Summary
Relativistic equations of motion are obtained from a general Lagrangian density when we have as constraints both ds 2=dz αdz α and the continuity equation. No Lagrange multipliers and, therefore, no arbitrary constants are introduced. The procedure is checked for a relativistic continuum with and without an electromagnetic field.
Riassunto
Si ottengono le equazioni di moto da una densità Lagrangiana quando sono presenti come vincoli, sia ds 2=dz αdz α che l'equazione di continuità. Non si introducono moltiplicatori di Lagrange e dunque nemmeno costanti arbitrarie. Il procedimento si applica al continuo relativistico con e senza il campo elettromagnetico.
Резюме
Получаются релятивистские уравнения движения из общей плотности Лагранжиана, когда мы имеем, в качестве ограничений, ds 2=dz αdz α и уравнение непрерывности. Не вводятся множители Лагранжа и, следовательно, никаких произвольных постоянных. Реализуется процедура проверки для релятивистского континуума с и без электромагнитного поля.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Bressan: inSymposia Mathematica (London, 1973), published by Istituto Nazionale di Alta Matematica, Roma.
B. Carter:Commun. Math. Phys.,30, 261 (1973).
G. Cavalleri andG. Spinelli:Nuovo Cimento,39 B, 93 (1977).
See for instance,G. Cavalleri andG. Spinelli:Phys. Rev. D,12, 2200, 2203 (1975). and references therein.
S. Deser:Gen. Rel. and Grav.,1, 9 (1970).
The barion conservation holds automatically also in Carter's method: see the introduction of ref. (2).
A conserved mass was also used, except inref. (8) by all the authors (I. M. Khalatnikov:Žurn. Ėksp. Teor. Fiz.,27, 529 (1954);A. H. Taub:Phys. Rev.,94, 1468 (1954);A. Lichnerowicz:Théories relativistes de la gravitation et de l'électromagnetisme (Paris, 1955), p. 74 and 101;Z. Koba:Prog. Theor. Phys.,14, 488 (1955);B. F. Schutz jr.:Phys. Rev. D,2, 2762 (1970);J. R. Ray:Journ. Math. Phys.,13, 1451, (1972)), who all dealt with the relativistic Lagrangian formulation for perfect fluids. These authors were also forced to introduce a constitutive equation. Moreover, some of them (Taub andSchutz) used Lagrange multipliers with nonholonomic constraints and thus introduced an arbitrary integration constant.
A nonconserved mass (because of the work performed by pressure) was used in our preceding paper:G. Cavalleri andG. Spinelli:Nuovo Cimento,25 B, 357 (1975).
G. A. Maugin andA. C. Eringen:Journ. Math. Phys.,13, 1788 (1972).
L. Infeld:Bull. Acad. Polon. Sci., Cl. III,5, 491 (1957);G. Kalman:Phys. Rev.,123, 384 (1961). See alsoI. M. Khalatnikov:Žurn. Ėksp. Teor. Fiz.,27, 529 (1954).
G. Cavalleri andG. Spinelli:Lett. Nuovo Cimento,9, 325 (1974).
We employ here point transformations which must not be confused with co-ordinate transformations. See, for instance,F. Plybon:Journ. Math. Phys.,12 57 (1971).
In such a reference frame we can writeS αβ=P αμ P βν t μν, wheret μν is a generalization oft rs(withr, s=1, 2, 3) witht 0ν arbitrary andP αβ=a αβ−ż α ż β. Such expression being written in a tensor form, they are valid in any reference system.
See, for instance,C. Moller:The Theory of Relativity, Sect.55 (Oxford, 1952).
(see for instanceM. Arzeliès:Compt. Rend.,270 A, 347 (1970)).
The variation of the co-ordinatesz α of the matter element does not affect the terms of the pure fields (including their mutual interactions), which are functions of thex α—co-ordinates.
Author information
Authors and Affiliations
Additional information
Перевебено ребакцией.
Rights and permissions
About this article
Cite this article
Cavalleri, G., Spinelli, G. Relativistic Lagrangian equations of motion with constraints: Check on the continuum. Nuovo Cim B 39, 87–92 (1977). https://doi.org/10.1007/BF02738178
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02738178