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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 mathematician Emmy Noether in 1918. [ 1 ]
Calculus of variations is concerned with variations of functionals, which are small changes in the functional's value due to small changes in the function that is its argument. The first variation [l] is defined as the linear part of the change in the functional, and the second variation [m] is defined as the quadratic part. [22]
No wandering domain theorem (ergodic theory) Noether's theorem (Lie groups, calculus of variations, differential invariants, physics) Noether's second theorem (calculus of variations, physics) Noether's theorem on rationality for surfaces (algebraic surfaces) Non-squeezing theorem (symplectic geometry) Norton's theorem (electrical networks)
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]
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 ...
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.
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]
For surfaces, the Hirzebruch–Riemann–Roch theorem is essentially the Riemann–Roch theorem for surfaces combined with the Noether formula. To see this, recall that for each divisor D on a surface there is an invertible sheaf L = O( D ) such that the linear system of D is more or less the space of sections of L .