Aspects of the inverse problem to the calculus of variations

Ian M. Anderson

Archivum Mathematicum (1988)

  • Volume: 024, Issue: 4, page 181-202
  • ISSN: 0044-8753

How to cite

top

Anderson, Ian M.. "Aspects of the inverse problem to the calculus of variations." Archivum Mathematicum 024.4 (1988): 181-202. <http://eudml.org/doc/18247>.

@article{Anderson1988,
author = {Anderson, Ian M.},
journal = {Archivum Mathematicum},
keywords = {Euler-Lagrange equations; Helmholtz equations; Cartan forms; Chern-Simons invariants; inverse problem; variational bicomplex; Noether's theorem},
language = {eng},
number = {4},
pages = {181-202},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Aspects of the inverse problem to the calculus of variations},
url = {http://eudml.org/doc/18247},
volume = {024},
year = {1988},
}

TY - JOUR
AU - Anderson, Ian M.
TI - Aspects of the inverse problem to the calculus of variations
JO - Archivum Mathematicum
PY - 1988
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 024
IS - 4
SP - 181
EP - 202
LA - eng
KW - Euler-Lagrange equations; Helmholtz equations; Cartan forms; Chern-Simons invariants; inverse problem; variational bicomplex; Noether's theorem
UR - http://eudml.org/doc/18247
ER -

References

top
  1. S. J. Aldersley, G. W. Horndeski, Conformally invariant tensorial concomitants of a pseudo-Riemannian metric, Utilitas Math. 17 (1980), 197-223. (1980) Zbl0441.53011MR0583141
  2. I. M. Anderson, On the structure of divergence-free tensors, J. Math. Phys. 19 (1978), 2570-2575. (1978) Zbl0429.53048MR0512978
  3. I. M. Anderson, Tensorial Euler-Lagrange expressions and conservation laws, Aequationes Math. 17 (1978), 255-291. (1978) Zbl0418.49041MR0493675
  4. I. M. Anderson, Natural variational principles on Riemannian manifolds, Annals of Math. 120 (1984), 329-370. (1984) Zbl0565.58019MR0763910
  5. I. M. Anderson, The variational bicomplex, (to appear). Zbl0881.35069
  6. I. M. Anderson, The minimal order solution to the inverse problem, (to appear). 
  7. I. M. Anderson, Natural differential operators on the variational bicomplex, (to appear). 
  8. I. M. Anderson, T. Duchamp, On the existence of global variational principles, Amer. J. Math. 102 (1980), 781-868. (1980) Zbl0454.58021MR0590637
  9. I. M. Anderson, T. Duchamp, Variational principles for second order quasi-linear scalar equations, J. Diff. Eqs. 51 (1984), 1-47. (1984) Zbl0533.49010MR0727029
  10. D. E. Betounes, Extensions of the classical Cartan form, Phys. Rev. D 29 (1984), 599-606. (1984) Zbl0546.49018MR0734285
  11. K. S. Cheng, W. T. Ni, Conditions for the local existence of metric in a generic affine manifold, Math. Proc. Camb. Phil. Soc. 87 (1980), 527-534. (1980) Zbl0442.53020MR0556932
  12. S. S. Chern, J. Simons, Characteristic forms and geometric invariants, Annals of Math. 99 (1974), 48-69. (1974) Zbl0283.53036MR0353327
  13. M. Crampin, Alternative Lagrangians in particle dynamics, (to appear). Zbl0666.58022MR0923340
  14. V. V. Dodonov V. I. Man'ko, V. D. Skarzhinsky, The inverse problem of the variational calculus and the nonuniqueness of the quantization of classical systems, Hadronic J. 4 (1981), 1734-1803. (1981) MR0632443
  15. V. V. Dodonov V. I. Man'ko, V. D. Skarzhinsky, Classically equivalent Hamiltonians and ambiguities of quantization: a particle in a magnetic field, Il Nuovo Cimento 69B (1982), 185-205. (1982) MR0669159
  16. J. Douglas, Solution to the inverse problem of the calculus of variations, Trans. Amer. Math. Soc. 50 (1941), 71-128. (1941) Zbl0025.18102MR0004740
  17. M. Ferraris, Fibered connections and Global Poincaré-Cartan forms in higher-order Calculus of Variations, in "Proc. of the Conference on Differential Geometry and its Applications, Nové Město na Moravě, Vol. II. Applications", Univerzita Karlova, Praga, 1984, pp. 61-91. (1984) Zbl0564.53013MR0793200
  18. V. N. Gusyatnikova A. M. Vinogradov V. A. Yumaguzhin, Secondary differential operators, J. Geom. Phys. 2 (1985), 23-65. (1985) Zbl0598.58009MR0845467
  19. M. Henneaux, Equations of motions, commutation relations and ambiguities in the Lagrangian formalism, Ann. Phys. 140 (1982), 45-64. (1982) Zbl0501.70020MR0660925
  20. M. Henneaux, L. C. Shepley, Lagrangians for spherically symmetric potentials, J. Math. Phys. 23 (1982), 2101-2104. (1982) Zbl0507.70022MR0680007
  21. M. Henneaux, On the inverse problem of the calculus of variations in field theory, J. Phys. A: Math. Gen. 17 (1984), 75-85. (1984) Zbl0557.70019MR0734109
  22. S. Hojman, H. Harleston, Equivalent Lagrangians: Multidimensional case, J. Math. Physics 22 (1981), 1414-1419. (1981) Zbl0522.70024MR0626131
  23. G. W. Horndeski, Differential operators associated with the Euler-Lagrange operator, Tensor 28 (1974), 303-318. (1974) Zbl0289.49045MR0356143
  24. J. Klein, Geometry of sprays. Lagrangian case. Principle of least curvature., in "Proc. IUTAM-ISIMM Symposium on Modern Developments in Analytical Mechanics, Vol. 1", (Benenti, Francavigilia, Lichnerowicz, eds.), Atti dela Accademia delle Scienze di Torino, 1983, pp. 177-196. (1983) Zbl0566.58012MR0773487
  25. I. Kolář, A geometrical version of the higher order Hamilton formalism in fibered manifolds, J. Geom. Phys. 1 (1984), 127-137. (1984) Zbl0595.58016MR0794983
  26. L. Littlejohn, On the classification of differential equations having orthogonal polynomial solutions, Annali di Mathematica pure ed applicata 138 (1984), 35-53. (1984) Zbl0571.34003MR0779537
  27. D. Lovelock, The Einstein tensor and its generalizations, J. Math. Physics 12 (1971), 498-501. (1971) Zbl0213.48801MR0275835
  28. J. M. Masqué, Poincaré-Cartan forms in higher order variational calculus on fibred manifolds, Revista Matematica Iberoamericana 1 (1985), 85-126. (1985) Zbl0606.49026MR0850411
  29. P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1986. (1986) Zbl0588.22001MR0836734
  30. P. J. Olver, Darboux's theorem for Hamiltonian differential operators, (to appear). 
  31. H. Rund, A Cartan form for the field theory of Carathéodory in the calculus of variations, in "Lecture Notes in Pure and Applied Mathematics No. 100: Differential Geometry, Calculus of Variations, and Their Applications", G. M. Rassias and T. M. Rassias (eds), Marcel Dekker, New York, 1985, pp. 455-470. (1985) Zbl0578.49025MR0822534
  32. W. Sarlet, Symmetries and alternative Lagrangians in higher-order mechanics, Phys. Lett. A 108 (1985), 14-18. (1985) MR0786789
  33. W. Sarlet F. Cantrijin, M. Crampin, A new look at second-order equations and Lagrangian mechanics, J. Phys. A: Math. Gen. 17 (1984), 1999-2009. (1984) Zbl0542.58011MR0763792
  34. F. Takens, Symmetries, conservation laws and variational principles, in "Lecture Notes in Mathematics No. 597", Springer-Verlag, New York, 1977, pp. 581-603. (1977) Zbl0368.49019MR0650304
  35. F. Takens, A global version of the inverse problem to the calculus of variations, J. Diff. Geom. 14 (1979), 543-562. (1979) Zbl0463.58015MR0600611
  36. G. Thompson, Second order equation fields and the inverse problem of Lagrangian dynamics, (to appear). Zbl0638.70013MR0917639
  37. E. Tonti, Inverse problem: Its general solution, in "Lecture Notes in Pure and Applied Mathematics No. 100: Differential Geometry, Calculus of Variations and Their Applications", Marcel Decker, New York, 1985, pp. 497-510. (1985) Zbl0583.49010MR0822537
  38. T. Tsujishita, On variation bicomplexes associated to differential equations, Osaka J. Math. 19 (1982), 311-363. (1982) Zbl0524.58041MR0667492
  39. W. M. Tulczyjew, The Euler-Lagrange resolution, in "Lecture Notes in Mathematics No. 836," Springer-Verlag, New York, 1980, pp. 22-48. (1980) Zbl0456.58012MR0607685
  40. A. M. Vinogradov, On the algebra-geometric foundation of Lagrangian field theory, Sov. Math. Dokl. 18 (1977), 1200-1204. (1977) Zbl0403.58005
  41. A. M. Vinogradov, The C-spectral sequence, Lagrangian formalism and conservation laws I, II, J. Math. Anal. Appl. 100 (1984), 1-129. (1984) Zbl0548.58015MR0739952
  42. E. Witten, Global aspects of current algebra, Nucl. Phys. B223 (1983), 422-432. (1983) MR0717915

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.