×

Geometry of the free-sliding Bernoulli beam. (English) Zbl 1397.58006

Summary: 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.

MSC:

58E30 Variational principles in infinite-dimensional spaces
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[1] Duchamp, I. M. Anderson and T.: On the existence of global variational principles. Amer. J. Math., 102, 5, 1980, 781-868, ISSN 0002-9327. DOI 10.2307/2374195. | · Zbl 0454.58021 · doi:10.2307/2374195
[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.
[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, | · Zbl 0052.32003
[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.
[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. · Zbl 0853.49001
[6] Janet, M.: Leçons sur les Systèmes d’Équations aux Dérivées Partielles. 1929, Gauthier-Villars,
[7] Krupka, D.: Of the structure of the Euler mapping. Arch. Math., 10, 1, 1974, 55-61, ISSN 0044-8753. · Zbl 0337.33012
[8] Krupka, D., Moreno, G., Urban, Z., Volná, J.: On a bicomplex induced by the variational sequence. Int. J. Geom. Methods Mod. Phys., 15, 5, 2015, 1550057. ISSN 0219-8878. DOI 10.1142/S0219887815500577. | | · Zbl 1322.58013 · doi:10.1142/S0219887815500577
[9] Lánczos, C.: The variational principles of mechanics. 4, 1970, University of Toronto Press, Toronto, Ont., Fourth Ed.. | · Zbl 0257.70001
[10] Love, A. E. H.: A treatise on the Mathematical Theory of Elasticity. 1944, Dover Publications, New York, Fourth Ed.. | · Zbl 0063.03651
[11] Moreno, G.: Condizioni di trasversalitá nel calcolo secondario. 2007, PhD thesis, University of Naples “Federico II” (2007).
[12] Moreno, G.: A C-spectral sequence associated with free boundary variational problems. Geometry, integrability and quantization, 2010, 146-156, Avangard Prima, Sofia, · Zbl 1382.58014
[13] Moreno, G.: The geometry of the space of cauchy data of nonlinear pdes. Central European Journal of Mathematics, 11, 11, 2013, 1960-1981, DOI 10.2478/s11533-013-0292-y. | | · Zbl 1292.35011 · doi:10.2478/s11533-013-0292-y
[14] Moreno, G., Stypa, M. E.: Natural boundary conditions in geometric calculus of variations. Math. Slovaca, 65, 6, 2015, 1531-1556, ISSN 0139-9918. DOI 10.1515/ms-2015-0105. | · Zbl 1389.58003 · doi:10.1515/ms-2015-0105
[15] Sardanashvily, G. A.: Gauge theory in jet manifolds. 1993, Hadronic Press Inc., Palm Harbor, FL, ISBN 0-911767-60-6.. | · Zbl 0811.58004
[16] Saunders, D. J.: The geometry of jet bundles. 142, 1989, Cambridge University Press, Cambridge, ISBN 0-521-36948-7. DOI 10.1017/CBO9780511526411. | | · Zbl 0665.58002 · doi:10.1017/CBO9780511526411
[17] Takens, F.: Symmetries, conservation laws and variational principles. Lect. Notes Math., 597, 1977, 581-604, Springer-Verlag, | | · Zbl 0368.49019 · doi:10.1007/BFb0085377
[18] Tsujishita, T.: On variation bicomplexes associated to differential equations. Osaka J. Math., 19, 2, 1982, 311-363, ISSN 0030-6126. | · Zbl 0524.58041
[19] Tulczyjew, W. M.: Sur la différentielle de Lagrange. C. R. Acad. Sci. Paris Sér. A, 280, 1975, 1295-1298, | · Zbl 0314.58018
[20] Brunt, B. van: The calculus of variations. 2004, Springer-Verlag, New York, ISBN 0-387-40247-0. · Zbl 1039.49001
[21] Vinogradov, A. M.: The C-spectral sequence, Lagrangian formalism, and conservation laws. I. The linear theory. J. Math. Anal. Appl., 100, 1, 1984, 1-40, ISSN 0022-247X. | · Zbl 0548.58014 · doi:10.1016/0022-247X(84)90071-4
[22] Vinogradov, A. M.: The C-spectral sequence, Lagrangian formalism, and conservation laws. II. The nonlinear theory. J. Math. Anal. Appl., 100, 1, 1984, 41-129, ISSN 0022-247X. | · Zbl 0548.58015 · doi:10.1016/0022-247X(84)90072-6
[23] Vinogradov, A. M., Moreno, G.: Domains in infinite jet spaces: the C-spectral sequence. Dokl. Akad. Nauk, 413, 2, 2007, 154-157, ISSN 0869-5652. DOI 10.1134/S1064562407020081. | · Zbl 1154.58002 · doi:10.1134/S1064562407020081
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.