New Mathematical Methods for Physics

$130.00

Series: Computational Mathematics and Analysis
BISAC: SCI040000

The concept of “group” has been introduced in mathematics for the first time by E. Galois (1830) and slowly passed from algebra to geometry with the work of S. Lie on Lie groups (1880) and Lie pseudogroups (1890) of transformations. The concept of a finite length differential sequence, now called the Janet sequence, had been described for the first time by M. Janet (1920). Then, the work of D. C. Spencer (1970) has been the first attempt to use the formal theory of systems of partial differential equations (PDE) in order to study the formal theory of Lie pseudogroups.

However, the linear and nonlinear Spencer sequences for Lie pseudogroups, though never used in physics, largely supersede the ”Cartan structure equations ” (1905) and are quite different from the ”Vessiot structure equations ” (1903), introduced for the same purpose but never acknowledged by E. Cartan or successors. Meanwhile, mixing differential geometry with homological algebra, M. Kashiwara (1970) created “algebraic analysis” in order to study differential modules and double duality. By chance, unexpected arguments have been introduced by the brothers E. and F. Cosserat (1909) in order to revisit elasticity and by H. Weyl (1918) in order to revisit electromagnetism through a unique differential sequence only depending on the structure of the conformal group of space-time.

The classical Galois theory deals with certain finite algebraic extensions and establishes a bijective order reversing correspondence between the intermediate fields and the subgroups of a group of permutations called the Galois group of the extension. It has been the dream of many mathematicians at the end of the nineteenth century to generalize these results to systems of linear or algebraic PDE and the corresponding finitely generated differential extensions, in order to be able to add the word differential in front of any classical statement.

The achievement of the Picard-Vessiot theory by E. Kolchin and coworkers between 1950 and 1970 is now well-known. However, the work of Vessiot on the differential Galois theory (1904), that is on the possibility to extend the classical Galois theory to systems of algebraic PDE and algebraic Lie pseudogroups, namely groups of transformations solutions for systems of algebraic PDE, has also never been acknowledged. His main idea has been to notice that the Galois theory (old and new) is a study of principal homogeneous spaces (PHS) for algebraic groups or pseudogroups described by what he called ”automorphic systems” of PDE.

The purpose of this book is first to revisit Gauge Theory and General Relativity in light of the latest developments just described and then to apply the differential Galois theory in order to revisit various domains of mechanics (Shell theory, Chain theory, Frenet-Serret formulas, Hamilton-Jacobi equations). All the results presented are new.

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Table of Contents

Table of Contents

Preface

Chapter 1. Algebraic Analysis and General Relativity (pp. 1-108)

Chapter 2. Differential Galois Theory and Mechanics (pp. 109-184)

About the Author (pp. 185-186)

Index (pp. 187)

References

Chapter 1:
[1] Adler, F.W.: ¨Uber die Mach-Lippmannsche Analogie zum zweiten
Hauptsatz, Anna. Phys. Chemie, [About the Mach-Lippmann analogy to
the second Main sentence, Anna. Phys. chemistry 22, 578-594 (1907)].
[2] Airy, G.B.: On the Strains in the Interior of Beams, Phil. Trans. Roy.
Soc. London, 153, 1863, 49-80 (1863).
[3] Arnold, V.: M´ethodes Math´ematiques de la M´ecanique Classique, Appendice
2 (G´eod´esiques des m´etriques invariantes `a gauche sur des
groupes de Lie et hydrodynamique des fluides parfaits) [Mathematical
Methods of the Classical Mechanics], Appendix 2 (Metaphysical Methods
of Metaphysics) invariant left on Lie groups and hydrodynamics of
perfect fluids), MIR, Moscow (1974,1976)].
[4] Assem, I.: Alg`ebres et Modules, [Algebra and Modules], Masson, Paris
(1997)].
[5] Beltrami, E.: Osservazioni sulla Nota Precedente, Atti Reale Accad. Naz.
Lincei Rend., [Remarks on the Previous Note, Acts Royal Accad. Nat.
Lincei Rend.] 5, 141-142 (1892).
[6] Birkhoff, G.: Hydrodynamics, Princeton University Press (1954).
[7] Bjork, J.E. (1993) Analytic D-Modules and Applications,Kluwer (1993).
[8] Bourbaki, N.: Alg`ebre, Ch. 10, Alg`ebre Homologique [Homological Algebra],
Masson, Paris (1980).
[9] de Broglie, L.: Thermodynamique de la Particule isol´ee [Thermodynamics
of the Isolated Particle], Gauthiers-Villars, Pris 1964).
[10] Choquet-Bruhat, Y.: Introduction to General Relativity, Black Holes and
Cosmology, Oxford University Press (2015).
[11] Chyzak, F.,Quadrat, A., Robertz, D.: Effective algorithms for parametrizing
linear control systems over Ore algebras, Appl. Algebra Engrg.
Comm. Comput., 16, 319-376, 2005.
[12] Chyzak, F., Quadrat, A., Robertz, D.: OREMODULES: A symbolic
package for the study of multidimensional linear systems, Springer,
Lecture Notes in Control and Inform. Sci., 352, 233-264, 2007.
http://wwwb.math.rwth-aachen.de/OreModules.
[13] Cosserat, E., & Cosserat, F.: Th´eorie des Corps D´eformables[Deformable
Body Theory, Hermann, Paris, 1909.
[14] Eisenhart, L.P.: Riemannian Geometry, Princeton University Press,
Princeton (1926).
[15] Foster, J., Nightingale, J.D.: A Short Course in General relativity, Longman
(1979).
[16] Gr ¨obner, W.: ¨Uber die Algebraischen Eigenschaften der Integrale
von Linearen Differentialgleichungen mit Konstanten Koeffizienten,
Monatsh. der Math. [On the Algebraic Properties of the Integrals of
Linear Differential Equations with Constants Coefficients, Monatsh. the
math.], 47, 247-284 (1939).
[17] Hu, S.-T.: Introduction to Homological Algebra, Holden-Day (1968).
[18] Hughston, L.P., Tod, K.P.: An Introduction to General Relativity, London
Math. Soc. Students Texts 5, Cambridge University Press (1990).
[19] Janet, M.: Sur les Syst`emes aux D´eriv´ees Partielles, Journal de Math.
[On Systems at Partial Rivals, Math Journal, 8, 65-151 (1920).
[20] Kashiwara,M.: Algebraic Study of Systems of Partial Differential Equations,
M´emoires de la Soci´et´e Math´ematique de France, [Memories of the
Mathematical Society of France] 63 (1995) (Transl. from Japanese of his
1970 Master’s Thesis).
[21] Kolchin, E.R.: Differential Algebra and Algebraic groups, Academic
Press, New York (1973).
[22] Kumpera, A., & Spencer, D.C.: Lie Equations, Ann. Math. Studies 73,
Princeton University Press, Princeton (1972).
[23] Kunz, E.: Introduction to Commutative Algebra and Algebraic Geometry,
Birkhaser (1985).
[24] Lippmann, G.: Extension du Principe de S. Carnot `a la Th´eorie des
P´enom`enes ´electriques, [Extension of S. Carnot’s Principle to the Theory
of Electric Phenomenes] C. R. Acad/ Sc. Paris, 82, 1425-1428 (1876).
[25] Lippmann, G.: ¨Uber die Analogie zwischen Absoluter Temperatur un
Elektrischem Potential [On the analogy between absolute temperature
and electrical potential], Ann. Phys. Chem., 23, 994-996 (1907).
[26] Macaulay, F.S.: The Algebraic Theory of Modular Systems, Cambridge
(1916).
[27] Mach, E.: Die Geschichte und die Wurzel des Satzes von der Erhaltung
der Arbeit [The story and the root of the sentence of the conservation of
work], p 54, Prag: Calve (1872).
[28] Mach, E.: Prinzipien der W¨armelehre, 2, Aufl. [Principles of Written
Teaching, 2, Edition], p 330, Leipzig: J.A. Barth (1900).
[29] Maxwell, J.C.: On Reciprocal Figures, Frames and Diagrams of Forces,
Trans. Roy. Soc. Edinburgh, 26, 1-40 (1870).
[30] Morera, G.: Soluzione Generale della Equazioni Indefinite dellEquilibrio
di un Corpo Continuo, [General Solution of the Indefinite Equations of
the Balance of a Continuous Body], Atti. Reale. Accad. dei Lincei 1,
137-141+233(1892).
[31] Nordstr ¨om, G.: Einstein’s Theory of Gravitation and Herglotz’sMechanics
of Continua, Proc. Kon. Ned. Akad. Wet., 19, 884-891 (1917).
[32] Northcott, D.G.: An Introduction to Homological Algebra, Cambridge
university Press (1966).
[33] Northcott, D.G.: Lessons on Rings Modules and Multiplicities, Cambridge
University Press (1968).
[34] Oberst, U.: Multidimensional Constant Linear Systems, Acta Appl.
Math., 20, 1-175 (1990).
[35] Oberst, U.: The Computation of Purity Filtrations over Commutative
Noetherian Rings of Operators and their Applications to Behaviours,
Multidim. Syst. Sign. Process. (MSSP) 26, 389-404 (2013).
http://dx.doi.org/10.1007/s11045-013-0253-4.
[36] Ougarov, V.: Th´eorie de la Relativit´e Restreinte [Theory of Relativity Restricted],
MIR, Moscow, 1969, (French translation, 1979).
[37] Poincar´e, H.: Sur une Forme Nouvelle des Equations de la M´ecanique
[On a New Form of the Magic Equations], C. R. Acad´emie des Sciences
Paris, 132 (7) (1901) 369-371.
[38] Pommaret, J.-F.: Systems of Partial Differential Equations and Lie Pseudogroups,
Gordon and Breach, New York, 1978; Russian translation:
MIR, Moscow, 1983.
[39] Pommaret, J.-F.: Differential Galois Theory, Gordon and Breach, New
York, 1983.
[40] Pommaret, J.-F.: Lie Pseudogroups and Mechanics, Gordon and Breach,
New York, 1988.
[41] Pommaret, J.-F.: Partial Differential Equations and Group Theory,
Kluwer, 1994. http://dx.doi.org/10.1007/978-94-017-2539-2.
[42] Pommaret, J.-F.: Franc¸ois Cosserat and the Secret of the Mathematical
Theory of Elasticity, Annales des Ponts et Chauss´ees, 82, 59-66 (1997)
(Translation by D.H. Delphenich).
[43] Pommaret, J.-F.: Group Interpretation of Coupling Phenomena, Acta Mechanica,
149 (2001) 23-39. http://dx.doi.org/10.1007/BF01261661.
[44] Pommaret, J.-F.: Partial Differential Control Theory, Kluwer, Dordrecht,
2001.
[45] Pommaret, J.-F.: Algebraic Analysis of Control Systems Defined by Partial
Differential Equations, in “Advanced Topics in Control Systems Theory”,
Springer, Lecture Notes in Control and Information Sciences 311
(2005) Chapter 5, pp. 155-223.
[46] Pommaret, J.-F.: Arnold’s Hydrodynamics Revisited, AJSEmathematiques,
1, 1, 2009, pp. 157-174.
[47] Pommaret, J.-F.: Parametrization of Cosserat Equations, ActaMechanica,
215 (2010) 43-55. http://dx.doi.org/10.1007/s00707-010-0292-y.
[48] Pommaret, J.-F.: Macaulay Inverse Systems revisited, Journal of Symbolic
Computation, 46, 1049-1069 (2011).
[49] Pommaret, J.-F.: Spencer Operator and Applications: From Continuum
Mechanics to Mathematical Physics, in “Continuum Mechanics-
Progress in Fundamentals and Engineering Applications”, Dr. Yong
Gan (Ed.), ISBN: 978-953-51-0447–6, InTech, 2012, Available from:
http://dx.doi.org/10.5772/35607.
[50] Pommaret, J.-F.: The Mathematical Foundations of General Relativity
Revisited, Journal of Modern Physics, 4 (2013) 223-239.
http://dx.doi.org/10.4236/jmp.2013.48A022
[51] Pommaret, J.-F.: The Mathematical Foundations of Gauge Theory
Revisited, Journal of Modern Physics, 5 (2014) 157-170.
http://dx.doi.org/10.4236/jmp.2014.55026.
[52] Pommaret, J.-F.: Relative Parametrization of Linear Multidimensional
Systems, Multidim. Syst. Sign. Process., 26, 405-437 2015). DOI
10.1007/s11045-013-0265-0.
[53] Pommaret,J.-F.: FromThermodynamics to Gauge Theory: theVirial Theorem
Revisited, pp. 1-46 in “Gauge Theories and Differential geometry”,
NOVA Science Publisher (2015).
[54] Pommaret, J.-F.: Airy, Beltrami, Maxwell, Einstein and Lanczos Potentials
revisited, Journal of Modern Physics, 7, 699-728 (2016).
http://dx.doi.org/10.4236/jmp.2016.77068.
[55] Pommaret, J.-F.: Deformation Theory of Algebraic and Geometric
Structures, Lambert Academic Publisher (LAP), Saarbrucken, Germany
(2016). A short summary can be found in ”Topics in Invariant Theory”,
S´eminaire P. Dubreil/M.-P. Malliavin, Springer Lecture Notes in Mathematics,
1478, 244-254 (1990). http://arxiv.org/abs/1207.1964
[56] Pommaret, J.-F. and Quadrat, A.: Localization and Parametrization of
Linear Multidimensional Control Systems, Systems & Control Letters,
37, 247-260 (1999).
[57] Pommaret, J.-F., Quadrat, A.: Algebraic Analysis of Linear Multidimensional
Control Systems, IMA Journal ofMathematical Control and Informations,
16, 275-297 (1999).
[58] Quadrat, A., Robertz, D.: Parametrizing all solutions of uncontrollable
multidimensional linear systems, Proceedings of the 16th IFAC World
Congress, Prague, July 4-8, 2005.
[59] Quadrat, A.: An Introduction to Constructive Algebraic Analysis and its
Applications,Les cours du CIRM, Journees Nationales de Calcul Formel,
1(2), 281-471 (2010).
[60] Quadrat, A., Robertz, R.: A Constructive Study of the Module Structure
of Rings of Partial Differential Operators, Acta Applicandae Mathematicae,
133, 187-234 (2014). http://hal-supelec.archives-ouvertes.fr/hal-
00925533.
[61] Rotman, J.J.: An Introduction to Homological Algebra, Pure and Applied
Mathematics, Academic Press (1979).
[62] Schneiders, J.-P.: An Introduction to D-Modules, Bull. Soc. Roy. Sci.
Li`ege, 63, 223-295 (1994).
[63] Spencer, D.C.: Overdetermined Systems of Partial Differential Equations,
Bull. Am. Math. Soc., 75 (1965) 1-114.
[64] Teodorescu, P.P.: Dynamics of Linear Elastic Bodies, Abacus Press, Tunbridge,
Wells (1975) (Editura Academiei, Bucuresti, Romania).
[65] Vessiot, E.: Sur la Th´eorie des Groupes Infinis [On the theme of the Infinite
Groups], Ann. Ec. Norm. Sup., 20, 411-451 (1903) (Can be obtained
from http://numdam.org).
[66] Weyl, H.: Space, Time, Matter, Springer, 1918, 1958; Dover, 1952.
[67] Zerz, E.: Topics in Multidimensional Linear Systems Theory, Lecture
Notes in Control and Information Sciences (LNCIS) 256, Springer
(2000).
[68] Zou, Z., Huang, P., Zang ,Y., Li, G.: Some Researches on Gauge Theories
of Gravitation, Scientia Sinica, XXII, 6, 628-636 (1979).
Chapter 2:
[1] Artin, E.: Galois Theory, Notre Dame Mathematical Lectures, 2, 1942,
1997.
[2] Bonnet, O.: M´emoire sur la Th´eorie des Surfaces Applicables sur une
Surface Donn´ee [Memory on the Threshold of Applicable Surfaces on a
Given Surface], Journal de l’Ecole Polytechnique, 25 (1867) 31-151.
[3] Byalinicki-Birula, A.: On the field of rational functions of Algebraic
Groups, Pacific Journal of Mathematics, 11 (1961) 1205-1210.
[4] Bialynicki-Birula, A.: On Galois Theory of Fields with Operators, Amer.
J. Math., 84 (1962) 89-109.
[5] Cartan, E.: Sur la Possibilit´e de Plonger une Surface Riemannienne dans
un Espace Euclidien [On the Possibility of Diving a Riemannian Surface
in an Euclidean Space], Ann. Soc. Pol. Math., 6 (1927) 1-7.
[6] Chevalley, C., Samuel, P.: Two Proofs of a Theorem on Algebraic groups,
Proc. Amer. Math. Soc., 2 (1951) 126-134;
[7] Ciarlet, P.G., Larsonneur, F.P.: On the Recovery of a Surface with Prescribed
First and Second Forms, J. Math. Pures Appl., 81 (2002) 167-
185.
[8] Codazzi, D.: Sulle Coordinate Curvilinee d’una Superficiedello Spazio
[On the Curvilinear Coordinates of a Space Superfielder], Ann. Math.
Pura Applicata, 2 (1868-1869) 101-19.
[9] Drach, J.: Th`ese de Doctorat, Ann. Ec. Normale Sup. (3) 15 (1898)
243-384.
[10] Gauss, K.F.: Disquitiones Generales circa Superficies Curvas, Comm.
Soc. Gott., 6 (1828).
[11] Han, Q.: Global Isometric Embedding of Surfaces in R3, Chapter 2 in
“Differential geometry and Coninuum mechanics”, Springer, 2015.
[12] Huaux, A.: Sur la S´eparation des Variables dans l’Equation aux d´eriv´ees
partielles de Hamilton-Jacobi [On the Separation of Variables in theHamilton-Jacobi Partial Equation Equation], Annali di Matematica Pura
ed Applicata, 108 (1976) 251-282.
[13] Janet, M.: Sur la Possibilit´e de Plonger une Surface Riemannienne dans
un Espace Euclidien [On the Possibility of Diving a Riemannian Surface
in an Euclidean Space], Ann. Soc. Pol. Math., 5 (1926) 38-43.
[14] Kaplanski, I.: An Introduction to Differential Algebra, Hermann, 1957,
1976.
[15] Kolchin, E.R.: Differential Algebra and Algebraic groups, Academic
Press, 1973.
[16] Kolchin, E.R.: Differential Algebraic Groups, Pure and Applied Mathematics,
114, Academic Press, 1985.
[17] Levi-Civitta, T.: Sulla Integrazione delle Equazione di Hamilton-Jacobi
per Separazione di Variabili [On the Integration of the Hamilton-Jacobi
Equation for the Separation of Variables], Math. Annalen, 59, (1904) 383
(Also “Opere Matematische”, 2 (1901-1907) 395-410).
[18] Mainardi, G.: Su la Theoria Generale delle Superficie [On the General
Theory of the Surface], Giornale dell’ Instituto Lombardo, 9 (1856) 385-
404.
[19] Pommaret, J.-F.: Systems of Partial Differential Equations and Lie Pseudogroups,
Gordon and Breach, New York, 1978 (Russian translation by
MIR, Moscow, 1983).
[20] Pommaret, J.-F.: Differential Galois Theory, Gordon and Breach, New
York, 1983 (750 pp).
[21] Pommaret, J.-F.: Lie Pseudogroups and Mechanics, Gordon and Breach,
New York, 1988 (590 pp).
[22] Pommaret, J.-F.: Partial Differential Equations and Group
Theory: New Perspectives for Applications, Kluwer, 1994.
http://dx.doi.org/10.1007/978-94-017-2539-2.
[23] Pommaret, J.-F.: Partial Differential Control Theory, Kluwer, 2001 (957
pp).
[24] Pommaret, J.-F.: Algebraic Analysis of Control Systems Defined by
Partial Differential Equations, in Advanced Topics in Control Systems
Theory, Lecture Notes in Control and Information Sciences LNCIS 311,
Chapter 5, Springer, 2005, 155-223.
[25] Pommaret, J.-F.: Relative Parametrization of Linear Multidimensional
Systems, Multidim. Syst. Sign. Process., 26 (2015) 405-437. DOI
10.1007/s11045-013-0265-0
[26] Pommaret, J.-F.: Deformation Theory of Algebraic and Geometric Structures,
Lambert Academic Publisher (LAP), Saarbrucken, Germany, 2016.
[27] Ritt, J.F.: Differential Algebra, Dover, 1950, 1966.
[28] Stewart, I.: Galois Theory, Chapman and Hall, 1973.
[29] Vessiot, E.: Sur la Th´eorie de Galois et ses Diverses G´en´eralisations [On
the Theory of Galois and its Diverse Generalizations], Ann. Ec. Normale
Sup., 21 (1904) 9-85 (Can be obtained from http://numdam.org).
[30] Zariski, O., Samuel, P.: Commutative Algebra, Van Nostrand, 1958.

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