The Hamilton-Cartan formalism in the calculus of variations

Hubert Goldschmidt; Shlomo Sternberg

Annales de l'institut Fourier (1973)

  • Volume: 23, Issue: 1, page 203-267
  • ISSN: 0373-0956

Abstract

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We give an exposition of the calculus of variations in several variables. The introduction of a linear differential form studied by Cartan makes possible an invariant treatment of the Hamiltonian formalism. Noether’s theorem, the Hamilton-Jacobi equation and the second variation are discussed and a Poisson bracket is defined.

How to cite

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Goldschmidt, Hubert, and Sternberg, Shlomo. "The Hamilton-Cartan formalism in the calculus of variations." Annales de l'institut Fourier 23.1 (1973): 203-267. <http://eudml.org/doc/74112>.

@article{Goldschmidt1973,
abstract = {We give an exposition of the calculus of variations in several variables. The introduction of a linear differential form studied by Cartan makes possible an invariant treatment of the Hamiltonian formalism. Noether’s theorem, the Hamilton-Jacobi equation and the second variation are discussed and a Poisson bracket is defined.},
author = {Goldschmidt, Hubert, Sternberg, Shlomo},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {1},
pages = {203-267},
publisher = {Association des Annales de l'Institut Fourier},
title = {The Hamilton-Cartan formalism in the calculus of variations},
url = {http://eudml.org/doc/74112},
volume = {23},
year = {1973},
}

TY - JOUR
AU - Goldschmidt, Hubert
AU - Sternberg, Shlomo
TI - The Hamilton-Cartan formalism in the calculus of variations
JO - Annales de l'institut Fourier
PY - 1973
PB - Association des Annales de l'Institut Fourier
VL - 23
IS - 1
SP - 203
EP - 267
AB - We give an exposition of the calculus of variations in several variables. The introduction of a linear differential form studied by Cartan makes possible an invariant treatment of the Hamiltonian formalism. Noether’s theorem, the Hamilton-Jacobi equation and the second variation are discussed and a Poisson bracket is defined.
LA - eng
UR - http://eudml.org/doc/74112
ER -

References

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  1. [1] C. CARATHEODORY, Variationsrechnung und partielle Differential-gleichungen erster Ordnung, Teubner, Leipzig, 1935. Zbl0011.35603JFM61.0547.01
  2. [2] E. CARTAN, Leçons sur les invariants intégraux, Hermann, Paris, 1922. JFM48.0538.02
  3. [3] Th. De DONDER, Théorie invariantive du calcul des variations, Hayez, Brussels, 1935. Zbl0013.16901
  4. [4] H. GOLDSCHMIDT, Integrability criteria for systems of non-linear partial differential equations, J. Differential Geometry, 1 (1967), 269-307. Zbl0159.14101MR37 #1746
  5. [5] R. HERMANN, Differential geometry and the calculus of variations, Academic Press, New York, London, 1968. Zbl0219.49023MR38 #1635
  6. [6] E.L. HILL, Hamilton's principle and the conservation theorems of mathematical physics, Rev. Modern Phys., 23 (1951), 253-260. Zbl0044.38509MR13,503g
  7. [7] Th. LEPAGE, Sur les champs géodésiques du calcul des variations, Acad. Roy. Belg. Bull. Cl. Sci., (5) 22 (1936), 716-729, 1036-1046 ; Sur le champ géodésique des intégrales multiples, Acad. Roy. Belg. Bull. Cl. Sci., (5) 27 (1941), 27-46 ; Champs stationnaires, champs géodésiques et formes intégrables, Acad. Roy. Belg. Bull. Cl. Sci., (5) 28 (1942), 73-92, 247-265. Zbl0016.26201JFM62.1329.01
  8. [8] J. MILNOR, Morse Theory, Annals of Mathematics Studies, n° 51, Princeton University Press. Princeton, N. J., 1963. Zbl0108.10401
  9. [9] C.B. MORREY, Jr., Multiple integrals in the calculus of variations, Springer-Verlag, Berlin, Heidelberg, New York, 1966. Zbl0142.38701MR34 #2380
  10. [10] S. SMALE, On the Morse index theorem, J. Math. Mech., 14 (1965), 1049-1055. Zbl0166.36102MR31 #6251
  11. [11] S. STERNBERG, Lectures on differential geometry, Prentice Hall, Englewood Cliffs, N. J., 1964. Zbl0129.13102MR33 #1797
  12. [12] L. VAN HOVE, Sur la construction des champs de De Donder-Weyl par la méthode des caractéristiques, Acad. Roy. Belg. Bull. Cl. Sci., (5) 31 (1945), 278-285. Zbl0033.12202
  13. [13] H. WEYL, Geodesic fields in the calculus of variations for multiple integrals, Ann. of Math., 36 (1935), 607-629. Zbl0013.12002JFM61.0554.04

Citations in EuDML Documents

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  1. Hubert Goldschmidt, Le formalisme de Hamilton-Cartan en calcul des variations
  2. A. Echeverría Enríquez, M. C. Muñoz Lecanda, Variational calculus in several variables : a hamiltonian approach
  3. L. Mangiarotti, M. Modugno, Some results on the calculus of variations on jet spaces
  4. Mauro Francaviglia, Demeter Krupka, The hamiltonian formalism in higher order variational problems
  5. Ivan Kolář, Fundamental vector fields on associated fiber bundles
  6. Demeter Krupka, A map associated to the Lepagian forms on the calculus of variations in fibred manifolds
  7. Emanuel López, Alberto Molgado, José A. Vallejo, The principle of stationary action in the calculus of variations
  8. Jordi Gaset, Pedro D. Prieto-Martínez, Narciso Román-Roy, Variational principles and symmetries on fibered multisymplectic manifolds
  9. Michel Bauderon, Le problème inverse du calcul des variations
  10. Josef Janyška, On the Lie algebra of vertical prolongation operators

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