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The calculus of variations on jet bundles as a universal approach for a variational formulation of fundamental physical theories

Abstract

As widely accepted, justified by the historical developments of physics, the background for standard formulation of postulates of physical theories leading to equations of motion, or even the form of equations of motion themselves, come from empirical experience. Equations of motion are then a starting point for obtaining specific conservation laws, as, for example, the well-known conservation laws of momenta and mechanical energy in mechanics. On the other hand, there are numerous examples of physical laws or equations of motion which can be obtained from a certain variational principle as Euler-Lagrange equations and their solutions, meaning that the \true trajectories" of the physical systems represent stationary points of the corresponding functionals.

It turns out that equations of motion in most of the fundamental theories of physics (as e.g. classical mechanics, mechanics of continuous media or fluids, electrodynamics, quantum mechanics, string theory, etc.), are Euler-Lagrange equations of an appropriately formulated variational principle. There are several well established geometrical theories providing a general description of variational problems of different kinds. One of the most universal and comprehensive is the calculus of variations on fibred manifolds and their jet prolongations. Among others, it includes a complete general solution of the so-called strong inverse variational problem allowing one not only to decide whether a concrete equation of motion can be obtained from a variational principle, but also to construct a corresponding variational functional. Moreover, conservation laws can be derived from symmetries of the Lagrangian defining this functional, or directly from symmetries of the equations.

In this paper we apply the variational theory on jet bundles to tackle some fundamental problems of physics, namely the questions on existence of a Lagrangian and the problem of conservation laws. The aim is to demonstrate that the methods are universal, and easily applicable to distinct physical disciplines: from classical mechanics, through special relativity, waves, classical electrodynamics, to quantum mechanics.

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Variations by generalized symmetries of local Noether strong currents equivalent to global canonical Noether currents

: Noether's second theorem in a general setting reducible gauge theories. J. Phys. A38 (2005) 5329-5344. [5] D. Bashkirov, G. Giachetta, L. Mangiarotti, G. Sardanashvily: The antifield Koszul-Tate complex of reducible Noether identities. J. Math. Phys. 46 (10) (2005). 103513, 19 pp. [6] E. Bessel-Hagen: Über die Erhaltungssätze der Elektrodynamik. Math. Ann. 84 (1921) 258-276. [7] A. Borowiec, M. Ferraris, M. Francaviglia, M. Palese: Conservation laws for non-global Lagrangians. Univ. Iagel. Acta Math. 41 (2003) 319

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Variational principles and symmetries on fibered multisymplectic manifolds

Abstract

The standard techniques of variational calculus are geometrically stated in the ambient of fiber bundles endowed with a (pre)multi-symplectic structure. Then, for the corresponding variational equations, conserved quantities (or, what is equivalent, conservation laws), symmetries, Cartan (Noether) symmetries, gauge symmetries and different versions of Noether's theorem are studied in this ambient. In this way, this constitutes a general geometric framework for all these topics that includes, as special cases, first and higher order field theories and (non-autonomous) mechanics.

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Geometry of the free-sliding Bernoulli beam

References [1] I. M. Anderson and T. Duchamp: On the existence of global variational principles. Amer. J. Math. 102 (5) (1980) 781-868. ISSN 0002-9327. DOI 10.2307/2374195 [2] A. V. Bocharov, V. N. Chetverikov, S. V. Duzhin, N. G. Khor'kova, I. S. Krasil'shchik, A. V. Samokhin, Yu. N. Torkhov, A. M. Verbovetsky, A. M. Vinogradov: Symmetries and conservation laws for differential equations of mathematical physics. American Mathematical Society, Providence, RI (1999). ISBN 0-8218-0958-X. Edited and with a preface by Krasil

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