Two comprehensions on Noether symmetry

Abstract

This paper presents two ways of comprehending the Noether symmetry for the Lagrange system.One is based on invariance of the Lagrangian, the other is based on invariance of the action. This paper proves that these two comprehensions are different from each other. We give the condition under which the invariance of the Lagrangian can become the invariance of the action,and the condition under which the invariance of the action can become the invariance of the Lagrangian is obtained. It is suitable that the Noether symmetry is comprehended as the invariance of the action.

Structure equation and Mei conserved quantity for Mei symmetry of Appell equation

“…Criterion 3 For Appell equation ( 8), if infinitesimal generators ξ 0 , ξ s make Eqs. (21)(22)(23) come into existence, then the invariance of Eq. ( 8) under the infinitesimal transformations Eq.…”

A Gauge field Induced by the Global Gauge Invariance of Action Integral

“…where φ is a real constant. For a long time, there are two ways of comprehending the gauge invariance of a Lagrange system [5]. One is based on the invariance of the Lagrangian; the other is based on the invariance of the action integral of the Lagrangian.…”

“…At present, there are two ways to comprehend the invariance of the Lagrange field. One is based on the Lagrangian; the other is based on the action integral of the Lagrangian [5]. For the gauge transformation, only the invariance of the Lagrangian is concerned in literatures.…”

Lie symmetry and its generation of conserved quantity of Appell equation in a dynamical system of the relative motion with Chetaev-type nonholonomic constraints

“…This kind of equation is not only applicable in holonomic systems, but also in nonholonomic systems, and it is not only applied in generalized coordinates, but also applied in quasi coordinates. The modern symmetry theories in a constrained mechanical system are mainly of three types: Noether symmetry, [1][2][3][4][5][6] Lie symmetry, [7][8][9][10][11][12][13][14][15][16] and Mei symmetry. [17][18][19][20][21][22][23][24][25][26] In recent decades, Chinese scholars have achieved important results in studying symmetry theories of Appell systems and their applications.…”