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Noether's Theorem and Symmetry
P.G.L. Leach and Andronikos Paliathanasis
2020
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In Noether's original presentation of her celebrated theorem of 1918, allowances were made for the dependence of the coefficient functions of the differential operator which generated the infinitesimal transformation of the Action Integral upon the derivatives of the dependent variable(s), the so-called generalized, or dynamical, symmetries. A similar allowance is to be found in the variables of the boundary function, often termed a gauge function by those who have not read the original paper. This generality was lost after texts such as those of Courant and Hilbert or Lovelock and Rund confined attention to only point transformations. In recent decades, this diminution of the power of Noether's Theorem has been partly countered, in particular, in the review of Sarlet and Cantrijn. In this Special Issue, we emphasize the generality of Noether's Theorem in its original form and explore the applicability of even more general coefficient functions by allowing for nonlocal terms. We also look at the application of these more general symmetries to problems in which parameters or parametric functions have a more general dependence upon the independent variables.
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Keywords
- action integral
- Analytic Mechanics
- approximate symmetry and solutions
- boundary term
- Boussinesq equation
- conservation law
- conservation laws
- continuous symmetry
- double dispersion equation
- energy-momentum tensor
- first integral
- first integrals
- FLRW spacetime
- Gauss-Bonnet cosmology
- generalized symmetry
- group-invariant solutions
- integrable nonlocal partial differential equations
- invariant
- invariant solutions
- Kelvin-Voigt equation
- Lagrange anchor
- lie symmetries
- Lie symmetry
- modified theories of gravity
- multiplier method
- n/a
- Noether operator identity
- Noether operators
- Noether symmetries
- Noether symmetry approach
- Noether’s theorem
- nonlocal transformation
- optimal system
- optimal systems
- Partial Differential equations
- quasi-Lagrangians
- quasi-Noether systems
- Roots
- spherically symmetric spacetimes
- symmetries
- symmetry reduction
- systems of ODEs
- variational principle
- viscoelasticity
- wave equation
Links
DOI: 10.3390/books978-3-03928-235-7Editions
