Geometry of the free-sliding Bernoulli beam

Giovanni Moreno; Monika Ewa Stypa

Communications in Mathematics (2016)

  • Volume: 24, Issue: 2, page 153-171
  • ISSN: 1804-1388

Abstract

top
If a variational problem comes with no boundary conditions prescribed beforehand, and yet these arise as a consequence of the variation process itself, we speak of the free boundary values variational problem. Such is, for instance, the problem of finding the shortest curve whose endpoints can slide along two prescribed curves. There exists a rigorous geometric way to formulate this sort of problems on smooth manifolds with boundary, which we review here in a friendly self-contained way. As an application, we study the particular free boundary values variational problem of the free-sliding Bernoulli beam. This paper is dedicated to the memory of prof. Gennadi Sardanashvily.

How to cite

top

Moreno, Giovanni, and Stypa, Monika Ewa. "Geometry of the free-sliding Bernoulli beam." Communications in Mathematics 24.2 (2016): 153-171. <http://eudml.org/doc/287882>.

@article{Moreno2016,
abstract = {If a variational problem comes with no boundary conditions prescribed beforehand, and yet these arise as a consequence of the variation process itself, we speak of the free boundary values variational problem. Such is, for instance, the problem of finding the shortest curve whose endpoints can slide along two prescribed curves. There exists a rigorous geometric way to formulate this sort of problems on smooth manifolds with boundary, which we review here in a friendly self-contained way. As an application, we study the particular free boundary values variational problem of the free-sliding Bernoulli beam. This paper is dedicated to the memory of prof. Gennadi Sardanashvily.},
author = {Moreno, Giovanni, Stypa, Monika Ewa},
journal = {Communications in Mathematics},
keywords = {Global Analysis; Calculus of Variations; Free Boundary Problems; Jet Spaces; Bernoulli Beam; global analysis; calculus of variations; free boundary problems; jet spaces; Bernoulli beam},
language = {eng},
number = {2},
pages = {153-171},
publisher = {University of Ostrava},
title = {Geometry of the free-sliding Bernoulli beam},
url = {http://eudml.org/doc/287882},
volume = {24},
year = {2016},
}

TY - JOUR
AU - Moreno, Giovanni
AU - Stypa, Monika Ewa
TI - Geometry of the free-sliding Bernoulli beam
JO - Communications in Mathematics
PY - 2016
PB - University of Ostrava
VL - 24
IS - 2
SP - 153
EP - 171
AB - If a variational problem comes with no boundary conditions prescribed beforehand, and yet these arise as a consequence of the variation process itself, we speak of the free boundary values variational problem. Such is, for instance, the problem of finding the shortest curve whose endpoints can slide along two prescribed curves. There exists a rigorous geometric way to formulate this sort of problems on smooth manifolds with boundary, which we review here in a friendly self-contained way. As an application, we study the particular free boundary values variational problem of the free-sliding Bernoulli beam. This paper is dedicated to the memory of prof. Gennadi Sardanashvily.
LA - eng
KW - Global Analysis; Calculus of Variations; Free Boundary Problems; Jet Spaces; Bernoulli Beam; global analysis; calculus of variations; free boundary problems; jet spaces; Bernoulli beam
UR - http://eudml.org/doc/287882
ER -

References

top
  1. Duchamp, I. M. Anderson and T., 10.2307/2374195, Amer. J. Math., 102, 5, 1980, 781-868, ISSN 0002-9327. DOI 10.2307/2374195. (1980) MR0590637DOI10.2307/2374195
  2. 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, Translations of Mathematical Monographs, 182, 1999, American Mathematical Society, Providence, RI, ISBN 0-8218-0958-X. Edited and with a preface by Krasil'shchik and Vinogradov, translated from 1997 Russian original by Verbovetsky and Krasil'shchik. (1999) MR1670044
  3. Dedecker, P., Calcul des variations, formes différentielles et champs géodésiques, Géométrie différentielle. Colloques Internationaux du Centre National de la Recherche Scientifique, Strasbourg, 1953, 17-34, Centre National de la Recherche Scientifique, Paris, (1953) Zbl0052.32003MR0062964
  4. Gel'fand, I. M., Dikiĭ, L. A., The calculus of jets and nonlinear Hamiltonian systems, Funkcional. Anal. i Priložen., 12, 2, 1978, 8-23, ISSN 0374-1990. (1978) MR0501129
  5. Giaquinta, M., Hildebrandt, S., Calculus of variations. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 310, 1996, Springer-Verlag, Berlin, ISBN 3-540-50625-X. The Lagrangian formalism. (1996) MR1385926
  6. Janet, M., Leçons sur les Systèmes d'Équations aux Dérivées Partielles, 1929, Gauthier-Villars, (1929) 
  7. Krupka, D., Of the structure of the Euler mapping, Arch. Math., 10, 1, 1974, 55-61, ISSN 0044-8753. (1974) MR0394755
  8. Krupka, D., Moreno, G., Urban, Z., Voln{á}, J., 10.1142/S0219887815500577, Int. J. Geom. Methods Mod. Phys., 15, 5, 2015, 1550057. ISSN 0219-8878. DOI 10.1142/S0219887815500577. (2015) Zbl1322.58013MR3349926DOI10.1142/S0219887815500577
  9. Lánczos, C., The variational principles of mechanics, 4, 1970, University of Toronto Press, Toronto, Ont., Fourth Ed.. (1970) Zbl0257.70001MR0431821
  10. Love, A. E. H., A treatise on the Mathematical Theory of Elasticity, 1944, Dover Publications, New York, Fourth Ed.. (1944) Zbl0063.03651MR0010851
  11. Moreno, G., Condizioni di trasversalitá nel calcolo secondario, 2007, PhD thesis, University of Naples ``Federico II'' (2007). (2007) 
  12. Moreno, G., A C-spectral sequence associated with free boundary variational problems, Geometry, integrability and quantization, 2010, 146-156, Avangard Prima, Sofia, (2010) MR2757930
  13. Moreno, G., 10.2478/s11533-013-0292-y, Central European Journal of Mathematics, 11, 11, 2013, 1960-1981, DOI 10.2478/s11533-013-0292-y. (2013) Zbl1292.35011MR3092791DOI10.2478/s11533-013-0292-y
  14. Moreno, G., Stypa, M. E., 10.1515/ms-2015-0105, Math. Slovaca, 65, 6, 2015, 1531-1556, ISSN 0139-9918. DOI 10.1515/ms-2015-0105. (2015) MR3458999DOI10.1515/ms-2015-0105
  15. Sardanashvily, G. A., Gauge theory in jet manifolds, 1993, Hadronic Press Inc., Palm Harbor, FL, ISBN 0-911767-60-6.. (1993) Zbl0811.58004MR1262598
  16. Saunders, D. J., 10.1017/CBO9780511526411, 142, 1989, Cambridge University Press, Cambridge, ISBN 0-521-36948-7. DOI 10.1017/CBO9780511526411. (1989) Zbl0665.58002MR0989588DOI10.1017/CBO9780511526411
  17. Takens, F., 10.1007/BFb0085377, Lect. Notes Math., 597, 1977, 581-604, Springer-Verlag, (1977) Zbl0368.49019MR0650304DOI10.1007/BFb0085377
  18. Tsujishita, T., On variation bicomplexes associated to differential equations, Osaka J. Math., 19, 2, 1982, 311-363, ISSN 0030-6126. (1982) Zbl0524.58041MR0667492
  19. Tulczyjew, W. M., Sur la différentielle de Lagrange, C. R. Acad. Sci. Paris Sér. A, 280, 1975, 1295-1298, (1975) Zbl0314.58018MR0377987
  20. Brunt, B. van, The calculus of variations, 2004, Springer-Verlag, New York, ISBN 0-387-40247-0. (2004) MR2004181
  21. Vinogradov, A. M., 10.1016/0022-247X(84)90071-4, J. Math. Anal. Appl., 100, 1, 1984, 1-40, ISSN 0022-247X. (1984) MR0739951DOI10.1016/0022-247X(84)90071-4
  22. Vinogradov, A. M., 10.1016/0022-247X(84)90072-6, J. Math. Anal. Appl., 100, 1, 1984, 41-129, ISSN 0022-247X. (1984) MR0739952DOI10.1016/0022-247X(84)90072-6
  23. Vinogradov, A. M., Moreno, G., 10.1134/S1064562407020081, Dokl. Akad. Nauk, 413, 2, 2007, 154-157, ISSN 0869-5652. DOI 10.1134/S1064562407020081. (2007) MR2456137DOI10.1134/S1064562407020081

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.