On the integrability conditions of the equations for Einsteinian gravitational fields

Summary Within the terms of the Einsteinian General Theory of Relativity is studied the behaviour of a material particle with a mass that is not too great in comparison with the total mass of the bodies which generate the field to which the particle itself is subject. The integration of the gravitational equations implies that the corpuscle is rigidly connected with a particular reference system of Fermi type, which is coordinated with the time line of the corpuscle itself.

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Sommario Si studia il comportamento, nella teoria einsteiniana della Relatività generale, di una particella materiale fornita di massa non troppo elevata rispetto alla massa complessiva dei corpi che generano il campo a cui è soggetta la particella stessa. L'integrazione delle equazioni gravitazionali implica che il corpuscolo sia solidale con un particolare sistema di riferimento di Fermi, coordinato alla linea oraria del corpuscolo stesso.

     

Hodograph relativistic equations of plane flows

Summary Various forms of hodograph equations for relativistic irrotational steady plane flows are worked out. The transformation from the hodograph to the physical plane is studied. Elementary solutions of these equations are found. An approximate hodograph equation for relativistic transonic flows is determined.

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Sommario Si ricavano, in diverse forme, le equazioni odografiche per le correnti relativistiche irrotazionali stazionarie piane. Si esamina il passaggio dal piano odografico al piano fisico. Si trovano alcune soluzioni elementari delle dette equazioni. Si determina un'equazione odografica approssimata per le correnti transoniche relativistiche.

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This work was done in the sphere of activity of the C.N.R. groups for mathematical research.     

On the equations of balance for mixtures constructed by means of classical statistical mechanics

Summary We use the techniques of an earlier paper to construct gross quantities and equations of balance for each constituent of a mixture as well as for the whole mixture. The equations of balance for mass, linear momentum and energy for the constituents are similar to those proposed by TRUESDELL, but here growth terms are absent and the peculiar stress tensor and heat flux vector have different interpretations than TRUESDELL's. We also show that the gross fields for the mixture, and the equations they satisfy, can be obtained by suitably adding their analogues for the constituents. In spite of the fact that the internal forces on the ^th constituent are described by means of a peculiar stress tensor T, we show that this theory does not give rise to the paradox considered and resolved by GURTIN, OLIVER & WILLIAMS.

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Sommario Mediante tecniche presentate in un precedente lavoro costruisco campi macroscopici ed equazioni di bilancio per ogni componente di una miscela cosi come per la miscela nel suo complesso. Le equazioni di bilancio per la massa, la quantità di moto e l'energia per i costituenti sono simili a quelle proposte da TRUESDELL, ma qui mancano i termini di sorgente, inoltre il tensore degli sforzi e il vettore flusso termico hanno interpretazioni diverse da quelle di TRUESDELL. Mostro che i campi macroscopici, e le equazioni che essi soddisfano, si ottengono sommando opportunamente gli analoghi costruiti per i componenti. Malgrado le forze interne su un componente vengano rappresentate mediante un tensore degli sforzi, mostro che questa teoria non da luogo al paradosso analizzato e risolto da GURTIN, OLIVER & WILLIAMS.

     

The Burnett equations for a relativistic gas

The Burnett constitutive equations for the dynamic pressure, heat flux and pressure deviator are obtained from the relativistic Boltzmann equation by using fourteen moment equations and an iteration method akin to the Maxwellian iteration procedure of the non-relativistic theory. The non-relativistic and ultra-relativistic limits of all Burnett coefficients are given and it is shown that the non-relativistic limit of the Burnett constitutive equations agree with those obtained from the non-relativistic Boltzmann equation. Received July 21, 2000     

On the kinetic theory of relativistic gases

Based on the relativistic Boltzmann equation and on the combined method of Chapman-Enskog and Grad, the laws of Navier-Stokes and Fourier are obtained for a single relativistic gas. Besides, it is shown that there exits only one potential function that corresponds to the relativistic Maxwellian potential function. Received: October 30, 1996     

The dilation group and dimensionless quantities in classical and relativistic hydrodynamics

Summary We examine from a group theoretical point of view the approach to the study of hydrodynamics given by dimensional analysis. We will study the action of the dilation group: it transforms a hydrodynamical problem into a “similar” one i.e. into a problem differring by the choice of the units. We want to find and relate to group properties the quantities whose constancy is sufficient to guarantee the similiarity of two problems. We introduce the concept of commensurable problems and we find that the dimensionless quantities which are commonly used in hydrodynamics (such as the Reynolds number e.g.) when restricted to this subclass of problems are sufficient to label the orbits of the dilation group i.e. the sets of the similar problems. The relativistic version of this analysis and some other extensions will be also considered.

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Sommario Si studia da un punto di vista gruppale l'approccio che l'analisi dimensionale fornisce allo studio della idrodinamica. Verrà esaminata l'azione del gruppo delle dilatazioni: esso trasforma un problema idrodinamico in uno “simile”, in uno cioè che differisce per la scelta delle unità. Verranno indicate e collegate a proprietà gruppali quelle grandezze la cui costanza è sufficiente ad assicurare che due problemi siano simili. Dopo aver introdotto il concetto di problemi commensurabili, troviamo che quelle quantità adimensionali, come il numero di Reynolds per esempio, che si usano normalmente in idrodinamica sono sufficienti a caratterizzare le orbite per il gruppo delle dilatazioni, cioè gli insiemi dei problemi simili, quando ristrette alla sottoclasse dei problemi commensurabili. Si considerano anche la versione relativistica ed alcune altre estensioni di questa analisi.

     

On pure extension

Summary A homogeneous deformation field, called pure extension is introduced. Various applications of this field are considered. Pure extension leads to a reasonable static interpretation of the strong-ellipticity condition in linearised elasticity theory. The response of a compressible elastic material to pure extension is related to the speeds of propagation of infinitesimal plane waves. Finally, a generalised Tension-Extension condition is introduced and discussed.     

On circulation in relativistic hydrodynamics

In this paper it is shown that the relativistic analogue of the circulation in classical hydrodynamics is the integral

$$C = \int {\left( {{\text{1 + }}\frac{i}{{c^{\text{2}} }}} \right)} {\text{ }}U_\alpha dx^\alpha ,$$     

On the contact problem of classical elasticity

We deal with the contact problem of homogeneous and isotropic linear elastostatics in the exterior of a bounded convex domain of ${\Bbb{R}}^{3}$ . We prove existence and uniqueness of a solution, provided the elasticity tensor is only strongly elliptic.     

Group properties of the equations of adiabatic motion of a medium in relativistic hydrodynamics

A group classification is presented and the complete set of invariant solutions is found for the equations of adiabatic motion of a medium in relativistic hydrodynamics.     

关于经典Appell-Hamel例——————分析力学札记之九

 不可积分微分约束系统动力学有Appell-Chetaev模型和Vacco模型,对经典Appell-Hamel例两个模型有同样解.     

关于 App ell方程1）--分析力学札记之二十八

Appell 方程是非完整系统独具特色的一类方程,凡涉及非完整力学的著作几乎都会介绍Appell方程. 本札记介绍Appell方程的形成和发展,包括Appell的原述,诸多名家对Appell方程的理解和表述,方程的称谓以及方程的后续发展.     

关于Appell方程——分析力学札记之二十八

Appell 方程是非完整系统独具特色的一类方程,凡涉及非完整力学的著作几乎都会介绍Appell方程. 本札记介绍Appell方程的形成和发展,包括Appell的原述,诸多名家对Appell方程的理解和表述,方程的称谓以及方程的后续发展.