A complete closed form vectorial solution to the Kepler problem

The paper gives an exact vectorial solution to the Kepler problem. A vectorial regularization that linearizes the Kepler problem is given using a Sundman transformation. Closed form expressions describing the Keplerian motion are deduced. A unified approach to the classic Kepler problem is offered, by studying both rectilinear and non-rectilinear Keplerian motions with the same instrument. The approach is an elementary one and only simple vectorial computations are involved.     

A group theoretical approach to a certain class of perturbations of the Kepler problem

We study a particular class of perturbations of the classical Kepler Hamiltonian, first in two, then in three and finally in n dimensions. At every stage of our investigation the group theoretical nature of our constructions is fully exposed.In particular we present a new regularization of the n-dimensional Kepler problem which is based on previous constructions of Guillemin & Sternberg (see [8]). This regularization is similarily related to Moser's (see [9]) as is Kustaanheimo-Stiefel's (see [4]) in three dimensions.     

The KS-transformation revisited

We show that the Kustaanheimo-Stiefel (KS) regularizing transformation for the perturbed Kepler motion is so deeply rooted to the Keplerian orbital elements as to yield the position vector of a particle on the osculating orbit as the effect of a peculiar roto-dilatation in the physical Euclidean space. Adopting the conventional vector formulation, quaternions and spinors are also involved. A key role is played by (i): a simple hodographical approach to the integrals of the Kepler motion (angular momentum vector, Runge-Lenz vector); (ii) a polarized outlook on the attitude frame of the Kepler orbit; (iii) a simple kinematical expression for the orbital elements. The mechanical energy, the bilinear relation, the gauge transformation — fundamental in the KS-theory — are naturally arrived at, acquiring interesting kinematical interpretations.

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Sommario Si mostra come la trasformazione di Kustaanheimo e Stiefel (KS) per regolarizzare il moto kepleriano perturbato sia cosí legata agli elementi orbitali da condurre, tramite una particolare roto-omotetia nello spazio fisico euclideo, al vettore posizione di una particella sull'orbita osculatrice. Avvalendosi del consueto calcolo vettoriale, si introducono solo all'occorrenza quaternioni e spinori. Fondamentali risultano (i): l'introduzione per via odografica degli integrali primi caratteristici del moto kepleriano (momento della quantità di moto, vettore di Runge-Lenz); (ii) l'interpretazione polarizzata della terna di assi che dà la posizione dell'orbita nello spazio; (iii) le semplici espressioni cinematiche degli elementi orbitali. L'energia meccanica e la relazione bilineare, basilari nella teoria KS, acquistano interessanti interpretazioni cinematiche.

     

Apollonius coordinates,theN-body problem,and continuation of periodic solutions

This paper treats theN-body problem and its relation to various restricted problems. For each solution of the Kepler problem a generalization of the pulsating coordinates used to express the Hamiltonian of the elliptic restricted three-body problem is given. These coordinates are called Apollonius coordinates. The method of symplectic scaling is used to give a precise derivation of the elliptic restricted problem showing the precise asymptotic relationship between the restricted problem and the full three-body problem. This derivation obviates the proof of the fact that a nondegenerate periodic solution of the elliptic restricted three-body problem can be continued into the full three-body problem under mild nonresonance assumptions. Also, the method of symplectic scaling is used to give a precise derivation of the elliptic Hill lunar equation showing the precise relationship between the elliptic Hill lunar equation and the full three-body problem. A similar continuation theorem is established.     

On “generalized Kepler’s third law” and mass dependence of periods of three-body orbits

Meccanica - Five years ago a generalization of Kepler’s third law was formulated (Dmitra?inovi? and ?uvakov in Phys Lett A 379:1939–1945, 2015) for three-body orbits in...     

Comparative analysis of the electrogravitational Kepler problem in GRT and RTG

In the framework of Einstein's General Relativity Theory and of the Relativistic Theory of Gravitation, the equations governing the trajectories of charged particles in the field created by a charged mass point are given. An analysis of the shape of the trajectories in both theories is presented. The first and the second order approximate solutions of the electrogravitational Kepler problem are found in the two theories and the results are compared with each other. I have pointed out the differences between the predictions in the two theories.     

Hohmann转移释放绳系卫星的方法研究

借助Ｈｏｈｍａｎｎ轨道转移的思想，研究一种绳系卫星释放方法并进行动力学分析．从轨道方程和Ｋｅ ｐｌｅｒ方程导出轨道转移条件．由数值计算对此方法加以说明．     

A torsion-based solution to the hyperbolic regime of the $$J_2$$ -problem

A popular intermediary in the theory of artificial satellites is obtained after the elimination of parallactic terms from the \(J_2\)-problem Hamiltonian. The resulting quasi-Keplerian system is in turn converted into the Kepler problem by a torsion. When this reduction process is applied to unbounded orbits, the solution is made of Keplerian hyperbolae. For this last case, we show that the torsion-based solution provides an effective alternative to the Keplerian approximation customarily used in flyby computations. Also, we check that the extension of the torsion-based solution to higher orders of the oblateness coefficient yields the expected convergence of asymptotic solutions to the true orbit.

     

Ultraconvergence for averaging discontinuous finite elements and its applications in Hamiltonian system

This paper discusses the k-degree averaging discontinuous finite element solution for the initial value problem of ordinary differential equations.When k is even,the averaging numerical flux (the average of left and right limits for the discontinuous finite element at nodes) has the optimal-order ultraconvergence 2k + 2.For nonlinear Hamiltonian systems (e.g.,Schro¨dinger equation and Kepler system) with momentum conservation,the discontinuous finite element methods preserve momentum at nodes.These properti...     

力学中的黄金分割率

黄金分割率0.618是一个神奇的数字,在建筑、绘画、雕塑、音乐等艺术领域和人类生活的许多领域里得到了广泛的应用. 本文讨论力学中存在黄金分割率的若干例子.     

On Shearing, Stretching and Spin

An analysis is presented of stretching, shearing and spin of material line elements in a continuous medium. It is shown how to determine all pairs of material line elements at a point x, at time t, which instantaneously are not subject to shearing. For a given pair not subject to shearing, a formula is presented for the determination of a third material line element such that all three form a triad not subject to shearing, instantaneously. It is seen that there is an infinity of such triads not subject to shearing. A new decomposition of the velocity gradient L is introduced. In place of the classical decomposition of Cauchy and Stokes, L=d+w, where d is the stretching tensor and w is the spin tensor, the new decomposition is L=?+ℳ, where ?, called the ldquo;modified” stretching tensor, is not symmetric, and ℳ, called the “modified” spin tensor, is skew-symmetric – the tensor ? being chosen so that it has three linearly independent real right (and left) eigenvectors. The physical interpretation of this decomposition is that the material line elements along the three linearly independent right eigenvectors of ? instantaneously form a triad not subject to shearing. They spin as a rigid body with angular velocity μ (say) associated with ℳ. Also, for each decomposition L=?+ℳ, there is a decomposition L=? ^T +, where is also skew-symmetric. The triad of material line elements along the right eigenvectors of ? ^T (the set reciprocal to the right eigenvectors of ?) is also instantaneously not subject to shearing and rotates with angular velocity (say) associated with . It is seen that the vorticity vector ω is the mean of the two angular velocities μ and , ω =(μ+)/2. For irrotational motion, ω =0, so that μ=-; any triad of material line elements suffering no shearing rotates with angular velocity equal and opposite to that of the reciprocal triad of material line elements. It is proved that provided d is not spherical, there is an infinity of choices for ? and ℳ in the decomposition L=?+ℳ. Two special types of decompositions are introduced. The first type is called “CCS-decomposition” (where CCS is an abbreviation for Central Circular Section). It is associated with the infinite family of triads (not subject to shearing) with a common edge along the normal to one plane of central circular section of an ellipsoid ? associated with the stretching tensor, and the two other edges arbitrary in the other plane of central circular section of ?. There are two such CCS-decompositions. The second type is called “triangular decomposition”, because, in a rectangular cartesian coordinate system, ? has three off-diagonal zero elements. There are six such decompositions. Received 14 November 2000 and accepted 2 August 2001     

Die Querschwingungen elastisch gebetteter Saiten,Stäbe,Membranen und Platten

Ohne ZusammenfassungHabilitationsschrift Karlsruhe. Ref.: Prof. Dr.Th. Pöschl, Korref.: Prof.K. v. Sanden.     

The Kepler problem spinor regularization and pre-quantization of the negative-energy manifold

Summary We extend the results of our two foregoing papers to the whole phase space of the Kepler problem, by: i) developing a natural spinor extension of the KS-regulari-zation of the Kepler problem, due to Kustaanheimo and Stiefel, and ii) applying the results to the pre-quantization (in the sense of Souriau) of the negative-energy manifold, the energy levels satisfying a well-known condition. We show that the quantized manifold is diffeomorphic to the symplectic productS ₃×S ₃ of two 3-spheres of the spinor phase space, factored through a suitable equívalence relation.

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Sommario Estendendo i risultati ottenuti in due prece denti lavori all'intero spazio delle fasi del problema di Keplero, si perviene nella presente nota ad una naturale estensione spinorialedella trasformazione KS (introdotta da Kustaanheimo e Stiefel per regolarizzare il problema di Keplero) la quale consente di prequantizzare, nel senso di Souriau, la varietà delle orbite kepleriane isoenergetiche, con energia negativa spddisfacente ad una ben nota condizione. Si mostra che la varietà quantica associata è diffeomorfa al prodotto simplettico S ₃×S ₃,opportunamente quozientato, di due 3-sfere dello spazio spinoriale delle fasi.

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Work performed under the auspices of G.N.F.M. of the Italian Research Council (C.N.R.).     

Left Ventricular Geometry,Tissue Composition,and Residual Stress in High Fat Diet Dahl-Salt Sensitive Rats

Background

Hypertension drives myocardial remodeling, leading to changes in structure, composition and mechanical behavior, including residual stress, which are linked to heart disease progression in a gender-specific manner. Emerging therapies are also targeting constituent-specific pathological features. All previous studies, however, have characterized remodeling in the intact tissue, rather than isolated tissue constituents, and did not include sex as a biological variable.

Objective

In this study we first identified the contribution of collagen fiber network and myocytes to the myocardial residual stress/strain in Dahl-Salt sensitive rats fed with high fat diet. Then, we quantified the effect of hypertension on the remodeling of the left ventricle (LV), as well as the existence of sex-specific remodeling features.

Methods

We performed mechanical tests (opening angle, ring-test) and histological analysis on isolated constituents and intact tissue of the LV. Based on the measurements from the tests, we performed a stress analysis to evaluate the residual stress distribution. Statistical analysis was performed to identify the effects of constituent isolation, elevated blood pressure, and sex of the animal on the experimental measurements and modeling results.

Results

Hypertension leads to reduced residual stress/strain in the intact tissue, isolated collagen fibers, and isolated myocytes in male and female rats. Collagen remains the largest contributor to myocardial residual stress in both normotensive and hypertensive animals. We identified sex-differences in both hypertensive and normotensive animals.

Conclusions

We observed both constituent- and sex-specific remodeling features in the LV of an animal model of hypertension.

     

Application of the coordinatewise descent method on a unit interval for weight optimization of structures made of composite materials

Problems of weight optimization of anisotropic structures with restrictions on strength, stability, etc., are solved by the coordinatewise descent method on a unit interval, combined with effective methods of nonlinear programming (dichotomy and golden section methods). __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 210–218, July–August, 2009.