woodlark Rise.

A Practical Mechanical Calculator for Spherical Trigonometry

INTRODUCTION THE SOLUTION OF SPHERICAL TRIANGLES is the basic mathematical problem of navigation; as a result, many methods have been evolved over the years to accomplish this fundamental task. The purpose of this paper is to add one more item to this long list in the hope that, for some applications at least, it will prove to be the most satisfactory method.

2782. A construction for the graphical solution of spherical triangles

2782. A construction for the graphical solution of spherical triangles

A Primitive Geometry

In this paper we exhibit some properties of a geometry which may be called 'primitive' for at least two reasons: first, the solution of triangles requires only one formula and second, a general transformation of coordinates may be written involving only one arbitrary constant. Our geometry will be based on axioms closely resembling those for vector spaces. The scalars to be used are denoted by capital letters and are consistently thought of as real numbers. Now we consider a set of elements, denoted by small letters, satisfying axioms to be specified. Any two elements x and y determine a unique element x+y, called their sum. Also, to every scalar A and every element x there corresponds a unique element Ax as product. Addition and scalar multiplication are governed by the usual axioms for vector spaces [2, pp. 3-4]. To prevent misunderstandings we use the letter 0 for the null element. The notation ((x, y) indicates the inner product of the ordered pair of elements x and y. It is a real number such that

Solution of Triangles on the Slide Rule

The following examples show how the law of sines may be used with the slide rule to solve triangles in case three sides, or two sides and the included angle, are given. Methods of solution by use of the law of sines in other cases are well known. Hence, subject to the limitations of the slide rule, the law of sines suffices to solve any triangle. The method involves successive approximations. The adjustment made at each step is estimated on the basis of the di1 erence between 1800 and the sum of the angles obtained by the previous approximation. Skill can be developed rapidly and the triangle usually solved in three or four settings. We take up first the case of three sides given.

References for Mathematics Teachers: The Teaching of Trigonometry

From the earliest times until the beginning of the eighteeenth century, trigonometry was regarded chiefly as the handmaiden to astronomy. Its development as a science may be said to have begun with Hipparchus about 150 b.c., but as a science it did not flourish with the Greeks. Not until some three centuries after Hipparchus did the Egyptian astronomer Ptolemy contribute to its advance by writing the famous Almagest. His influence upon trigonometry was destined to be felt for the next 1300 years, down to the time of Galileo. Throughout this entire period the solution of spherical triangles held the attention of mathematicians and astronomers. Much of the progress made from time to time was due chiefly to the Arabs and the Hindus.

Geometrical Solution of Spherical Triangles

Geometrical Solution of Spherical Triangles

Geometrical Solution of Spherical Triangles

Geometrical Solution of Spherical Triangles