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Geometric formulation of classical mechanics and field theory

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References

  1. E. Cartan:Leçons sur les invariants intégraux (Paris, 1922).

  2. Ŕ. Abraham andJ. E. Marsden:Foundations of Mechanics (New York, N. Y., 1967); II edition (New York, N. Y., 1978). The second edition may be considered as a new book and contains an extensive bibliography.

  3. W. Thirring:Classical Dynamical Systems (New York, N. Y., and Wien, 1978).

  4. C. C. Wang:Mathematical Principles of Mechanics and Electromagnetism (Part A) (New York, N. Y., and London, 1979).

  5. In this context see, for instance,R. Hermann: inTopics in Quantum Field Theory, edited byJ. A. de Azcárraga (New York, N. Y., 1978);C. N. Yang:Ann. N. Y. Acad. Sci.,294, 86 (1977), and references therein.

  6. P. Dedecker:Calcul des variations, formes différentielles et champs géodésiques, inColloque Internationale de Géometrie Différentielle, Strasbourg, 1953 (Paris, 1954);Calculs des variations et topologie algébrique, inMémories de la Societé Royale des Sciences de Liége, 4ème série, XIX, fasc. I. See alsoOn the Generalization of Symplectic Geometry to Multiple Integrals in the Calculus of Variations, inLecture Notes in Mathematics,570 (1977), p. 395.

  7. A. Trautman:Commun. Math. Phys.,6, 248 (1967).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. P. L. García:Collect. Math.,19, 73, 155 (1968).

    MathSciNet  MATH  Google Scholar 

  9. R. Hermann:Vector Bundles in Mathematical Physics, Vol.1 (New York, N. Y., 1970);Geometry, Physics and Systems (New York, N. Y., 1973).

  10. Y. Choquet-Bruhat, C. De Witt-Morette andM. Dillard-Bleick:Analysis, Manifolds and Physics (Amsterdam, 1978).

  11. J. Śniatycki:Proc. Cambridge Philos. Soc.,68, 475 (1970).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. H. Goldschmidt andS. Sternberg:Ann. Inst. Fourier (Grenoble),23, 203 (1973);H. Goldschmidt: inGéometrie différentielle (1972) Colloque, Santiago de Compostela, inLecture Notes in Mathematics, Vol.392 (Berlin, 1974).

    Article  MathSciNet  MATH  Google Scholar 

  13. P. L. García:Symp. Math.,14, 219 (1974).

    Google Scholar 

  14. P. L. García andA. Pérez-Rendón:Arch. Ration. Mech. Anal.,43, 101 (1971).

    Article  MATH  Google Scholar 

  15. D. Krupka:Folia Fac. Sci. Nat. Univ. Purkinianae Brunensis (Physica),14, 1 (1975).

    Google Scholar 

  16. D. Krupka:J. Math. Anal. Appl.,49, 180, 469 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  17. W. Szczyrba:Ann. Pol. Math.,32, 145 (1976).

    MathSciNet  MATH  Google Scholar 

  18. J. Kijowski andW. Szczyrba:Commun. Math. Phys.,46, 183 (1976).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. D. Krupka:Arch. Math. 2, Scripta Fac. Sci. Nat. UJEP Brunensis,12, 99 (1976).

    MathSciNet  Google Scholar 

  20. P. Rodrigues:J. Math. Phys. (N. Y.),18, 1720 (1977).

    Article  ADS  MATH  Google Scholar 

  21. V. Aldaya andJ. A. de Azcárraga:J. Math. Phys. (N. Y.),19, 1869 (1978).

    Article  ADS  MATH  Google Scholar 

  22. C. Godbillon:Géometrie différentielle et mécanique analytique (Paris, 1969).

  23. P. Malliavin:Géometrie différentielle intrinseque (Paris, 1972).

  24. J. M. Souriau:Structure des systèmes dynamiques (Paris, 1969).

  25. V. I. Arnold:Mathematical Methods of Classical Mechanics (Berlin, 1978) (Russian edition, 1974).

  26. S. Mac Lane:Am. Math. Monthly,77, 570 (1970), reprinted inSelected Papers, edited byI. Kaplanski (New York, N. Y., and Heidelberg, 1979).

    Article  MathSciNet  Google Scholar 

  27. R. M. Santilli:Foundations of Theoretical Mechanics (I) (New York, N. Y., 1978).

  28. R. Jost:Rev. Mod. Phys.,36, 572 (1964).

    Article  ADS  Google Scholar 

  29. E. T. Whittaker:A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (Cambridge, 1937).

  30. W. M. Tulczyjew:Ann. Inst. Henri Poincaré,27, 101 (1977).

    MathSciNet  MATH  Google Scholar 

  31. See,e.g.,J. M. Lévy-Leblond: inGroup Theory and its Applications, edited byE. M. Loebl, Vol.2 (New York, N. Y., 1971), and references therein.

  32. M. Pauri andG. M. Prosperi:J. Math. Phys. (N. Y.),7, 366 (1966);8, 2256 (1967);9, 1146 (1968).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  33. E. C. G. Sudarshan andN. Mukunda:Classical Dynamics: A Modern Perspective (New York, N. Y., 1974).

  34. V. Bargmann:Ann. Math.,59, 1 (1954). See alsoM. Hamermesh:Group Theory and its Applications to Physical Problems (New York, N. Y., 1962).

    Article  MathSciNet  MATH  Google Scholar 

  35. E. P. Wigner:Ann. Math.,40, 149 (1939).

    Article  MathSciNet  Google Scholar 

  36. L. D. Landau andE. M. Lifschitz:Mechanics (New York, N. Y., 1960).

  37. J. M. Lévy-Leblond:Commun. Math. Phys.,12, 64 (1969).

    Article  ADS  MATH  Google Scholar 

  38. J. Kijowski:Commun. Math. Phys.,30, 99 (1973).

    Article  MathSciNet  ADS  Google Scholar 

  39. J. Kijowski andW. Szczyrba:Commun. Math. Phys.,46, 183 (1976).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  40. J. Eells:Bull. Am. Math. Soc.,72, 751 (1966).

    Article  MathSciNet  Google Scholar 

  41. R. S. Palais:Foundations of Global Nonlinear Analysis (New York, N. Y., 1968).

  42. S. Lang:Differentiable Manifolds (New York, N. Y., 1972).

  43. Pham Mau Quan:Introduction a la géométrie des variétés différentiables (Paris, 1969).

  44. F. Hirzebruch:Topological Methods in Algebraic Geometry (Berlin, 1966).

  45. G. Mack andA. Salam:Ann. Phys. (N. Y.),53, 174 (1969);S. Coleman andA. Jackiw:Ann. Phys. (N. Y.),67, 552 (1971),. For the structure of the conformal group see alsoH. A. Kastrup:Phys. Rev.,142, 1060 (1966).

    Article  MathSciNet  ADS  Google Scholar 

  46. D. Husemoller:Fibre Bundles, second edition (New York, N. Y., 1975);N. Steenrod:The Topology of Fibre Bundles (Princeton, N. J., 1951).

  47. J. L. Koszul:Lectures on Fibre Bundles and Differential Geometry (Bombay, 1960).

  48. G. H. Thomas:Riv. Nuovo Cimento,3, No. 4 (1980).

  49. W. Drechsler andM. E. Mayer:Fibre bundle techniques in gauge theories, inLecture Notes in Physics, Vol.67 (Berlin, 1977).

  50. A. Trautman:Rep. Math. Phys.,1, 29 (1970).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  51. M. C. Ehresmann:C. R. Acad. Sci. Paris,233, 598, 777, 1081 (1951).

    MathSciNet  MATH  Google Scholar 

  52. J. F. Pommaret:Systems of Partial Differential Equations and Lie Pseudogroups (New York, N. Y., and Paris, 1978).

  53. J. C. Tougeron:Idéaux des fonctions différentiables, inErgebnisse der Mathematik, Vol.71 (Berlin, 1972).

  54. R. N. Sen:Bundle representations and their applications, Ben Gurion University preprint (1980).

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Aldaya, V., de Azcárraga, J.A. Geometric formulation of classical mechanics and field theory. Riv. Nuovo Cim. 3, 1–66 (1980). https://doi.org/10.1007/BF02906204

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