Noether’s theorem presents a correspondence between each principle of conservation of a physical magnitude (thus, energy, momentum, momentum) and a formal invariance of the laws of physics. In other words, for all continuous symmetry (for example, a spatial rotation) of the Lagrangian of the system, there is a conserved magnitude throughout its evolution. It is a central result in theoretical physics that expresses the existence of certain abstract symmetries in a physical system involving the existence of conservation laws . It is named after the mathematician Emmy Noether ( 1982 – 1935 ) who formulated it in 1915 where he could prove that all laws of symmetry, both in Classical Mechanics and in Quantum Mechanics.
In addition to allowing practical physical applications, this theorem also constitutes an explanation of why there are laws of conservation and physical magnitudes that do not change throughout the temporal evolution of a physical system.
Summary
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- 1 Definition
- 2 Proof of Noether’s theorem
- 1 Definition
- 3 Sources
Definition
Noether’s theorem relates pairs of basic ideas of physics , one is the invariance of the form that a physical law takes with respect to any (generalized) transformation that preserves the coordinate system (spatial and temporal aspects taken into account), and the other is the law of conservation of a physical quantity.
Informally, Noether’s theorem can be established as: To each (continuous) symmetry, there corresponds a conservation law and vice versa.
The formal statement of the theorem derives an expression for the physical quantity that is conserved (and therefore also defines it), from the invariance condition only. For example:
- The invariance of physical systems with respect to translation (simply stated, the laws of physics do not vary with location in space) gives the law of conservation of linear momentum;
- The invariance with respect to the (direction of the axis of) rotation gives the law of conservation of angular momentum;
- The invariance with respect to (translation in) time gives the well-known law of conservation of energy, etc.
Going up to quantum field theory , the invariance with respect to the general gauge transformation gives the law of conservation of electric charge. Thus, the result is a very important contribution to physics in general, as it helps to provide powerful insights into any general theory in physics , simply by analyzing the various transformations that would make the form of the laws involved invariant.
Proof of Noether’s theorem
Noether’s theorem provides the method for constructing conserved currents. These currents derive from the difeomorphisms in the phase space of a dynamic system that leave the Lagrangian invariant.
Definition
A uniparametric family of symmetries of a Lagrangian, L: TQ → R (TQ is the (fibred) space tangent to the configuration variety), is a differentiable application (1):
((s, q) → q n where q 0 = q and Γ = {q: R → Q} is a path space in the variety of system configuration. That is, q 0 = q (s, q) is a function of the parameter s that will parameterize a certain curve in Q.), so that there is a function f (q, q ‘) such that (2):
for some f: TQ → R.
Expressed in another way, it must be fulfilled for all roads q n (3):
If the Lagrangian remains invariant against the transformation, we will have
