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

J 2016

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

MUSILOVÁ, Jana a Stanislav HRONEK

Základní údaje

Originální název

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

Autoři

MUSILOVÁ, Jana (203 Česká republika, garant, domácí) a Stanislav HRONEK (203 Česká republika, domácí)

Vydání

Communications in mathematics, 2016, 1804-1388

Další údaje

Jazyk

angličtina

Typ výsledku

Článek v odborném periodiku

Obor

10301 Atomic, molecular and chemical physics

Stát vydavatele

Česká republika

Utajení

není předmětem státního či obchodního tajemství

Kód RIV

RIV/00216224:14310/16:00088597

Organizační jednotka

Přírodovědecká fakulta

DOI

http://dx.doi.org/10.1515/cm-2016-0012

Klíčová slova anglicky

fibred manifolds; calculus of variations; equations of motion;inverse problem;symmetries; conservation laws; variational physical theories

Štítky

AKR, rivok

Příznaky

Mezinárodní význam, Recenzováno

Změněno: 14. 4. 2017 13:59, Ing. Andrea Mikešková

Anotace

V originále

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.

Návaznosti

GA14-02476S, projekt VaV

Název: Variace, geometrie a fyzika

Investor: Grantová agentura ČR, Variace, geometrie a fyzika

Citovat

MUSILOVÁ, Jana a Stanislav HRONEK. The calculus of variations on jet bundles as a universal approach for a variational formulation of fundamental physical theories. Communications in mathematics. 2016, roč. 24, č. 2, s. 173-193. ISSN 1804-1388. Dostupné z: https://dx.doi.org/10.1515/cm-2016-0012.

@article{1369263,
   author = {Musilová, Jana and Hronek, Stanislav},
   article_number = {2},
   doi = {http://dx.doi.org/10.1515/cm-2016-0012},
   keywords = {fibred manifolds; calculus of variations; equations of motion;inverse problem;symmetries; conservation laws; variational physical theories},
   language = {eng},
   issn = {1804-1388},
   journal = {Communications in mathematics},
   title = {The calculus of variations on jet bundles as a universal approach for a variational formulation of fundamental physical theories},
   volume = {24},
   year = {2016}
}

TY  - JOUR
ID  - 1369263
AU  - Musilová, Jana - Hronek, Stanislav
PY  - 2016
TI  - The calculus of variations on jet bundles as a universal approach for a variational formulation of fundamental physical theories
JF  - Communications in mathematics
VL  - 24
IS  - 2
SP  - 173-193
EP  - 173-193
SN  - 18041388
KW  - fibred manifolds
KW  - calculus of variations
KW  - equations of motion;inverse problem;symmetries
KW  - conservation laws
KW  - variational physical theories
N2  - 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.
ER  -

MUSILOVÁ, Jana a Stanislav HRONEK. The calculus of variations on jet bundles as a universal approach for a variational formulation of fundamental physical theories. \textit{Communications in mathematics}. 2016, roč.~24, č.~2, s.~173-193. ISSN~1804-1388. Dostupné z: https://dx.doi.org/10.1515/cm-2016-0012.

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