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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]
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 ...
Being Lagrangian symmetries, gauge symmetries of a Lagrangian satisfy Noether's first theorem, but the corresponding conserved current takes a particular superpotential form = + where the first term vanishes on solutions of the Euler–Lagrange equations and the second one is a boundary term, where is called a superpotential.
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.
[1] [2] [3] In classical mechanics for instance, in the action formulation, extremal solutions to the variational principle are on shell and the Euler–Lagrange equations give the on-shell equations. Noether's theorem regarding differentiable symmetries of physical action and conservation laws is another on-shell theorem.
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)
According to Noether's theorem, each symmetry of a system is associated a conserved quantity. [ 1 ] [ 2 ] For example, the rotational invariance of a system implies the conservation of its angular momentum , or spacetime invariance implies the conservation of energy–momentum.