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Download as PDF; Printable version; In other projects ... Fundamental theorem; Limits; Continuity; ... Template:Calculus is used to give a brief outline of calculus ...
This is the first of two theorems (see Noether's second theorem) published by the mathematician Emmy Noether in 1918. [1] The action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can be determined by the principle of least action .
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.
Her work on differential invariants in the calculus of variations, Noether's theorem, has been called "one of the most important mathematical theorems ever proved in guiding the development of modern physics". [11] In the second epoch (1920–1926), she began work that "changed the face of [abstract] algebra". [12]
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 ...
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.