Differential-geometrical approach to the dynamics of dissipationless incompressible Hall magnetohydrodynamics: I. Lagrangian mechanics on semidirect product of two volume preserving diffeomorphisms and conservation laws

General introduction with a review of the principles of Hamiltonian and Lagrangian mechanics. The connection between symmetries and conservation laws, with a presentation of Noether’s theorem, is included.

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Symmetries and the explanation of conservation laws in the light of the inverse problem in Lagrangian mechanics

Studies in History and Philosophy of Science Part B Studies in History and Philosophy of Modern Physics ◽

10.1016/j.shpsb.2007.12.001 ◽

2008 ◽

Vol 39 (2) ◽

pp. 325-345 ◽

Cited By ~ 4

Author(s):

Sheldon R. Smith

Keyword(s):

Inverse Problem ◽

Conservation Laws ◽

Lagrangian Mechanics

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Symmetries & Lagrangian-Hamilton-Jacobi Theory

10.1093/oso/9780198822370.003.0011 ◽

2018 ◽

Author(s):

Peter Mann

Keyword(s):

Conservation Laws ◽

Final Section ◽

Lagrangian Mechanics ◽

Conserved Quantities ◽

Fundamental Theory ◽

Continuous Symmetry ◽

Noether’S Theorem ◽

Noether's Theorem ◽

Symmetry Transformations ◽

Hamilton Jacobi Equation

This chapter discusses conservation laws in Lagrangian mechanics and shows that certain conservation laws are just particular examples of a more fundamental theory called ‘Noether’s theorem’, after Amalie ‘Emmy’ Noether, who first discovered it in 1918. The chapter starts off by discussing Noether’s theorem and symmetry transformations in Lagrangian mechanics in detail. It then moves on to gauge theory and surface terms in the action before isotropic symmetries. continuous symmetry, conserved quantities, conjugate momentum, cyclic coordinates, Hessian condition and discrete symmetries are discussed. The chapter also covers Lie algebra, spontaneous symmetry breaking, reduction theorems, non-dynamical symmetries and Ostrogradsky momentum. The final section of the chapter details Carathéodory–Hamilton–Jacobi theory in the Lagrangian setting, to derive the Hamilton–Jacobi equation on the tangent bundle!

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A Basis of Casimirs in 3D Magnetohydrodynamics

International Mathematics Research Notices ◽

10.1093/imrn/rnz393 ◽

2020 ◽

Cited By ~ 2

Author(s):

Boris Khesin ◽

Daniel Peralta-Salas ◽

Cheng Yang

Keyword(s):

Lie Algebra ◽

Semidirect Product ◽

Dual Space ◽

Magnetic Helicity ◽

Coadjoint Action ◽

Integral Invariants ◽

Volume Preserving

Abstract We prove that any regular Casimir in 3D magnetohydrodynamics (MHD) is a function of the magnetic helicity and cross-helicity. In other words, these two helicities are the only independent regular integral invariants of the coadjoint action of the MHD group $\textrm{SDiff}(M)\ltimes \mathfrak X^*(M)$, which is the semidirect product of the group of volume-preserving diffeomorphisms and the dual space of its Lie algebra.

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Lagrangian mechanics for fields

10.1093/oso/9780192895646.003.0028 ◽

2021 ◽

pp. 449-459

Author(s):

Andrew M. Steane

Keyword(s):

Conservation Laws ◽

Energy Tensor ◽

Lagrangian Mechanics ◽

Lagrange Equations ◽

Flat Spacetime ◽

Lagrangian Methods ◽

Classical Fields ◽

Proca Equations ◽

Lagrangian Densities ◽

Hilbert Action

An introduction to Lagrangian methods for classical fields in flat spacetime and then in curved spacetime. The Euler-Lagrange equations for Lagrangian densities are obtained, and applied to the wave, Klein-Gordan, Weyl, Dirac, Maxwell and Proca equations. The canonical energy tensor is obtained. Conservation laws and Noether’s theorem are described. An example of the treatment of Interactions is given by presenting the the QED Lagrangian. Finally, covariant Lagrangian methods are described, and the Einstein field eqution is derived from the Einstein-Hilbert action.

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