Jets and the variational calculus
Communications in Mathematics (2021)
- Issue: 1, page 91-114
- ISSN: 1804-1388
Access Full Article
topAbstract
topHow to cite
topSaunders, David J.. "Jets and the variational calculus." Communications in Mathematics (2021): 91-114. <http://eudml.org/doc/297658>.
@article{Saunders2021,
abstract = {We review the approach to the calculus of variations using Ehresmann's theory of jets. We describe different types of jet manifold, different types of variational problem and different cohomological structures associated with such problems.},
author = {Saunders, David J.},
journal = {Communications in Mathematics},
keywords = {Jets; Calculus of variations},
language = {eng},
number = {1},
pages = {91-114},
publisher = {University of Ostrava},
title = {Jets and the variational calculus},
url = {http://eudml.org/doc/297658},
year = {2021},
}
TY - JOUR
AU - Saunders, David J.
TI - Jets and the variational calculus
JO - Communications in Mathematics
PY - 2021
PB - University of Ostrava
IS - 1
SP - 91
EP - 114
AB - We review the approach to the calculus of variations using Ehresmann's theory of jets. We describe different types of jet manifold, different types of variational problem and different cohomological structures associated with such problems.
LA - eng
KW - Jets; Calculus of variations
UR - http://eudml.org/doc/297658
ER -
References
top- Anderson, I.M., The variational bicomplex, Technical report of Utah State University, 1989, available at https://ncatlab.org/nlab/files/AndersonVariationalBicomplex.pdf. (1989)
- Anderson, I.M., Duchamp, T.E., 10.1016/0022-0396(84)90100-1, J. Diff. Eq., 51, 1, 1984, 1-47, (1984) MR0727029DOI10.1016/0022-0396(84)90100-1
- Betounes, D.E., 10.1103/PhysRevD.29.599, Phys. Rev. D, 29, 1984, 599-606, (1984) MR0734285DOI10.1103/PhysRevD.29.599
- Borel, É., Sur quelques points de la théorie des fonctions, Ann. scient. de l'École Norm. Sup., 3, 12, 1895, 9-55, (1895) MR1508908
- Carathéodory, C., Über die Variationsrechnung bei mehrfachen Integralen, Acta Szeged Sect. Scient. Mathem., 4, 1929, 193-216, (1929)
- Crampin, M., Saunders, D.J., The Hilbert--Carathéodory and Poincaré--Cartan forms for higher-order multiple-integral variational problems, Houston J. Math., 30, 3, 2004, 657-689, (2004) MR2083869
- Do, T., The inverse problem in the calculus of variations via exterior differential systems, 2016, Ph.D. Thesis, La Trobe University, Melbourne, Australia. (2016)
- Do, T., Prince, G., 10.1016/j.difgeo.2016.01.005, Diff. Geom. Appl., 45, 2016, 148-179, (2016) MR3457392DOI10.1016/j.difgeo.2016.01.005
- Ehresmann, C., Les prolongements d'une variété différentiable: calcul des jets, prolongement principal, C. R. Acad. Sci. Paris, 233, 1951, 598-600, (1951) MR0044198
- Ehresmann, C., Les prolongements d’une variété différentiable: l’espace des jets d’ordre de dans , C. R. Acad. Sci. Paris, 233, 1951, 777-779, (1951) MR0045435
- Ehresmann, C., Les prolongements d'une variété différentiable: transitivité des prolongements, C. R. Acad. Sci. Paris, 233, 1951, 1081-1083, (1951) MR0045436
- Ehresmann, C., Les prolongements d'une variété différentiable: éléments de contact et éléments d'enveloppe, C. R. Acad. Sci. Paris, 234, 1952, 1028-1030, (1952) MR0046737
- Ehresmann, C., Les prolongements d'une variété différentiable: covariants différentiels et prolongements d'une structure infinitésimale, C. R. Acad. Sci. Paris, 234, 1952, 1424-1425, (1952) MR0047385
- Frölicher, A., Nijenhuis, A., Theory of vector valued differential forms. Part I, Ind. Math., 18, 1056, 338-360, (1056) MR0082554
- Goldschmidt, H., Sternberg, S., 10.5802/aif.451, Ann. Inst, Fourier, 23, 1, 1973, 203-267, (1973) MR0341531DOI10.5802/aif.451
- Kolář, I., Michor, P.W., Slovak, J., Natural operations in differential geometry, 1993, Springer, (1993) MR1202431
- Kolář, I., Natural operators related with the variational calculus, Differential Geometry and its Applications, Opava 1992, 1993, 461-472, Silesian University, Opava, (1993) MR1255562
- Kolář, I., Weil bundles as generalized jet space, Handbook of Global Analysis, ed. D. Krupka and D.J. Saunders, 2007, 625-665, Elsevier, (2007) MR2389643
- Kolář, I., 10.1016/j.difgeo.2011.03.007, Diff. Geom. Appl., 29, 5, 2011, 647-652, (2011) Zbl1229.58005MR2831821DOI10.1016/j.difgeo.2011.03.007
- Kriegl, A., Michor, P., The Convenient Setting of Global Analysis, 1997, AMS Mathematical Surveys and Monographs 53, (1997) Zbl0889.58001MR1471480
- Krbek, M., Musilová, J., 10.1007/s10440-005-4980-x, Acta Appl. Math., 88, 2005, 177-199, (2005) MR2169038DOI10.1007/s10440-005-4980-x
- Krupka, D., 10.21136/CMJ.1977.101449, Czech. Math. J., 27, 1977, 114-118, (1977) MR2523431DOI10.21136/CMJ.1977.101449
- Krupka, D., Variational sequences on finite order jet spaces, Differential Geometry and its Applications, Brno 1989, 1990, 236-254, World Scientific, (1990) MR1062026
- Krupka, D., 10.1016/0926-2245(95)92849-Z, Diff. Geom. Appl., 5, 1995, 257-276, (1995) MR1353059DOI10.1016/0926-2245(95)92849-Z
- Krupka, D., Musilová, J., 10.1016/S0926-2245(98)00023-0, Diff. Geom. Appl., 9, 1998, 293-305, (1998) MR1661189DOI10.1016/S0926-2245(98)00023-0
- Libermann, P., Introduction to the theory of semi-holonomic jets, Arch. Math. (Brno), 33, 1997, 173-189, (1997) MR1478771
- Libermann, P., Charles Ehresmann's concepts in differential geometry, Geometry and Topology of Manifolds, Banach Center Publications, vol. 76, 2007, 35-50, Institute of Mathematics, Polish Academy of Sciences, Warszawa, (2007) MR2342844
- Manno, G., Vitolo, R., Variational sequences on finite order jets of submanifolds, Differential Geometry and its Applications, Opava, 2001, 2001, 435-447, Silesian University at Opava, (2001) MR1978797
- Olver, P.J., 10.1007/BF00990539, Acta Appl. Math., 31, 1993, 99-136, (1993) MR1223167DOI10.1007/BF00990539
- Saunders, D.J., The geometry of jet bundles, 1989, Cambridge, (1989) Zbl0665.58002MR0989588
- Saunders, D.J., 10.1016/0926-2245(91)90025-5, Diff. Geom. Appl., 1, 1991, 109-122, (1991) MR1244438DOI10.1016/0926-2245(91)90025-5
- Saunders, D.J., Jet manifolds and natural bundles, Handbook of Global Analysis, ed. D. Krupka and D.J. Saunders, 2007, 1037-1070, Elsevier, (2007) MR2389651
- Saunders, D.J., 10.1016/j.geomphys.2009.03.001, J. Geom. Phys., 59, 2009, 727-739, (2009) Zbl1168.58006MR2510165DOI10.1016/j.geomphys.2009.03.001
- Saunders, D.J., Horizontal forms on jet bundles, Balkan J. Geom. Appl., 15, 1, 2010, 149-154, (2010) MR2608517
- Saunders, D.J., 10.1016/j.difgeo.2011.11.006, Diff. Geom. Appl., 30, 1, 2012, 59-64, (2012) MR2871705DOI10.1016/j.difgeo.2011.11.006
- Saunders, D.J., On Lagrangians with reduced-order Euler--Lagrange equations, SIGMA, 14, 2018, 089, 13 pages, (2018) MR3846852
- Urban, Z., Krupka, D., 10.5486/PMD.2013.5265, Publ. Math. Debrecen, 82, 1, 2013, 59-76, (2013) MR3034368DOI10.5486/PMD.2013.5265
- Vitolo, R., Variational sequences, Handbook of Global Analysis, ed. D. Krupka and D.J. Saunders, 2007, 1117-1160, Elsevier, (2007) MR2389653
- Weil, A., Théorie des points proches sur les variétés différentielles, Colloque de topologie et géométrie différentielle, 1953, 111-117, Strasbourg, (1953) MR0061455
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.