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Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law.This is the first of two theorems (see Noether's second theorem) published by the mathematician Emmy Noether in 1918. [1]
In accordance with Noether's second theorem, there is one-to-one correspondence between the gauge symmetries of a Lagrangian and the Noether identities which the Euler–Lagrange operator satisfies. Consequently, gauge symmetries characterize the degeneracy of a Lagrangian system. [5]
For a given genus g, the moduli space for curves C of genus g should contain a dense subset parameterizing those curves with the minimum in the way of special divisors. One goal of the theory is to 'count constants', for those curves: to predict the dimension of the space of special divisors (up to linear equivalence) of a given degree d, as a function of g, that must be present on a curve of ...
An application of the second isomorphism theorem identifies projective linear groups: for example, the group on the complex projective line starts with setting = (), the group of invertible 2 × 2 complex matrices, = (), the subgroup of determinant 1 matrices, and the normal subgroup of scalar matrices = {():}, we have = {}, where is ...
Noether identities need not be independent, but satisfy first-stage Noether identities, which are subject to the second-stage Noether identities and so on. Higher-stage Noether identities also are separated into the trivial and non-trivial once. A degenerate Lagrangian is called reducible if there exist non-trivial higher-stage Noether identities.
In mathematical analysis, Schwarz's theorem (or Clairaut's theorem on equality of mixed partials) [9] named after Alexis Clairaut and Hermann Schwarz, states that for a function : defined on a set , if is a point such that some neighborhood of is contained in and has continuous second partial derivatives on that neighborhood of , then for all i ...
Envelope theorem (calculus of variations) Isoperimetric theorem (curves, calculus of variations) Minimax theorem (game theory) Mountain pass theorem (calculus of variations) Noether's second theorem (calculus of variations, physics) Parthasarathy's theorem (game theory) Sion's minimax theorem (game theory) Tonelli's theorem (functional analysis)
Together with the Grunwald–Wang theorem, the Albert–Brauer–Hasse–Noether theorem implies that every central simple algebra over an algebraic number field is cyclic, i.e. can be obtained by an explicit construction from a cyclic field extension L/K.