# The geometry of the space of Cauchy data of nonlinear PDEs

Open Mathematics (2013)

- Volume: 11, Issue: 11, page 1960-1981
- ISSN: 2391-5455

## Access Full Article

top## Abstract

top## How to cite

topGiovanni Moreno. "The geometry of the space of Cauchy data of nonlinear PDEs." Open Mathematics 11.11 (2013): 1960-1981. <http://eudml.org/doc/269260>.

@article{GiovanniMoreno2013,

abstract = {First-order jet bundles can be put at the foundations of the modern geometric approach to nonlinear PDEs, since higher-order jet bundles can be seen as constrained iterated jet bundles. The definition of first-order jet bundles can be given in many equivalent ways - for instance, by means of Grassmann bundles. In this paper we generalize it by means of flag bundles, and develop the corresponding theory for higher-oder and infinite-order jet bundles. We show that this is a natural geometric framework for the space of Cauchy data for nonlinear PDEs. As an example, we derive a general notion of transversality conditions in the Calculus of Variations.},

author = {Giovanni Moreno},

journal = {Open Mathematics},

keywords = {Geometry of PDEs; Cauchy data; Lagrangian formalism; Fiber bundles; Jet spaces; Flags; flag bundles; geometry of partial differential equations; infinite-order bundles},

language = {eng},

number = {11},

pages = {1960-1981},

title = {The geometry of the space of Cauchy data of nonlinear PDEs},

url = {http://eudml.org/doc/269260},

volume = {11},

year = {2013},

}

TY - JOUR

AU - Giovanni Moreno

TI - The geometry of the space of Cauchy data of nonlinear PDEs

JO - Open Mathematics

PY - 2013

VL - 11

IS - 11

SP - 1960

EP - 1981

AB - First-order jet bundles can be put at the foundations of the modern geometric approach to nonlinear PDEs, since higher-order jet bundles can be seen as constrained iterated jet bundles. The definition of first-order jet bundles can be given in many equivalent ways - for instance, by means of Grassmann bundles. In this paper we generalize it by means of flag bundles, and develop the corresponding theory for higher-oder and infinite-order jet bundles. We show that this is a natural geometric framework for the space of Cauchy data for nonlinear PDEs. As an example, we derive a general notion of transversality conditions in the Calculus of Variations.

LA - eng

KW - Geometry of PDEs; Cauchy data; Lagrangian formalism; Fiber bundles; Jet spaces; Flags; flag bundles; geometry of partial differential equations; infinite-order bundles

UR - http://eudml.org/doc/269260

ER -

## References

top- [1] Bocharov A.V., Chetverikov V.N., Duzhin S.V., Khor’kova N.G., Krasil’shchik I.S., Samokhin A.V., Torkhov Yu.N., Verbovetsky A.M., Vinogradov A.M., Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Transl. Math. Monogr., 182, American Mathematical Society, Providence, 1999
- [2] Bott R., Tu L.W., Differential Forms in Algebraic Topology, Grad. Texts in Math., 82, Springer, New York-Berlin, 1982 http://dx.doi.org/10.1007/978-1-4757-3951-0 Zbl0496.55001
- [3] van Brunt B., The Calculus of Variations, Universitext, Springer, New York, 2004 Zbl1039.49001
- [4] Bryant R.L., Chern S.S., Gardner R.B., Goldschmidt H.L., Griffiths P.A., Exterior Differential Systems, Math. Sci. Res. Inst. Publ., 18, Springer, New York, 1991 http://dx.doi.org/10.1007/978-1-4613-9714-4
- [5] Giaquinta M., Hildebrandt S., Calculus of Variations. I, Grundlehren Math. Wiss., 310, Springer, Berlin, 1996 Zbl0853.49001
- [6] Kijowski J., A simple derivation of canonical structure and quasi-local Hamiltonians in general relativity, Gen. Relativity Gravitation, 1997, 29(3), 307–343 http://dx.doi.org/10.1023/A:1010268818255 Zbl0873.53070
- [7] Krasil’shchik J., Verbovetsky A., Geometry of jet spaces and integrable systems, J. Geom. Phys., 2011, 61(9), 1633–1674 http://dx.doi.org/10.1016/j.geomphys.2010.10.012 Zbl1230.58005
- [8] Krupka D., Of the structure of the Euler mapping, Arch. Math. (Brno), 1974, 10(1), 55–61 Zbl0337.33012
- [9] Michor P.W., Manifolds of Differentiable Mappings, Shiva Mathematics Series, 3, Shiva Publishing, Nantwich, 1980 Zbl0433.58001
- [10] Moreno G., A C-spectral sequence associated with free boundary variational problems, In: Geometry, Integrability and Quantization, Avangard Prima, Sofia, 2010, 146–156
- [11] Vinogradov A.M., Many-valued solutions, and a principle for the classification of nonlinear differential equations, Dokl. Akad. Nauk SSSR, 1973, 210, 11–14 (in Russian)
- [12] Vinogradov A.M., The C-spectral sequence, Lagrangian formalism, and conservation laws. I. The linear theory, J. Math. Anal. Appl., 1984, 100(1), 1–40 http://dx.doi.org/10.1016/0022-247X(84)90071-4
- [13] Vinogradov A.M., The C-spectral sequence, Lagrangian formalism, and conservation laws. II. The nonlinear theory, J. Math. Anal. Appl., 1984, 100(1), 41–129 http://dx.doi.org/10.1016/0022-247X(84)90072-6
- [14] Vinogradov A.M., Geometric singularities of solutions of nonlinear partial differential equations, In: Differential Geometry and its Applications, Brno, 1986, Math. Appl. (East European Ser.), 27, Reidel, Dordrecht, 1987, 359–379
- [15] Vinogradov A.M., Cohomological Analysis of Partial Differential Equations and Secondary Calculus, Transl. Math. Monogr., 204, American Mathematical Society, Providence, 2001 Zbl1152.58308
- [16] Vinogradov A.M., Moreno G., Domains in infinite jet spaces: the C-spectral sequence, Dokl. Math., 2007, 75(2), 204–207 http://dx.doi.org/10.1134/S1064562407020081 Zbl1154.58002
- [17] Vitagliano L., Secondary calculus and the covariant phase space, J. Geom. Phys., 2009, 59(4), 426–447 http://dx.doi.org/10.1016/j.geomphys.2008.12.001 Zbl1171.53057
- [18] Vitagliano L., private communication, 2010

## Citations in EuDML Documents

top## NotesEmbed ?

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