Noether theorem and first integrals of constrained Lagrangean systems

First Applications of Noether ’ s Theorem

and the last two (actually Lorentz transformations already include both) can be written x → x = ∑ ν Λ μ νx ν = Λνx ν , (using the Einstein summation convention), where the matrix Λ is a real matrix satisfying the pseudoorthogonality condition ΛμνgμρΛ ρ τ = gντ , which is required so that the length of a four-vector is preserved, x′ := xxμ = x . All together, this symmetry group is called the in...

Noether, Partial Noether Operators and First Integrals for the Coupled Lane-emden System

Systems of Lane-Emden equations arise in the modelling of several physical phenomena, such as pattern formation, population evolution and chemical reactions. In this paper we construct Noether and partial Noether operators corresponding to a Lagrangian and a partial Lagrangian for a coupled Lane-Emden system. Then the first integrals with respect to Noether and partial Noether operators are obt...

1 The First Law of Isolated Horizons via Noether Theorem

A general recipe proposed elsewhere to define, via Noether theorem, the variation of energy for a natural field theory is applied to Einstein-Maxwell theory. The electromagnetic field is analysed in the geometric framework of natural bundles. Einstein-Maxwell theory turns then out to be natural rather than gauge-natural. As a consequence of this assumption a correction termà la Regge-Teitelboim...

The Fundamental Theorem on Quadratic First Integrals.

The First Mean Value Theorem for Integrals

For simplicity, we use the following convention: X is a non empty set, S is a σ-field of subsets of X, M is a σ-measure on S, f , g are partial functions from X to R, and E is an element of S. One can prove the following three propositions: (1) If for every element x of X such that x ∈ dom f holds f(x) ≤ g(x), then g − f is non-negative. (2) For every set Y and for every partial function f from...