Invariant Variation Problems

By Emmy Noether, M. Tavel. 2005

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Abstract The problems in variation here concerned are such as to admit a continuous group (in Lie's sense); the conclusions that emerge for the corresponding differential equations find their most general expression in the theorems formulated in Section I and proved in following sections. Concerning these differential equations that arise from problems of variation, far more precise statements can be made than about arbitrary differential equations admitting of a group, which are the subject of Lie's researches. What is to follow, therefore, represents a combination of the methods of the formal calculus of variations with those of Lie's group theory. For special groups and problems in variation, this combination of methods is not new; I may cite Hamel and Herglotz for special finite groups, Lorentz and his pupils (for instance Fokker, Weyl and Klein)1 for special infinite groups. Especially Klein's second Note and the present developments have been mutually influenced by each other, in which regard I may ...

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

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Citations in the corpus

Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data (2020, background): “Noether’s Hamiltonian theorem states that for each continuous symmetry in the Hamiltonian of a dynamical system there exists a corresponding conserved quantity (Noether, 1971; Butterﬁeld, 2006).”
Equivariant Hamiltonian Flows (2019, background): “By Noether’s ﬁrst Theorem [23], if T is a symmetry then the generator g commutes with the Hamiltonian function, that is { H, g } =0 3 .”
Covariant phase space with boundaries (2019, background): “This definition however needs some improvement for theories with continuous local symmetries, since for such theories the initial value problem is not well-defined [41].”
The Long-Baseline Neutrino Experiment: Exploring Fundamental Symmetries of the Universe (2013, background): “◦ Conservation laws arise from underlying symmetries in Nature [101].”
Snowmass White Paper: Generalized Symmetries in Quantum Field Theory and Beyond (2022, background): “Examples include Noether’s theorems [1, 2] connecting symmetries and conservation laws, and Landau’s symmetry-based theory of phase transitions [3].”
Hamiltonian Neural Networks (2019, background): “Since almost all physical laws can be expressed as conservation laws, our approach is quite general [27].”