New integrable problem of classical mechanics

Complete integrability in Liouville's sense is proven for rotation of an arbitrary rigid body with a fixed point in a Newtonian field with an arbitrary homogeneous quadratic potential. A consequences is the complete integrability of rotation of a rigid body with fixed center of mass in the field of arbitrary sufficiently remote objects (in the second approximation). Explicit formulae are obtained expressing angular velocities of the rigid body in terms of -functions for Riemannian surfaces. Integrable cases are found for rotation of a rigid body in nonlinear Newtonian potential fields.     

Formulation and solution of the classical seashell problem

Summary Unlike its predecessor this second paper formulates the problem of seashell geometry entirely in real spaceE ³, presenting those equations of most use for parctical digital computer similations. The mathematics dilates previously mentioned clockspring “wires” (growth trajectoriesY(ϕ)) into tubular spiral surfacesr(σ, ϕ) complete with orthoclinal growth lines and simple periodic corrugations or flaresQ(σ, ϕ). It is seen that second-order theory requires a new boundary condition, the famous HLOR growth vectorr(σ, 0), which is absent from classical analyses. Thus it is demonstrated that simple periodic surface features, of a kind occurring widely in nature, obey precise Cauchy boundary conditions which may be related to the quantized cyclicities of metabolic, and geophysical, rhythms associated with biological shell growth.

---

Riassunto A differenza del precedente quest'articolo formula il problema della geometria delle conchiglie completamente in spazio realeE ³, quindi presentando quelle equazioni piú utili nella simulazione pratica del computer numerico. In questa discussione i “fili” elastici a molla d'orologio (traiettorie di crescitaY(ϕ)) sono dilatari quindi diventano superfici a spirale tubolarer(σ, ϕ) e completi di anelli ortoclinati (orientati perpendicolari alla direzione di crescita) e onduleQ(σ, ϕ) periodiche semplici. Si vede che nella teoria di secondo ordine è richiesta una condizione di confine nuova, il ben conosciuto HLOR vettore di crescitar(σ, ϕ), che è assente nell'analisi classica. Quindi è dimostrato che le caratteristiche periodiche semplici delle superfici, simili a quelle molto frequenti in natura, obbediscono a precise condizioni di confine di Cauchy le quali possono essere collegate con ciclicità pulsanti di ritmi metabolici e geofisici associati con la crescita biologica delle conchiglie.

---

Резюме В отличие от первой части в этатье формулируется проблема геометрии морской раковины полностью в реальном пространствеE ³, с целью получения уравнений, удобных для практического компьютерного моделирования. Математический аппарат позволяет растянуть ранее предложенные растущие траекторииT(ϕ) в цилиндрические поверхностиr(σ, ϕ), которые заканчиваются ортоклинными растущими линиями и простыми периодическими складками или раструбамиQ(σ, ϕ). Отмечается, что теория второго порядка требует нового граничного условия, известного HLOR вектораr(σ, 0), который отсутствует в классическом анализе. Показывается, что характеристики простых периодических поверхностей, широко встречающихся в природе, подчиняются граничным условиям Коши, котрые могут быть связаны с квантованными цикличностями метаболических и геофизических ритмов, связанныу с ростом биологических раковин.

     

Formulation and solution of the classical seashell problem

Summary Despite an extensive scholarly literature dating back to classical times, seashell geometries have hitherto resisted rigorous theoretical analysis, leaving applied scientists to adopt a directionless empirical approach toward classification. The voluminousness of recent palaeontological literature demonstrates the importance of this problem to applied scientists, but in no way reflects corresponding conceptual or theoretical advances beyond the XIX century thinking which was so ably summarized by Sir D’Arcy Wentworh Thompson in 1917. However, in this foundation paper for the newly emerging science of theoretical conchology, unifying theoretical considerations for the first time, permits a rigorous formulation and a complete solution of the problem of biological shell geometries. Shell coilingabout the axis of symmetry can be deduced from first principles using energy considerations associated with incremental growth. The present paper shows that those shell apertures which are incurved (?cowrielike?), outflared (?stromblike?) or even backturned (?Opisthostomoidal?) are merely special cases of a much broader spectrum of ?allowable? energy-efficient growth trajectories (tensile elastic clockspring spirals), many of which were widely used by Cretaceous ammonites. Energy considerations also dictate shell growthalong the axis of symmetry, thus seashell spires can be understood in terms of certain special figures of revolution (M?bius elastic conoids), the better-known coeloconoidal and cyrtoconoidal shell spires being only two special cases arising from a whole class of topologically possible, energy efficient and biologically observed geometries. The ?wires? and ?conoids? of the present paper are instructive conceptual simplifications sufficient for present purposes. A second paper will later deal with generalized tubular surfaces in three dimensions.

---

Riassunto Malgrado un’ampia e dotta letteratura che risale ai tempi classici, la geometria delle conchiglie ha resistito fino ad ora ad analisi teoriche rigorose, quindi gli scienziati che si cimentano in questo campo hanno adottato un metodo empirico senza direttiva per quanto riguarda la classificazione. L’abbondanza della recente letteratura paleontologica dimostra l’importanza di questo problema per gli scienziati di questo campo, ma non riflette in alcun modo i corrispondenti progressi concettuali o teorici rispetto al pensiero del diciannovesimo secolo che venne cosí abilmente riassunto da Sir D’Arcy Wentworth Thompson nel 1917. Tuttavia, in questo lavoro fondamentale per la nuova scienza emergente di conchigliologia teorica, l’unificazione delle considerazioni teoriche permette una rigorosa formulazione e una completa soluzione del problema della geometria biologica delle conchiglie. L’avvolgimento delle conchiglieintorno all’asse di simmetria si deduce dai primi princípi usando considerazioni sull’energia associata alla crescita per aumento di dimensioni. Questo lavoro mostra che le aperture delle conchiglie che sono incurvate (di tipo ?cowrie?), allargate verso l’esterno (di tipo ?strombe?) o anche rivoltate all’indietro (di tipo ?opistostomoideo?) sono solamente casi speciali di uno spettro piú ampio di traiettorie di crescita efficienti d’energia ?permesse? (spirali tensili, elastiche a molla d’orologio), molte delle quali vennero estesamente usate dagli ammoniti del Cretaceo. Considerazioni d’energia dettano anche la crescita della conchiglialungo l’asse di simmetria, cosí le spirali delle conchiglie marine possono essere comprese nei termini di certe speciali figure di rivoluzione (conoidi elastici di M?bius), essendo i gusci meglio conosciuti delle conchiglie coeloconoidali e cyrtoconoidali soltanto due casi speciali che derivano da una intera classe di geometria topologicamente possibili, efficienti di energia e biologicamente osservate. I ?fili? e i ?conoidi? di questo lavoro sono istruttive semplificazioni concettuali sufficienti a questo scopo. Un secondo lavoro tratterà successivamente superfici tubolari generalizzate in tre dimensioni.

---

Резюме Несмотря на большое количество литературы, до настоящего времени отсутствует строгий теоретический анализ геометрией морских раковин. Многотомность существующей палеонтологической литературы демонстрирует важность этой проблемы для прикладных ученых ине отражает соответствующих концептуальных и теоретических достижений после XIX века, которые были систематизированы Томсоном в 1917 г. Однако в этой фундаментальной работе по теоретической конхиологии были объединены теоретические рассмотрения, которые впервые позволили сформулировать и полностью решить проблему биологических геометрий морских раковин. Свертывание раковин спиралью вокруг оси симметрии можно получить из первых принципов, используя энергетические рассуждения, связанные с дифференциальным ростом. В настоящей статье показывается, что те отверстия раковин, которые являются искривленными, в форме раструба, или даже повернутыми назад, представляют просто частные случаи более широкого спектра ?разрешенных? траекторий роста. Энергетические рассуждения ткаже диктуют рост раковин вдоль оси симметрии, тонкое острие морских раковин можно понять в терминах некоторых специальных фигур вращения. ?Провода? и ?коноиды? настоящей статьи представляют поучительные концептуальные упрощения, достаточные для наших целей. Вторая статья будет посвящена трубчатым поверхностям в трех измерениях.

     

The post-post-Newtonian problem in classical electromagnetic theory

We find the Lagrangian to order c^?4 for two charged bodies (with $\text{e}{}_1\text{m}{}_1\text{=}\text{e}{}_2\text{m}{}_2$) in electromagnetic theory. This Lagrangian contains acceleration terms in its final form and we show why it is incorrect to eliminate these terms by using the equationsof motion in the Lagrangian as was done by Golubenkov and Smorodinskii, and by Landau and Lifshitz. We find the center of inertia and show that the potential energy term does not split equally between particles 1 and 2 as it does in the Darwin Lagrangian (Lagrangian to order c^?2). In addition to the infinite self-energy terms in the electromagnetic energy-momentum tensor, which are eliminated using Gupta's method, some new type of divergent terms are found in the moment of electromagnetic field energy and in the electromagnetic field momentum which cancel in the final conservation law for the center of inertia.     

The inverse problem in classical statistical mechanics

We address the problem of whether there exists an external potential corresponding to a given equilibrium single particle density of a classical system. Results are established for both the canonical and grand canonical distributions. It is shown that for essentially all systems without hard core interactions, there is a unique external potential which produces any given density. The external potential is shown to be a continuous function of the density and, in certain cases, it is shown to be differentiable. As a consequence of the differentiability of the inverse map (which is established without reference to the hard core structure in the grand canonical ensemble), we prove the existence of the Ornstein-Zernike direct correlation function. A set of necessary, but not sufficient conditions for the solution of the inverse problem in systems with hard core interactions is derived.Work partially supported by NSF grant PHY-8117463Work partially supported by NSF grant PHY-8116101 A01     

A resolution of the classical wave-particle problem

The classical wave-particle problem is resolved in accord with Newton's concept of the particle nature of light by associating particle density and flux with the classical wave energy density and flux. Point particles flowing along discrete trajectories yield interference and diffraction patterns, as illustrated by Young's double pinhole interference. Bound particle motion is prescribed by standing waves. Particle motion as a function of time is presented for the case of a particle in a box. Initial conditions uniquely determine the subsequent motion. Some discussion regarding quantum theory is preseted.     

The quantum measurement problem and selection of classical states

The problem of selection of preferred basis during passage from quantum to classical systems is treated with the help of a simple example of a 2-state system like the sugar molecule. A simple principle leading to this selection is stated and demonstrated in case of the chosen example. The principle, stated simply is that the preferred basis is the one in which the system environment interaction hamiltonian is diagonal. Talk given at the International Symposium on Theoretical Physics, Bangalore, November 1984.     

On the classical limit of the Kepler problem

We introduce coherent states on the dynamical group of the nonrelativistic Kepler (hydrogen atom) problem. In the limit of high excitation these states are well concentrated wavepackets which move along classical trajectories.     

An orbit-preserving discretization of the classical Kepler problem

We present a remarkable discretization of the 3-dimensional classical Kepler problem which preserves its trajectories and all integrals of motion. The points of any discrete orbit belong to some exact continuous trajectory.     

对一个力学碰撞问题的讨论

对普通物理力学中有关碰撞的一个问题,从恢复系数e的取值角度出发,进行了详细求解和讨论,得到了更完整的结果.     

On the global symmetry of the classical Kepler problem

A model for the classical Kepler problem is presented in which both the temporal evolution and the symmetry group act globally in a simple and canonical way. These actions are generated by the Hamiltonian function, the angular momentum and the Runge-Lenz vector. The symmetry group is SO(4) for negative and SO(1,3)₀ for positive energy.     

On the inverse FPR problem: Quantum is classical

The notion of quantum supports introduced by Foulis, Piron, and Randall can be used to construct combinatorial versions of contextualist hidden-variable models for finite quantum logics. The original logic can be uniquely recovered from appropriate such models as a solution of a combinatorial inverse problem. One can thus set up a classical ontology for a finite quantum logics that completely specifies it. Computer studies are used to explore the ideas.     

Modification of hyperspherical coordinates in the classical three-particle problem

A new version of the hyperspherical coordinates in the classical three-particle problem is considered, that leads to a modification of the Delves coordinates. A type of the kinetic and potential energy is obtained for the system. The problem is reduced in these coordinates for different cases of the classical motion. From geometrical reasonings a formula is chosen for the reduced mass of the three-body system. The triangle of masses and the relevant basic quantities and relations are introduced.     

Correspondences between the classical electrostatic Thomson problem and atomic electronic structure

Correspondences between the Thomson problem and atomic electron shell-filling patterns are observed as systematic non-uniformities in the distribution of potential energy necessary to change configurations of N ≤ 100 electrons into discrete geometries of neighboring N ? 1 systems. These non-uniformities yield electron energy pairs, intra-subshell pattern similarities with empirical ionization energy, and a salient pattern that coincides with size-normalized empirical ionization energies. Spatial symmetry limitations on discrete charges constrained to a spherical volume are conjectured as underlying physical mechanisms responsible for shell-filling patterns in atomic electronic structure and the Periodic Law.