×

zbMATH — the first resource for mathematics

The calculus of variations on jet bundles as a universal approach for a variational formulation of fundamental physical theories. (English) Zbl 1397.70036
Summary: 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.
MSC:
70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems
70S10 Symmetries and conservation laws in mechanics of particles and systems
49N45 Inverse problems in optimal control
49S05 Variational principles of physics
58E30 Variational principles in infinite-dimensional spaces
PDF BibTeX XML Cite
Full Text: Link
References:
[1] Anderson, I. M.: The Variational Bicomplex. 1989, Book preprint, technical report of the Utah State University.
[2] Rodrigues, M. de León,P. R.: Generalized Classical Mechanics and Field Theory. 1985, North-Holland, Amsterdam, | · Zbl 0581.58015
[3] Giachetta, G., Mangiarotti, L., Sardanashvily, G.: New Lagrangian and Hamiltonian Methods in Field Theory. 1997, World Scientific, Singapore, | · Zbl 0913.58001
[4] Krasilschik, I. S., Vinogradov, V. V. Lychagin,A. M.: Geometry of Jet Spaces and Differential Equations. 1986, Gordon and Breach,
[5] Krbek, M., Musilov?, J.: Representation of the variational sequence by differential forms. Acta Appl. Math., 88, 2, 2005, 177-199, | · Zbl 1085.58014
[6] Krupka, D.: Introduction to Global Variational Geometry. Atlantis Studies in Variational Geometry, 2015, Atlantis Press, | · Zbl 1310.49001
[7] Krupka, D., Musilov?, J.: Trivial lagrangians in field theory. Differential Geometry and its Applications, 9, 1998, 293-305, | · Zbl 0932.58024
[8] Krupkov?, O.: The Geometry of Ordinary Differential Equations. Lecture Notes in Mathematics, 1678, 1997, Springer-Verlag, Berlin, Heidelberg, | · Zbl 0936.70001
[9] Landau, L. D., Lifshitz, E. M.: Quantum Mechanics: Non-Relativistic Theory. 3, 1977, Pergamon Press, 3rd ed.. · Zbl 0178.57901
[10] Landau, L. D., Lifshitz, E. M.: he Classical Theory of Fields. 2, 1975, Pergamon Press, 3rd ed.. · Zbl 0178.28704
[11] Palese, M., Rossi, O., Winterroth, E., Musilov?, J.: Variational Sequences, Representation Sequences and Applications in Physics. SIGMA, 12, 2016, 1-44, · Zbl 1347.70043
[12] Rossi, O., Musilov?, J.: The relativistic mechanics in a nonholonomic setting: a unified approach to particles with non-zero mass and massless particles. J. Phys. A: Math. Theor., 45, 2012, 255202, 27 pp.. | · Zbl 1348.70051
[13] Sardanashvily, G.: Noether’s Theorems, Applications in Mechanics and Field Theory. 2016, Atlantis Studies in Variational Geometry, Atlantis Press,
[14] Saunders, D. J.: The Geometry of Jet Bundles. 1989, Cambridge University Press, Cambridge, | · Zbl 0665.58002
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.