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A geometrical approach to the Hamilton-Jacobi form of dynamics and its generalizations

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References

  1. R. Abraham andE. J. Marsden:Foundations of Mechanics (Benjamin-Cummings, Reading, Mass., 1978).

    MATH  Google Scholar 

  2. W. R. Hamilton:Phil. Trans. Roy. Soc. Part II, 247 (1834);Phil Trans. Roy. Soc. Part I, 95 (1835). In «The Mathematical Papers of Sir W. R. Hamilton» (Cambridge University Press, 1940).

  3. C. G. J. Jacobi:Crelle's J.,17, 97 (1837).

    Article  MathSciNet  MATH  Google Scholar 

  4. C. G. J. Jacobi:Liouville's J.,3, 60, 161 (1837).

    Google Scholar 

  5. V. I. Arnold:Geometrical Methods in the Theory of Ordinary Differential Equations (Springer, Berlin, 1983).

    Book  MATH  Google Scholar 

  6. C. Carathèodory:Calculus of Variations and Partial Differential Equations of the First Order (Holden-Day, San Francisco, Cal., 1965).

    MATH  Google Scholar 

  7. P. A. M. Dirac:J. Math.,2, 129 (1950).

    MathSciNet  MATH  Google Scholar 

  8. H. Goldstein:Classical Mechanics (Addison-Wesley, New York, 1980).

    MATH  Google Scholar 

  9. C. Lanczos:The Variational Principles of Mechanics (University of Toronto Press, 1970).

  10. E. T. Whittaker:A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th Edition (Cambridge University Press, Cambridge, 1959).

    Google Scholar 

  11. W. R. Hamilton:Trans. R. Irish Acad.,15, 69 (1828).

    Google Scholar 

  12. M. Born andE. Wolff:Principles of Optics (Pergamon Press, London, 1965).

    Google Scholar 

  13. P. Fermat: inOeuvres de P. de Fermat, vol.2 (Paris, 1891), p. 354.

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

    MATH  Google Scholar 

  15. V. I. Arnold:Ordinary Differential Equations (MIT Press, Cambridge, Mass., 1978).

    Google Scholar 

  16. A. Chodos andC. M. Sommerfeld:J. Math. Phys. (N.Y.),24, 371 (1983).

    Article  Google Scholar 

  17. E. G. Peter Rowe:Am J. Phys.,53, 997 (1985).

    Article  ADS  Google Scholar 

  18. S. Benenti andW. M. Tulczjew:C.R. Acad. Sci., Sez I,294, 677 (1982).

    MATH  Google Scholar 

  19. C. E. Delaunay:Sur une nouvelle théorie analytique du mouvement de la Lune (Paris, 1846).

  20. V. I. Arnold:Mathematical Methods of Classical Mechanics (Springer, Berlin, 1978).

    Book  MATH  Google Scholar 

  21. G. Gallavotti:The Integrability Problem and the Hamilton-Jacobi Equation. A Review (Les Houches, 1983).

  22. A. Einsten:Sitzungsber. Preuss. Akad. Wiss.,2, 606 (1917).

    Google Scholar 

  23. A. Einstein:Verh. Dtsch. Phys. Ges.,19, 82 (1917).

    Google Scholar 

  24. R. Hermann:Differential Geometry and the Calculus of Variations (Academic Press, New York, N.Y., 1968).

    MATH  Google Scholar 

  25. P. Maillavin:Géometrie differentielle intrinseque (Hermann, Paris, 1972).

    Google Scholar 

  26. P. Liberman andC. M. Marle:Symplectic Geometry and Analytical Mechanics (Reidel, Dordrecht, 1987).

    Book  Google Scholar 

  27. G. Marmo, E. J. Saletan, A. Simoni andB. Vitale:Dynamical Systems. A Differential Geometric Approach to Symmetry and Reduction (J. Wiley, New York, N.Y., 1985).

    MATH  Google Scholar 

  28. P. A. M. Dirac:Can. J. Math.,3, 1 (1951).

    Article  MathSciNet  MATH  Google Scholar 

  29. W. M. Tulczyjew:Ann. Inst. Henry Poincaré,27, 101 (1977).

    MathSciNet  MATH  Google Scholar 

  30. G. Marmo andN. Mukunda:Nuovo Cimento B,92, 1 (1986).

    Article  MathSciNet  ADS  Google Scholar 

  31. N. Mukunda:Proc. Ind. Acad. Sci. A,87, 85 (1978).

    MathSciNet  MATH  Google Scholar 

  32. S. Benenti:Separability structures on Riemannian manifolds, in:Differential Geometric Methods in Mathematical Physics (Springer, Berlin, 1979).

    Google Scholar 

  33. S. Benenti:Symplectic relations in analytical mechanics, in:Proceedings of the IUTAM-ISIMM Symposium on «Modern Developments in Analytical Mechanics», Turin, 1982.

  34. S. Benenti:The Hamilton-Jacobi equation for a Hamiltonian action, in:Proceedings of the 4th Seminar in Mathematical Physics, edited byA. Prastaro (Pitagora, Bologna, 1983).

    Google Scholar 

  35. S. Benenti:Relazioni Simplettiche: la Trasformazione di Legendre e la Teoria di Hamilton-Jacobi (Pitagora, Bologna, 1988).

    MATH  Google Scholar 

  36. S. Benenti andM. Francaviglia:The theory of separability of the Hamilton-Jacobi equation and its applications in general relativity in:General Relativity and Gravitation, edited byW. Held (Plenum, New York, N.Y., 1980).

    Google Scholar 

  37. S. Benenti andW. M. Tulczyjew:The geometrical meaning and globalization of the Hamilton-Jacobi equation, in:Differential Geometric Methods in Mathematical Physics (Springer, Berlin, 1979).

    Google Scholar 

  38. S. Benenti andW. M. Tulczyjew:C.R. Acad. Sci.,300, 119 (1985).

    MathSciNet  MATH  Google Scholar 

  39. S. H. Benton:The Hamilton-Jacobi Equation (Academic Press, New York, N.Y., 1977).

    MATH  Google Scholar 

  40. F. Cardin:Sulla geometria delle soluzioni dell'equazione di Hamilton-Jacobi. preprint (Dept. of Math., Padova, 1988).

    Google Scholar 

  41. P. Dazord andT. Delzant:J. Diff. Geom.,26, 223 (1987).

    MathSciNet  MATH  Google Scholar 

  42. D. Dominici, G. Longhi, J. Gomis andJ. M. Pons:J. Math. Phys. (N.J.),25, 2439 (1978).

    Article  MathSciNet  ADS  Google Scholar 

  43. C. M. Marle:J. Math. Pures Appl.,59, 133 (1980).

    MathSciNet  ADS  MATH  Google Scholar 

  44. C. M. Marle:Contact manifolds, canonical manifolds and the Hamilton-Jacobi equation, inProceedings of the IUTAM-ISIMM Symposium, Turin, 1982; Atti Acc. Sci. Torino,117 (1983).

  45. C. M. Marle:A propos des systemes Hamiltoniens completement integrables et des variables actions-angles inJournées Relativistes (Lyon, 1982).

  46. C. M. Marle:Asterisques,107–108, 69 (1983).

    MathSciNet  Google Scholar 

  47. W. M. Tulczyew:C. R. Acad. Sci.,283, 15 (1976).

    Google Scholar 

  48. W. M. Tulczyjew:C. R. Acad. Sci. A,283, 15 (1976).

    MathSciNet  MATH  Google Scholar 

  49. V. I. Arnold:Russ. Math. Surv.,23, 1 (1968).

    Article  Google Scholar 

  50. V. I. Arnold:Funct. Anal. Appl.,15, 325 (1981).

    Google Scholar 

  51. V. I. Arnold:Sov. Phys. Usp.,26, 1025 (1983).

    Article  ADS  Google Scholar 

  52. V. J. Arnold, A. Varchenko andS. Goussein-Zadé:Singularités des applications differentiables, (MIR, Moscow, 1986).

    Google Scholar 

  53. J. J. Duistermaat:Commun. Pure Appl. Math.,27, 207 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  54. J. Guckenheimer:Ann. Inst. Fourier, Grenoble,23, 31 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  55. J. Guckenheimer:Topology,13, 127 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  56. V. Guillemin andS. Sternberg:Am. Math. Soc. Survey,14 (Am. Math. Soc. Providence, R.I., 1977).

    Google Scholar 

  57. V. Guillemin andS. Sternberg:Symplectic Techniques in Physics (Cambridge University Press, Cambridge, 1984).

    MATH  Google Scholar 

  58. S. Janeczko:J. Geom. Phys.,2, 33 (1985).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  59. Yu. A. Kravstov andYu. I. Orlov:Sov. Phys. Usp.,26, 1038 (1983).

    Article  ADS  Google Scholar 

  60. J. Leray:Lagrangian Analysis and Quantum Mechanics, (MIT Press, Cambridge, Mass., 1981).

    MATH  Google Scholar 

  61. V. P. Maslov andM. V. Fedoriuk:Semi-Classical Approximations in Quantum Mechanics (Reidel, Dordrecht, 1981).

    Book  Google Scholar 

  62. J. Sniatycki andW. M. Tulczyjew:Ind. Univ. Math. J.,22, 267 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  63. W. M. Tulczyew:C. R. Acad. Sci., Sez. A,281, 545 (1975).

    Google Scholar 

  64. A. Weinstein:Singularities of families of functions, inDifferentialgeometrie in der Grossen, edited byV. Klingberg (Bibliographisches Institut, Mannheim, 1971).

    Google Scholar 

  65. G. Morandi:Geometrical Aspects of the Hamilton-Jacobi Theory, in:Recent Developments in Theoretical Physics, edited byE. C. G. Sudarshan, K. Srinivasa Rao andR. Sridhar (World Scientific, Singapore, 1987).

    Google Scholar 

  66. V. I. Arnold:Funct. Anal. Appl.,1, 1 (1967).

    Article  Google Scholar 

  67. A. Weinstein:Adv. Math.,6, 329 (1971).

    Article  MATH  Google Scholar 

  68. A. Weinstein:Bull. Am. Math. Soc.,5, 1 (1981).

    Article  MATH  Google Scholar 

  69. V. I. Arnold:Commun. Pure Appl. Math.,29, 557 (1976).

    Article  ADS  Google Scholar 

  70. G. Marmo:Function groups and reduction of Hamiltonian systems, inProceedings of the IUTAM-ISIMM Symposium, Turin, 1982; Atti Acc. Sci. Torino,117, 273 (1983).

  71. J. Hayden andE. C. Zeeman:Math. Proc. Camb. Phil. Soc.,89, 193 (1980).

    Article  MathSciNet  Google Scholar 

  72. R. Abraham andJ. Robbin:Transversal Mappings and Flows (Benjamin-Cummings, Reading, Mass., 1967).

    MATH  Google Scholar 

  73. M. Postnikov:Lectures in Geometry. Semester V:Lie Groups and Lie Algebras (MIR, Moscow, 1986).

    Google Scholar 

  74. O. Espinola, N. L. Teixeira andM. L. Espinola:J. Math. Phys. (N.Y.),27 151 (1986).

    Article  ADS  Google Scholar 

  75. S. MacLane:Am. Math. Monthly,77, 570 (1970).

    Article  MathSciNet  Google Scholar 

  76. J. Schwinger:Quantum Theory of Gauge Fields, inCargese Lectures (1963).

  77. A. Truman:J. Math. Phys. (N.Y.),17, 1852 (1976).

    Article  MathSciNet  ADS  Google Scholar 

  78. A. Truman:J. Math. Phys. (N.Y.),18, 1499 (1977).

    Article  MathSciNet  ADS  Google Scholar 

  79. A. Truman:J. Math. Phys., (N.Y.),18, 2308 (1977).

    Article  MathSciNet  ADS  MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Scienze Fisiche dell'Università, Napoli

    G. Marmo

  2. INFN, Mostra d'Oltremare Pad. 19, I-80125, Napoli, Italy

    G. Marmo

  3. Dipartimento di Fisica dell'Università, Ferrara

    G. Morandi

  4. GNSM-CISM and INFN, Via Paradiso 12, I-44100, Ferrara, Italy

    G. Morandi

  5. Centre for Theoretical Studies and Department of Physics, Indian Institute of Science, 560012, Bangalore, India

    N. Mukunda (Jawaharlal Nehru Fellow)

Authors
  1. G. Marmo
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  2. G. Morandi
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  3. N. Mukunda
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Marmo, G., Morandi, G. & Mukunda, N. A geometrical approach to the Hamilton-Jacobi form of dynamics and its generalizations. Riv. Nuovo Cim. 13, 1–74 (1990). https://doi.org/10.1007/BF02832785

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