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

Autoři 2016 Článek v odborném periodiku Communications in mathematics http://dx.doi.org/10.1515/cm-2016-0012 Teoretická fyzika fibred manifolds; calculus of variations; equations of motion;inverse problem;symmetries; conservation laws; variational physical theories As widely accepted, justied 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 specic 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 uids, 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 dierent kinds. One of the most universal and comprehensive is the calculus of variations on bred 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 dening 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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