Noethers Symmetry Theorem

This theorem [7, 8] was published in 1918 and is named afl i Noether, The theorem states that a symmetry in a system leads to a conserved quantity. 

Applications of Noether s theorem deal with the invariance of physical laws against translations of space, rotations in space, and transformation of time. The consequences of invariance or, in other words, symmetry are summarized in Table 15.1. 

if a physical law is not changing if the time is transformed, it is always the same at any instant of time. From this property, the conservation of energy can be concluded. 

A simple illustration of Noether s theorem has been presented by Baez [9], For a single particle, its position should be represented by a generalized coordinate q. The generalized velocity of the particle is q. In terms of Lagrangian mechanics, the generalized momentum p and the generalized force F are as follows  

As in ordinary mechanics, the Euler - Lagrange equations demand p = F, which arises from a variational problem. Now we want to transform the position q into another new position. We may express this as the position g is a function of some parameter s, i.e., q = q s). In the same way the velocity transforms when the position is changed, q = q s). When the Lagrangian is invariant for such a transformation, we address this as a kind of symmetry, here the translational homogeneity of space. This means, changing the parameter 5 by a small value, the Lagrangian should remain the same  

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We recall, from elementary classical mechanics, that symmetry properties of the Lagrangian (or Hamiltonian) generally imply the existence of conserved quantities. If the Lagrangian is invariant under time displacement, for example, then the energy is conserved similarly, translation invariance implies momentum conservation. More generally, Noether s Theorem states that for each continuous N-dimensional group of transformations that commutes with the dynamics, there exist N conserved quantities. 

The source of electric charge in this view is a symmetry of the action in Noether s theorem, a symmetry that means that ([> must be complex, that is, that there must be two fields ... 

Within this context, ordinary differential equations are viewed as vector fields on manifolds or configuration spaces [2]. For example, Newton s equations are second-order differential equations describing smooth curves on Riemannian manifolds. Noether s theorem [4] states that a diffeomorphism,3 < ), of a Riemannian manifold, C, indices a diffeomorphism, D< >, of its tangent4 bundle,5 TC. If 4> is a symmetry of Newton s equations, then Dt(> preserves the Lagrangian o /Jc ) = jSf. As opposed to equations of motion in conventional... 

The close connection between symmetry transformations and conservation laws was first noted by Jacobi, and later formulated as Noether s theorem invariance of the Lagrangian under a one-parameter transformation implies the existence of a conserved quantity associated with the generator of the transformation [304], The equations of motion imply that the time derivative of any function 3(p, q) is... 

Noether s theorem for gauge symmetry For a local infinitesimal gauge transformation about a solution of the field equations,... 

The conservation of energy, momentum and angular momentum, considered before, in all cases is a consequence of Noether s theorem [17]. An important new result is obtained by applying the theorem to internal symmetry transformations. This result is illustrated by the theory of the complex... 

According to Noether s theorem (Arnold (1989)) symmetries of a mechanical system are always accompanied by constants of the motion. According to Section 3.1, system symmetries can be obvious (e.g. geometric) or hidden . Examples for obvious symmetries that lead to constants of the motion are invariance with respect to time translations, spatial translations and rotations. Invariance with respect to time leads to the conservation of energy, spatial and rotational symmetries lead to the conservation of linear and angular momentum, respectively (see, e.g., Landau and Lifschitz (1970)). Hidden symmetries cannot be associated with... 

It is useful for the following discussion to consider the symmetries of the Lagrangian (2.1) in order to analyse the conservation laws of a system characterised by (2.1) on the most general level, i.e. without further specifying F", and their consequences for the structure of a density functional approach to (2.1). We first consider continuous symmetries which in the field theoretical context are usually discussed on the basis of Noether s theorem (see e.g. [26, 28]). The most obvious symmetry of the Lagrangian (2.1), its gauge invariance (2.9), directly reflects current conservation,... 

To see the value of this alternative view of superposition, we shall derive the conserved charge corresponding to this symmetry implied by Noether s theorem. Using the fact that x / and ( ) both obey the wave equation, the Lagrangian density (Equation 4.51) transforms under Equation 4.75 as... 

In field theory, electric charge [6] is a symmetry of action, because it is a conserved quantity. This requirement leads to the consideration of a complex scalar field . The simplest possibility [U(l)] is that have two components, but in general it may have more than two as in the internal space of 0(3) electrodynamics which consists of the complex basis ((1),(2),(3)). The first two indices denote complex conjugate pairs, and the third is real-valued. These indices superimposed on the 4-vector give a 12-vector. In U(l) theory, the indices (1) and (2) are superimposed on the 4-vector, 4M in free space, so, 4M in U(l) electrodynamics in free space is considered as transverse, that is, determined by (1) and (2) only. These considerations lead to the conclusion that charge is not a point localized on an electron rather, it is a symmetry of action dictated ultimately by the Noether theorem [6]. 

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